""" This code is automatically generated. Never edit it manually. For details of generating the code see `rubi_parsing_guide.md` in `parsetools`. """ from sympy.external import import_module matchpy = import_module("matchpy") if matchpy: from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match from sympy.integrals.rubi.utility_function import ( Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor, PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral, CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial, Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant, Negative, Quotient ) from sympy.core.add import Add from sympy.core.mod import Mod from sympy.core.mul import Mul from sympy.core import EulerGamma from sympy.core.numbers import (Float, I, Integer) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.functions.elementary.complexes import (Abs, sign) from sympy.functions.elementary.miscellaneous import sqrt from sympy.integrals.integrals import Integral from sympy.logic.boolalg import (And, Or) from sympy.simplify.simplify import simplify from sympy.integrals.rubi.symbol import WC from sympy.core.symbol import symbols, Symbol from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf) from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch) from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2) from sympy.core.numbers import pi as Pi A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz'] a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']] i, ii, Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u') _UseGamma = False ShowSteps = False StepCounter = None def miscellaneous_algebraic(): from sympy.integrals.rubi.constraints import cons800, cons2, cons3, cons8, cons52, cons4, cons5, cons20, cons19, cons801, cons29, cons50, cons127, cons54, cons802, cons27, cons803, cons804, cons151, cons805, cons502, cons806, cons650, cons807, cons808, cons21, cons48, cons809, cons810, cons70, cons811, cons812, cons813, cons814, cons815, cons816, cons817, cons818, cons819, cons820, cons821, cons822, cons823, cons454, cons824, cons825, cons826, cons827, cons828, cons829, cons830, cons831, cons832, cons833, cons834, cons835, cons836, cons837, cons838, cons839, cons840, cons841, cons842, cons843, cons844, cons845, cons846, cons847, cons848, cons849, cons850, cons851, cons852, cons853, cons854, cons210, cons211, cons66, cons855, cons68, cons856, cons857, cons466, cons858, cons859, cons860, cons55, cons13, cons139, cons861, cons862, cons150, cons246, cons165, cons863, cons523, cons864, cons865, cons866, cons86, cons867, cons36, cons37, cons868, cons470, cons471, cons869, cons870, cons38, cons871, cons872, cons873, cons874, cons875, cons876, cons877, cons878, cons879, cons880, cons881, cons882, cons883, cons884, cons885, cons886, cons887, cons888, cons889, cons890, cons891, cons892, cons893, cons894, cons895, cons896, cons897, cons898, cons899, cons900, cons901, cons902, cons903, cons904, cons905, cons906, cons676, cons907, cons483, cons908, cons909, cons484, cons910, cons911, cons912, cons913, cons914, cons915, cons916, cons917, cons918, cons87, cons33, cons96, cons919, cons198, cons369, cons358, cons491, cons543, cons25, cons920, cons556, cons921, cons554, cons57, cons496, cons59, cons60, cons61, cons62, cons922, cons923, cons924, cons925, cons926, cons597, cons73, cons927, cons588, cons89, cons130, cons928, cons929, cons930, cons931, cons932, cons47, cons316, cons228, cons933, cons934, cons935, cons936, cons937, cons938, cons939, cons940, cons941, cons942, cons943, cons944, cons945, cons946, cons947, cons948, cons284, cons949, cons65, cons721, cons950, cons951, cons952, cons75, cons953, cons704, cons149, cons954, cons955, cons798, cons956, cons957, cons958, cons959, cons960, cons961, cons962, cons963, cons964, cons965, cons966, cons967, cons968, cons71, cons969, cons970, cons971, cons972, cons973, cons974, cons975, cons976, cons977, cons514, cons978, cons979, cons980, cons981, cons982, cons669, cons983, cons984, cons799, cons985, cons986, cons987, cons988, cons989, cons990, cons95, cons90, cons991, cons992, cons993, cons994, cons995, cons996, cons997, cons998, cons999, cons1000, cons40, cons1001, cons1002, cons1003, cons1004, cons1005, cons1006, cons1007, cons1008, cons1009, cons1010, cons1011, cons1012, cons385, cons1013, cons1014, cons1015, cons1016, cons1017, cons1018, cons1019, cons1020, cons359, cons1021, cons248, cons1022, cons1023, cons1024, cons1025, cons1026, cons1027, cons1028, cons1029, cons1030, cons1031, cons1032, cons1033, cons1034, cons1035, cons1036, cons1037, cons1038, cons1039, cons1040, cons1041, cons1042, cons1043, cons1044, cons1045, cons299, cons1046, cons1047, cons1048, cons1049, cons1050, cons707, cons384, cons1051, cons1052, cons699, cons711, cons155, cons1053, cons1054, cons1055, cons1056, cons1057, cons1058, cons1059, cons1060, cons1061, cons226, cons1062, cons517, cons1063, cons1064, cons1065, cons1066, cons1067, cons1068, cons1069, cons1070, cons1071, cons1072, cons1073, cons45, cons481, cons482, cons1074, cons1075, cons1076, cons1077, cons1078, cons1079, cons1080, cons1081, cons1082, cons1083, cons1084, cons1085, cons1086, cons1087, cons1088, cons1089, cons1090, cons1091 pattern1476 = Pattern(Integral(((x_**n_*WC('c', S(1)))**q_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons52, cons4, cons5, cons800) rule1476 = ReplacementRule(pattern1476, replacement1476) pattern1477 = Pattern(Integral(x_**WC('m', S(1))*((x_**n_*WC('c', S(1)))**q_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons19, cons4, cons5, cons52, cons800, cons20) rule1477 = ReplacementRule(pattern1477, replacement1477) pattern1478 = Pattern(Integral(x_**WC('m', S(1))*((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('r', S(1))*WC('e', S(1)))**p_*((c_ + x_**WC('n', S(1))*WC('d', S(1)))**s_*WC('f', S(1)))**q_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons52, cons54, cons802, cons801) rule1478 = ReplacementRule(pattern1478, replacement1478) pattern1479 = Pattern(Integral(((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_*WC('u', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons27) rule1479 = ReplacementRule(pattern1479, replacement1479) pattern1480 = Pattern(Integral(((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_*WC('u', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons803, cons804) rule1480 = ReplacementRule(pattern1480, replacement1480) pattern1481 = Pattern(Integral(((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons151, cons805) rule1481 = ReplacementRule(pattern1481, With1481) pattern1482 = Pattern(Integral(x_**WC('m', S(1))*((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons151, cons502) rule1482 = ReplacementRule(pattern1482, With1482) pattern1483 = Pattern(Integral(u_**WC('r', S(1))*((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons806, cons151, cons805, cons650) rule1483 = ReplacementRule(pattern1483, With1483) pattern1484 = Pattern(Integral(u_**WC('r', S(1))*x_**WC('m', S(1))*((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons806, cons151, cons805, cons807) rule1484 = ReplacementRule(pattern1484, With1484) pattern1485 = Pattern(Integral(((WC('c', S(1))/x_)**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons4, cons5, cons808) rule1485 = ReplacementRule(pattern1485, replacement1485) pattern1486 = Pattern(Integral(x_**WC('m', S(1))*((WC('c', S(1))/x_)**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons4, cons5, cons20) rule1486 = ReplacementRule(pattern1486, replacement1486) pattern1487 = Pattern(Integral((x_*WC('d', S(1)))**m_*((WC('c', S(1))/x_)**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons21) rule1487 = ReplacementRule(pattern1487, replacement1487) pattern1488 = Pattern(Integral(((WC('d', S(1))/x_)**n_*WC('b', S(1)) + (WC('d', S(1))/x_)**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons48) rule1488 = ReplacementRule(pattern1488, replacement1488) pattern1489 = Pattern(Integral(x_**WC('m', S(1))*(a_ + (WC('d', S(1))/x_)**n_*WC('b', S(1)) + (WC('d', S(1))/x_)**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons48, cons20) rule1489 = ReplacementRule(pattern1489, replacement1489) pattern1490 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + (WC('d', S(1))/x_)**n_*WC('b', S(1)) + (WC('d', S(1))/x_)**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons48, cons21) rule1490 = ReplacementRule(pattern1490, replacement1490) pattern1491 = Pattern(Integral((x_**WC('n2', S(1))*WC('c', S(1)) + (WC('d', S(1))/x_)**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons809, cons810) rule1491 = ReplacementRule(pattern1491, replacement1491) pattern1492 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)) + (WC('d', S(1))/x_)**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons809, cons810, cons20) rule1492 = ReplacementRule(pattern1492, replacement1492) pattern1493 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)) + (WC('d', S(1))/x_)**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons809, cons21, cons810) rule1493 = ReplacementRule(pattern1493, replacement1493) pattern1494 = Pattern(Integral(u_**m_, x_), cons19, cons70, cons811) rule1494 = ReplacementRule(pattern1494, replacement1494) pattern1495 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('n', S(1)), x_), cons19, cons4, cons812, cons813) rule1495 = ReplacementRule(pattern1495, replacement1495) pattern1496 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('n', S(1))*w_**WC('p', S(1)), x_), cons19, cons4, cons5, cons814, cons815) rule1496 = ReplacementRule(pattern1496, replacement1496) pattern1497 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('n', S(1))*w_**WC('p', S(1))*z_**WC('q', S(1)), x_), cons19, cons4, cons5, cons52, cons816, cons817) rule1497 = ReplacementRule(pattern1497, replacement1497) pattern1498 = Pattern(Integral(u_**p_, x_), cons5, cons818, cons819) rule1498 = ReplacementRule(pattern1498, replacement1498) pattern1499 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('p', S(1)), x_), cons19, cons5, cons70, cons820, cons821) rule1499 = ReplacementRule(pattern1499, replacement1499) pattern1500 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('n', S(1))*w_**WC('p', S(1)), x_), cons19, cons4, cons5, cons812, cons822, cons823) rule1500 = ReplacementRule(pattern1500, replacement1500) pattern1501 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1)), x_), cons5, cons52, cons454, cons824) rule1501 = ReplacementRule(pattern1501, replacement1501) pattern1502 = Pattern(Integral(u_**p_, x_), cons5, cons825, cons826) rule1502 = ReplacementRule(pattern1502, replacement1502) pattern1503 = Pattern(Integral(u_**WC('p', S(1))*(x_*WC('c', S(1)))**WC('m', S(1)), x_), cons8, cons19, cons5, cons825, cons826) rule1503 = ReplacementRule(pattern1503, replacement1503) pattern1504 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1)), x_), cons5, cons52, cons827, cons828, cons829) rule1504 = ReplacementRule(pattern1504, replacement1504) pattern1505 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1))*x_**WC('m', S(1)), x_), cons19, cons5, cons52, cons827, cons828, cons829) rule1505 = ReplacementRule(pattern1505, replacement1505) pattern1506 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('p', S(1))*w_**WC('q', S(1)), x_), cons19, cons5, cons52, cons830, cons828, cons831, cons832) rule1506 = ReplacementRule(pattern1506, replacement1506) pattern1507 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1))*x_**WC('m', S(1))*z_**WC('r', S(1)), x_), cons19, cons5, cons52, cons54, cons833, cons828, cons834, cons835) rule1507 = ReplacementRule(pattern1507, replacement1507) pattern1508 = Pattern(Integral(u_**p_, x_), cons5, cons836, cons837) rule1508 = ReplacementRule(pattern1508, replacement1508) pattern1509 = Pattern(Integral(u_**WC('p', S(1))*x_**WC('m', S(1)), x_), cons19, cons5, cons836, cons837) rule1509 = ReplacementRule(pattern1509, replacement1509) pattern1510 = Pattern(Integral(u_**p_, x_), cons5, cons838, cons839) rule1510 = ReplacementRule(pattern1510, replacement1510) pattern1511 = Pattern(Integral(u_**WC('p', S(1))*(x_*WC('d', S(1)))**WC('m', S(1)), x_), cons29, cons19, cons5, cons838, cons839) rule1511 = ReplacementRule(pattern1511, replacement1511) pattern1512 = Pattern(Integral(u_**WC('q', S(1))*v_**WC('p', S(1)), x_), cons5, cons52, cons825, cons840, cons841) rule1512 = ReplacementRule(pattern1512, replacement1512) pattern1513 = Pattern(Integral(u_**WC('q', S(1))*v_**WC('p', S(1)), x_), cons5, cons52, cons825, cons842, cons843) rule1513 = ReplacementRule(pattern1513, replacement1513) pattern1514 = Pattern(Integral(u_**WC('p', S(1))*x_**WC('m', S(1))*z_**WC('q', S(1)), x_), cons19, cons5, cons52, cons844, cons838, cons845) rule1514 = ReplacementRule(pattern1514, replacement1514) pattern1515 = Pattern(Integral(u_**WC('p', S(1))*x_**WC('m', S(1))*z_**WC('q', S(1)), x_), cons19, cons5, cons52, cons844, cons825, cons846) rule1515 = ReplacementRule(pattern1515, replacement1515) pattern1516 = Pattern(Integral(u_**p_, x_), cons5, cons847, cons848) rule1516 = ReplacementRule(pattern1516, replacement1516) pattern1517 = Pattern(Integral(u_**WC('p', S(1))*x_**WC('m', S(1)), x_), cons19, cons5, cons847, cons848) rule1517 = ReplacementRule(pattern1517, replacement1517) pattern1518 = Pattern(Integral(u_**WC('p', S(1))*z_, x_), cons5, cons844, cons847, cons849, cons850) rule1518 = ReplacementRule(pattern1518, replacement1518) pattern1519 = Pattern(Integral(u_**WC('p', S(1))*x_**WC('m', S(1))*z_, x_), cons19, cons5, cons844, cons847, cons849, cons850) rule1519 = ReplacementRule(pattern1519, replacement1519) pattern1520 = Pattern(Integral(x_**WC('m', S(1))*(e_ + x_**WC('n', S(1))*WC('h', S(1)) + x_**WC('q', S(1))*WC('f', S(1)) + x_**WC('r', S(1))*WC('g', S(1)))/(a_ + x_**WC('n', S(1))*WC('c', S(1)))**(S(3)/2), x_), cons2, cons8, cons50, cons127, cons210, cons211, cons19, cons4, cons851, cons852, cons853, cons854) rule1520 = ReplacementRule(pattern1520, replacement1520) pattern1521 = Pattern(Integral((d_*x_)**WC('m', S(1))*(e_ + x_**WC('n', S(1))*WC('h', S(1)) + x_**WC('q', S(1))*WC('f', S(1)) + x_**WC('r', S(1))*WC('g', S(1)))/(a_ + x_**WC('n', S(1))*WC('c', S(1)))**(S(3)/2), x_), cons2, cons8, cons29, cons50, cons127, cons210, cons211, cons19, cons4, cons853, cons851, cons852, cons854) rule1521 = ReplacementRule(pattern1521, replacement1521) pattern1522 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**m_*(a_ + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons66, cons151, cons855) rule1522 = ReplacementRule(pattern1522, With1522) pattern1523 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons19, cons4, cons5, cons68, cons856, cons857) rule1523 = ReplacementRule(pattern1523, replacement1523) pattern1524 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons66, cons466, cons858) rule1524 = ReplacementRule(pattern1524, replacement1524) pattern1525 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons19, cons4, cons66, cons859) rule1525 = ReplacementRule(pattern1525, replacement1525) pattern1526 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons4, cons66, cons859) rule1526 = ReplacementRule(pattern1526, replacement1526) pattern1527 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons19, cons4, cons5, cons860, cons502) rule1527 = ReplacementRule(pattern1527, replacement1527) pattern1528 = Pattern(Integral(Pq_*(c_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons19, cons4, cons5, cons860, cons502) rule1528 = ReplacementRule(pattern1528, replacement1528) pattern1529 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons19, cons4, cons66, cons55, cons13, cons139) rule1529 = ReplacementRule(pattern1529, replacement1529) pattern1530 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons29, cons19, cons4, cons5, cons66, cons861) rule1530 = ReplacementRule(pattern1530, replacement1530) pattern1531 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons66, cons861, cons862) rule1531 = ReplacementRule(pattern1531, replacement1531) pattern1532 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons66, cons150, cons246, cons165, cons863) rule1532 = ReplacementRule(pattern1532, With1532) pattern1533 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons66, cons523, cons13, cons165) rule1533 = ReplacementRule(pattern1533, With1533) pattern1534 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons66, cons523, cons13, cons165) rule1534 = ReplacementRule(pattern1534, With1534) pattern1535 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons66, cons150, cons13, cons139, CustomConstraint(With1535)) rule1535 = ReplacementRule(pattern1535, replacement1535) pattern1536 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons66, cons150, cons13, cons139, cons864) rule1536 = ReplacementRule(pattern1536, replacement1536) pattern1537 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)) + x_*WC('e', S(1)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons865) rule1537 = ReplacementRule(pattern1537, replacement1537) pattern1538 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons29, cons127, cons210, cons865) rule1538 = ReplacementRule(pattern1538, replacement1538) pattern1539 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_*WC('e', S(1)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons29, cons50, cons210, cons865) rule1539 = ReplacementRule(pattern1539, replacement1539) pattern1540 = Pattern(Integral(x_**S(2)*(x_**S(4)*WC('h', S(1)) + x_*WC('f', S(1)) + WC('e', S(0)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons50, cons127, cons211, cons866) rule1540 = ReplacementRule(pattern1540, replacement1540) pattern1541 = Pattern(Integral(x_**S(2)*(x_**S(4)*WC('h', S(1)) + WC('e', S(0)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons50, cons211, cons866) rule1541 = ReplacementRule(pattern1541, replacement1541) pattern1542 = Pattern(Integral((d_ + x_**S(6)*WC('h', S(1)) + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)) + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons211, cons866, cons865) rule1542 = ReplacementRule(pattern1542, replacement1542) pattern1543 = Pattern(Integral((d_ + x_**S(6)*WC('h', S(1)) + x_**S(4)*WC('g', S(1)) + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons29, cons50, cons210, cons211, cons866, cons865) rule1543 = ReplacementRule(pattern1543, replacement1543) pattern1544 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons66, cons150, cons13, cons139, CustomConstraint(With1544)) rule1544 = ReplacementRule(pattern1544, replacement1544) pattern1545 = Pattern(Integral(Pq_*x_**m_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons66, cons150, cons13, cons139, cons86) rule1545 = ReplacementRule(pattern1545, With1545) pattern1546 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons860, cons150, cons20, CustomConstraint(With1546)) rule1546 = ReplacementRule(pattern1546, replacement1546) pattern1547 = Pattern(Integral((A_ + x_*WC('B', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons867) rule1547 = ReplacementRule(pattern1547, replacement1547) pattern1548 = Pattern(Integral((A_ + x_*WC('B', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons868, cons470) rule1548 = ReplacementRule(pattern1548, With1548) pattern1549 = Pattern(Integral((A_ + x_*WC('B', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons868, cons471) rule1549 = ReplacementRule(pattern1549, With1549) pattern1550 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons869, cons870) rule1550 = ReplacementRule(pattern1550, replacement1550) pattern1551 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons871) rule1551 = ReplacementRule(pattern1551, With1551) pattern1552 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons872) rule1552 = ReplacementRule(pattern1552, With1552) pattern1553 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons873) rule1553 = ReplacementRule(pattern1553, With1553) pattern1554 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons874) rule1554 = ReplacementRule(pattern1554, With1554) pattern1555 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons875) rule1555 = ReplacementRule(pattern1555, With1555) pattern1556 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons876) rule1556 = ReplacementRule(pattern1556, With1556) pattern1557 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons877) rule1557 = ReplacementRule(pattern1557, With1557) pattern1558 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons878) rule1558 = ReplacementRule(pattern1558, With1558) pattern1559 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons879) rule1559 = ReplacementRule(pattern1559, With1559) pattern1560 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons880) rule1560 = ReplacementRule(pattern1560, With1560) pattern1561 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons881) rule1561 = ReplacementRule(pattern1561, With1561) pattern1562 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons882) rule1562 = ReplacementRule(pattern1562, With1562) pattern1563 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons883) rule1563 = ReplacementRule(pattern1563, With1563) pattern1564 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons884) rule1564 = ReplacementRule(pattern1564, With1564) pattern1565 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons885) rule1565 = ReplacementRule(pattern1565, With1565) pattern1566 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons886) rule1566 = ReplacementRule(pattern1566, With1566) pattern1567 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons887) rule1567 = ReplacementRule(pattern1567, With1567) pattern1568 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons888) rule1568 = ReplacementRule(pattern1568, With1568) pattern1569 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons889) rule1569 = ReplacementRule(pattern1569, With1569) pattern1570 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons890) rule1570 = ReplacementRule(pattern1570, With1570) pattern1571 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons891) rule1571 = ReplacementRule(pattern1571, With1571) pattern1572 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons892) rule1572 = ReplacementRule(pattern1572, With1572) pattern1573 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons893) rule1573 = ReplacementRule(pattern1573, With1573) pattern1574 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons894) rule1574 = ReplacementRule(pattern1574, With1574) pattern1575 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons895) rule1575 = ReplacementRule(pattern1575, replacement1575) pattern1576 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons896) rule1576 = ReplacementRule(pattern1576, replacement1576) pattern1577 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons897) rule1577 = ReplacementRule(pattern1577, replacement1577) pattern1578 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons898) rule1578 = ReplacementRule(pattern1578, With1578) pattern1579 = Pattern(Integral(x_*(x_*WC('C', S(1)) + WC('B', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons899) rule1579 = ReplacementRule(pattern1579, With1579) pattern1580 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons900) rule1580 = ReplacementRule(pattern1580, With1580) pattern1581 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons901) rule1581 = ReplacementRule(pattern1581, With1581) pattern1582 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons902) rule1582 = ReplacementRule(pattern1582, With1582) pattern1583 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons903) rule1583 = ReplacementRule(pattern1583, With1583) pattern1584 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons868, cons904, cons905, CustomConstraint(With1584)) rule1584 = ReplacementRule(pattern1584, replacement1584) pattern1585 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons904, cons905, CustomConstraint(With1585)) rule1585 = ReplacementRule(pattern1585, replacement1585) pattern1586 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons904, cons905, CustomConstraint(With1586)) rule1586 = ReplacementRule(pattern1586, replacement1586) pattern1587 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons37, cons38, cons868, cons904, cons906, CustomConstraint(With1587)) rule1587 = ReplacementRule(pattern1587, replacement1587) pattern1588 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons37, cons38, cons904, cons906, CustomConstraint(With1588)) rule1588 = ReplacementRule(pattern1588, replacement1588) pattern1589 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons36, cons38, cons904, cons906, CustomConstraint(With1589)) rule1589 = ReplacementRule(pattern1589, replacement1589) pattern1590 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons8, cons19, cons66, cons676, cons907, CustomConstraint(With1590)) rule1590 = ReplacementRule(pattern1590, replacement1590) pattern1591 = Pattern(Integral(Pq_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons66, cons676, cons907, CustomConstraint(With1591)) rule1591 = ReplacementRule(pattern1591, replacement1591) pattern1592 = Pattern(Integral((c_ + x_*WC('d', S(1)))/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons483, cons908) rule1592 = ReplacementRule(pattern1592, With1592) pattern1593 = Pattern(Integral((c_ + x_*WC('d', S(1)))/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons483, cons909) rule1593 = ReplacementRule(pattern1593, With1593) pattern1594 = Pattern(Integral((c_ + x_*WC('d', S(1)))/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons484, cons910) rule1594 = ReplacementRule(pattern1594, With1594) pattern1595 = Pattern(Integral((c_ + x_*WC('d', S(1)))/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons484, cons911) rule1595 = ReplacementRule(pattern1595, With1595) pattern1596 = Pattern(Integral((c_ + x_**S(4)*WC('d', S(1)))/sqrt(a_ + x_**S(6)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons912) rule1596 = ReplacementRule(pattern1596, With1596) pattern1597 = Pattern(Integral((c_ + x_**S(4)*WC('d', S(1)))/sqrt(a_ + x_**S(6)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons913) rule1597 = ReplacementRule(pattern1597, With1597) pattern1598 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))/sqrt(a_ + x_**S(8)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons914) rule1598 = ReplacementRule(pattern1598, replacement1598) pattern1599 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))/sqrt(a_ + x_**S(8)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons915) rule1599 = ReplacementRule(pattern1599, replacement1599) pattern1600 = Pattern(Integral(Pq_/(x_*sqrt(a_ + x_**n_*WC('b', S(1)))), x_), cons2, cons3, cons66, cons150, cons916) rule1600 = ReplacementRule(pattern1600, replacement1600) pattern1601 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons66, cons676, cons917) rule1601 = ReplacementRule(pattern1601, With1601) pattern1602 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons66, cons676, cons917) rule1602 = ReplacementRule(pattern1602, With1602) pattern1603 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons66, cons150, cons918) rule1603 = ReplacementRule(pattern1603, replacement1603) pattern1604 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons8, cons19, cons66, cons87) rule1604 = ReplacementRule(pattern1604, replacement1604) pattern1605 = Pattern(Integral(Pq_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons66, cons87) rule1605 = ReplacementRule(pattern1605, replacement1605) pattern1606 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons66, cons150, cons33, cons96, cons919, CustomConstraint(With1606)) rule1606 = ReplacementRule(pattern1606, replacement1606) pattern1607 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons66, cons150, CustomConstraint(With1607)) rule1607 = ReplacementRule(pattern1607, replacement1607) pattern1608 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons66, cons150, CustomConstraint(With1608)) rule1608 = ReplacementRule(pattern1608, replacement1608) pattern1609 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons66, cons198, cons20) rule1609 = ReplacementRule(pattern1609, With1609) pattern1610 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons66, cons198, cons369) rule1610 = ReplacementRule(pattern1610, With1610) pattern1611 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons66, cons198, cons358) rule1611 = ReplacementRule(pattern1611, With1611) pattern1612 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons19, cons5, cons66, cons491) rule1612 = ReplacementRule(pattern1612, With1612) pattern1613 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons66, cons491) rule1613 = ReplacementRule(pattern1613, With1613) pattern1614 = Pattern(Integral(Pq_*(c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons66, cons491) rule1614 = ReplacementRule(pattern1614, replacement1614) pattern1615 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons19, cons4, cons5, cons860, cons543, cons25) rule1615 = ReplacementRule(pattern1615, replacement1615) pattern1616 = Pattern(Integral(Pq_*(c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons5, cons860, cons543, cons25) rule1616 = ReplacementRule(pattern1616, replacement1616) pattern1617 = Pattern(Integral((A_ + x_**WC('m', S(1))*WC('B', S(1)))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons36, cons37, cons19, cons4, cons5, cons55) rule1617 = ReplacementRule(pattern1617, replacement1617) pattern1618 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons19, cons4, cons5, cons920) rule1618 = ReplacementRule(pattern1618, replacement1618) pattern1619 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons4, cons5, cons920) rule1619 = ReplacementRule(pattern1619, replacement1619) pattern1620 = Pattern(Integral(Pq_*u_**WC('m', S(1))*(a_ + v_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons19, cons4, cons5, cons556, cons921) rule1620 = ReplacementRule(pattern1620, replacement1620) pattern1621 = Pattern(Integral(Pq_*(a_ + v_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons554, cons921) rule1621 = ReplacementRule(pattern1621, replacement1621) pattern1622 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**WC('p', S(1)), x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons5, cons66, cons57, cons496) rule1622 = ReplacementRule(pattern1622, replacement1622) pattern1623 = Pattern(Integral(Pq_*(a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**WC('p', S(1)), x_), cons59, cons60, cons61, cons62, cons4, cons5, cons66, cons57, cons496) rule1623 = ReplacementRule(pattern1623, replacement1623) pattern1624 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**WC('p', S(1)), x_), cons59, cons60, cons61, cons62, cons8, cons19, cons4, cons5, cons66, cons57) rule1624 = ReplacementRule(pattern1624, replacement1624) pattern1625 = Pattern(Integral(Pq_*(a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**WC('p', S(1)), x_), cons59, cons60, cons61, cons62, cons4, cons5, cons66, cons57) rule1625 = ReplacementRule(pattern1625, replacement1625) pattern1626 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('p', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)) + x_**WC('n2', S(1))*WC('g', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons48, cons922, cons923) rule1626 = ReplacementRule(pattern1626, replacement1626) pattern1627 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('p', S(1))*(e_ + x_**WC('n2', S(1))*WC('g', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons210, cons4, cons5, cons48, cons924, cons923) rule1627 = ReplacementRule(pattern1627, replacement1627) pattern1628 = Pattern(Integral((x_*WC('h', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('p', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)) + x_**WC('n2', S(1))*WC('g', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons211, cons19, cons4, cons5, cons48, cons925, cons926, cons68) rule1628 = ReplacementRule(pattern1628, replacement1628) pattern1629 = Pattern(Integral((x_*WC('h', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('p', S(1))*(e_ + x_**WC('n2', S(1))*WC('g', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons210, cons211, cons19, cons4, cons5, cons48, cons597, cons926, cons68) rule1629 = ReplacementRule(pattern1629, replacement1629) pattern1630 = Pattern(Integral((A_ + x_**WC('m', S(1))*WC('B', S(1)))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(x_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons36, cons37, cons19, cons4, cons5, cons52, cons73, cons55) rule1630 = ReplacementRule(pattern1630, replacement1630) pattern1631 = Pattern(Integral(Px_**WC('q', S(1))*((c_ + x_*WC('d', S(1)))**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons927, cons588, cons89) rule1631 = ReplacementRule(pattern1631, With1631) pattern1632 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons19, cons4, cons5, cons48, cons860, cons55) rule1632 = ReplacementRule(pattern1632, replacement1632) pattern1633 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons48, cons66, cons130) rule1633 = ReplacementRule(pattern1633, replacement1633) pattern1634 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons4, cons48, cons66, cons130) rule1634 = ReplacementRule(pattern1634, replacement1634) pattern1635 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**WC('n', S(1))*WC('e', S(1)) + x_**WC('n2', S(1))*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons48, cons928, cons929) rule1635 = ReplacementRule(pattern1635, replacement1635) pattern1636 = Pattern(Integral((d_ + x_**WC('n2', S(1))*WC('f', S(1)))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons127, cons4, cons5, cons48, cons924, cons930) rule1636 = ReplacementRule(pattern1636, replacement1636) pattern1637 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**WC('n', S(1))*WC('e', S(1)) + x_**WC('n2', S(1))*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons48, cons931, cons932, cons68) rule1637 = ReplacementRule(pattern1637, replacement1637) pattern1638 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(d_ + x_**WC('n2', S(1))*WC('f', S(1)))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons127, cons210, cons19, cons4, cons5, cons48, cons597, cons930, cons68) rule1638 = ReplacementRule(pattern1638, replacement1638) pattern1639 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons48, cons66, cons47, cons316) rule1639 = ReplacementRule(pattern1639, replacement1639) pattern1640 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons4, cons5, cons48, cons66, cons47, cons316) rule1640 = ReplacementRule(pattern1640, replacement1640) pattern1641 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons5, cons48, cons860, cons228, cons502) rule1641 = ReplacementRule(pattern1641, replacement1641) pattern1642 = Pattern(Integral(Pq_*(d_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons48, cons860, cons228, cons502) rule1642 = ReplacementRule(pattern1642, replacement1642) pattern1643 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons48, cons66, cons861) rule1643 = ReplacementRule(pattern1643, replacement1643) pattern1644 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons4, cons5, cons48, cons66, cons861, cons862) rule1644 = ReplacementRule(pattern1644, replacement1644) pattern1645 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)) + x_**WC('n2', S(1))*WC('f', S(1)) + x_**WC('n3', S(1))*WC('g', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons48, cons933, cons228, cons934, cons935) rule1645 = ReplacementRule(pattern1645, replacement1645) pattern1646 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**WC('n2', S(1))*WC('f', S(1)) + x_**WC('n3', S(1))*WC('g', S(1))), x_), cons2, cons3, cons8, cons29, cons127, cons210, cons4, cons5, cons48, cons933, cons228, cons936, cons937) rule1646 = ReplacementRule(pattern1646, replacement1646) pattern1647 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)) + x_**WC('n3', S(1))*WC('g', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons210, cons4, cons5, cons48, cons933, cons228, cons934, cons938) rule1647 = ReplacementRule(pattern1647, replacement1647) pattern1648 = Pattern(Integral((d_ + x_**WC('n3', S(1))*WC('g', S(1)))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons210, cons4, cons5, cons48, cons933, cons228, cons936, cons939) rule1648 = ReplacementRule(pattern1648, replacement1648) pattern1649 = Pattern(Integral(x_**WC('m', S(1))*(e_ + x_**WC('q', S(1))*WC('f', S(1)) + x_**WC('r', S(1))*WC('g', S(1)) + x_**WC('s', S(1))*WC('h', S(1)))/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons50, cons127, cons210, cons211, cons19, cons4, cons48, cons940, cons941, cons942, cons228, cons943, cons854) rule1649 = ReplacementRule(pattern1649, replacement1649) pattern1650 = Pattern(Integral((d_*x_)**WC('m', S(1))*(e_ + x_**WC('q', S(1))*WC('f', S(1)) + x_**WC('r', S(1))*WC('g', S(1)) + x_**WC('s', S(1))*WC('h', S(1)))/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons211, cons19, cons4, cons48, cons940, cons941, cons942, cons228, cons943, cons854) rule1650 = ReplacementRule(pattern1650, replacement1650) pattern1651 = Pattern(Integral(Pq_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons48, cons66, cons228, cons150, cons13, cons139, CustomConstraint(With1651)) rule1651 = ReplacementRule(pattern1651, replacement1651) pattern1652 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)) + x_*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons228, cons944) rule1652 = ReplacementRule(pattern1652, replacement1652) pattern1653 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons127, cons210, cons228, cons944) rule1653 = ReplacementRule(pattern1653, replacement1653) pattern1654 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons210, cons228, cons944) rule1654 = ReplacementRule(pattern1654, replacement1654) pattern1655 = Pattern(Integral(x_**S(2)*(x_**S(4)*WC('h', S(1)) + x_**S(2)*WC('g', S(1)) + x_*WC('f', S(1)) + WC('e', S(0)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons50, cons127, cons210, cons211, cons228, cons945, cons946) rule1655 = ReplacementRule(pattern1655, replacement1655) pattern1656 = Pattern(Integral(x_**S(2)*(x_**S(4)*WC('h', S(1)) + x_**S(2)*WC('g', S(1)) + WC('e', S(0)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons50, cons210, cons211, cons228, cons945, cons946) rule1656 = ReplacementRule(pattern1656, replacement1656) pattern1657 = Pattern(Integral((d_ + x_**S(6)*WC('h', S(1)) + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)) + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons211, cons228, cons945, cons947) rule1657 = ReplacementRule(pattern1657, replacement1657) pattern1658 = Pattern(Integral((d_ + x_**S(6)*WC('h', S(1)) + x_**S(3)*WC('f', S(1)) + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons211, cons228, cons945, cons948) rule1658 = ReplacementRule(pattern1658, replacement1658) pattern1659 = Pattern(Integral(Pq_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons48, cons66, cons228, cons150, cons13, cons139, CustomConstraint(With1659)) rule1659 = ReplacementRule(pattern1659, replacement1659) pattern1660 = Pattern(Integral(Pq_*x_**m_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons48, cons66, cons228, cons150, cons13, cons139, cons86, CustomConstraint(With1660)) rule1660 = ReplacementRule(pattern1660, replacement1660) pattern1661 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons48, cons860, cons228, cons150, cons20, CustomConstraint(With1661)) rule1661 = ReplacementRule(pattern1661, replacement1661) pattern1662 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))/(a_ + x_**n2_*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons19, cons48, cons860, cons228, cons150, cons284) rule1662 = ReplacementRule(pattern1662, replacement1662) pattern1663 = Pattern(Integral(Pq_/(a_ + x_**n2_*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons8, cons48, cons860, cons228, cons150, cons949) rule1663 = ReplacementRule(pattern1663, replacement1663) pattern1664 = Pattern(Integral(Pq_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons66, cons228, cons65, CustomConstraint(With1664)) rule1664 = ReplacementRule(pattern1664, replacement1664) pattern1665 = Pattern(Integral(Pq_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons66, cons228, cons721, cons950, CustomConstraint(With1665)) rule1665 = ReplacementRule(pattern1665, replacement1665) pattern1666 = Pattern(Integral(Pq_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons66, cons228, cons721, cons951, CustomConstraint(With1666)) rule1666 = ReplacementRule(pattern1666, replacement1666) pattern1667 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons19, cons5, cons48, cons860, cons228, cons150, CustomConstraint(With1667)) rule1667 = ReplacementRule(pattern1667, replacement1667) pattern1668 = Pattern(Integral(Pq_*(a_ + x_**n2_*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons48, cons860, cons228, cons150, CustomConstraint(With1668)) rule1668 = ReplacementRule(pattern1668, replacement1668) pattern1669 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons19, cons5, cons48, cons66, cons228, cons150, cons952) rule1669 = ReplacementRule(pattern1669, With1669) pattern1670 = Pattern(Integral(Pq_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons48, cons66, cons228, cons150, cons952) rule1670 = ReplacementRule(pattern1670, With1670) pattern1671 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons8, cons29, cons19, cons48, cons66, cons228, cons150) rule1671 = ReplacementRule(pattern1671, replacement1671) pattern1672 = Pattern(Integral(Pq_/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons8, cons48, cons66, cons228, cons150) rule1672 = ReplacementRule(pattern1672, replacement1672) pattern1673 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons48, cons66, cons228, cons198, cons20) rule1673 = ReplacementRule(pattern1673, With1673) pattern1674 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons48, cons66, cons228, cons198, cons369) rule1674 = ReplacementRule(pattern1674, With1674) pattern1675 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons19, cons5, cons48, cons66, cons228, cons198, cons358) rule1675 = ReplacementRule(pattern1675, With1675) pattern1676 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons48, cons66, cons228, cons491) rule1676 = ReplacementRule(pattern1676, With1676) pattern1677 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons5, cons48, cons66, cons228, cons491) rule1677 = ReplacementRule(pattern1677, With1677) pattern1678 = Pattern(Integral(Pq_*(d_*x_)**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons48, cons66, cons228, cons491, cons75) rule1678 = ReplacementRule(pattern1678, replacement1678) pattern1679 = Pattern(Integral(Pq_*(d_*x_)**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons48, cons66, cons228, cons491, cons953) rule1679 = ReplacementRule(pattern1679, replacement1679) pattern1680 = Pattern(Integral(Pq_*(d_*x_)**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons19, cons5, cons48, cons66, cons228, cons491) rule1680 = ReplacementRule(pattern1680, replacement1680) pattern1681 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons4, cons5, cons48, cons860, cons228, cons543, cons25) rule1681 = ReplacementRule(pattern1681, replacement1681) pattern1682 = Pattern(Integral(Pq_*(d_*x_)**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons19, cons5, cons48, cons860, cons228, cons543, cons25) rule1682 = ReplacementRule(pattern1682, replacement1682) pattern1683 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons48, cons66, cons228) rule1683 = ReplacementRule(pattern1683, With1683) pattern1684 = Pattern(Integral(Pq_/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons8, cons4, cons48, cons66, cons228) rule1684 = ReplacementRule(pattern1684, With1684) pattern1685 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons19, cons4, cons48, cons66, cons704) rule1685 = ReplacementRule(pattern1685, replacement1685) pattern1686 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons4, cons48, cons66, cons704) rule1686 = ReplacementRule(pattern1686, replacement1686) pattern1687 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons48, cons920) rule1687 = ReplacementRule(pattern1687, replacement1687) pattern1688 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons4, cons5, cons48, cons920) rule1688 = ReplacementRule(pattern1688, replacement1688) pattern1689 = Pattern(Integral(Pq_*u_**WC('m', S(1))*(a_ + v_**n_*WC('b', S(1)) + v_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons19, cons4, cons5, cons48, cons556, cons921) rule1689 = ReplacementRule(pattern1689, replacement1689) pattern1690 = Pattern(Integral(Pq_*(a_ + v_**n_*WC('b', S(1)) + v_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons4, cons5, cons48, cons554, cons921) rule1690 = ReplacementRule(pattern1690, replacement1690) pattern1691 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons798, cons4, cons5, cons149, cons954, cons955) rule1691 = ReplacementRule(pattern1691, replacement1691) pattern1692 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons798, cons4, cons149, cons954, cons956, cons13, cons139) rule1692 = ReplacementRule(pattern1692, replacement1692) pattern1693 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons798, cons4, cons5, cons149, cons954, cons956, cons957) rule1693 = ReplacementRule(pattern1693, replacement1693) pattern1694 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons149, cons958, cons959, cons165, cons960) rule1694 = ReplacementRule(pattern1694, replacement1694) pattern1695 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons149, cons958, cons959, cons165, cons961) rule1695 = ReplacementRule(pattern1695, replacement1695) pattern1696 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons149, cons958, cons959, cons139, cons962) rule1696 = ReplacementRule(pattern1696, replacement1696) pattern1697 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons149, cons958, cons959, cons139) rule1697 = ReplacementRule(pattern1697, replacement1697) pattern1698 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons798, cons4, cons963, cons954, cons964) rule1698 = ReplacementRule(pattern1698, replacement1698) pattern1699 = Pattern(Integral(S(1)/sqrt(x_**S(2)*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons4, cons965) rule1699 = ReplacementRule(pattern1699, replacement1699) pattern1700 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons798, cons4, cons721, cons954, cons964) rule1700 = ReplacementRule(pattern1700, replacement1700) pattern1701 = Pattern(Integral(S(1)/sqrt(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons966, cons967) rule1701 = ReplacementRule(pattern1701, replacement1701) pattern1702 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons798, cons4, cons5, cons149, cons954, cons968) rule1702 = ReplacementRule(pattern1702, replacement1702) pattern1703 = Pattern(Integral((u_**WC('j', S(1))*WC('a', S(1)) + u_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons798, cons4, cons5, cons70, cons71) rule1703 = ReplacementRule(pattern1703, replacement1703) pattern1704 = Pattern(Integral(x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons798, cons19, cons4, cons5, cons149, cons954, cons969, cons55) rule1704 = ReplacementRule(pattern1704, replacement1704) pattern1705 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons5, cons149, cons954, cons970, cons971) rule1705 = ReplacementRule(pattern1705, replacement1705) pattern1706 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons149, cons954, cons972, cons13, cons139, cons971) rule1706 = ReplacementRule(pattern1706, replacement1706) pattern1707 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons5, cons149, cons954, cons972, cons973, cons974) rule1707 = ReplacementRule(pattern1707, replacement1707) pattern1708 = Pattern(Integral((c_*x_)**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons5, cons149, cons954, cons972) rule1708 = ReplacementRule(pattern1708, replacement1708) pattern1709 = Pattern(Integral(x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons798, cons19, cons4, cons5, cons149, cons954, cons969, cons502, cons975) rule1709 = ReplacementRule(pattern1709, replacement1709) pattern1710 = Pattern(Integral((c_*x_)**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons5, cons149, cons954, cons969, cons502, cons975) rule1710 = ReplacementRule(pattern1710, replacement1710) pattern1711 = Pattern(Integral((x_*WC('c', S(1)))**m_*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons149, cons976, cons959, cons974, cons165, cons977) rule1711 = ReplacementRule(pattern1711, replacement1711) pattern1712 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons149, cons958, cons959, cons974, cons165, cons514) rule1712 = ReplacementRule(pattern1712, replacement1712) pattern1713 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons149, cons976, cons959, cons974, cons139, cons978) rule1713 = ReplacementRule(pattern1713, replacement1713) pattern1714 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons149, cons958, cons959, cons974, cons139) rule1714 = ReplacementRule(pattern1714, replacement1714) pattern1715 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons149, cons966, cons959, cons974, cons979, cons514) rule1715 = ReplacementRule(pattern1715, replacement1715) pattern1716 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons149, cons966, cons959, cons974, cons980) rule1716 = ReplacementRule(pattern1716, replacement1716) pattern1717 = Pattern(Integral(x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons798, cons19, cons4, cons5, cons149, cons954, cons969, cons68, cons543, cons25) rule1717 = ReplacementRule(pattern1717, replacement1717) pattern1718 = Pattern(Integral((c_*x_)**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons5, cons149, cons954, cons969, cons68, cons543, cons25) rule1718 = ReplacementRule(pattern1718, replacement1718) pattern1719 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons963, cons954, cons981, cons971) rule1719 = ReplacementRule(pattern1719, replacement1719) pattern1720 = Pattern(Integral(x_**WC('m', S(1))/sqrt(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons798, cons4, cons982, cons954) rule1720 = ReplacementRule(pattern1720, replacement1720) pattern1721 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons721, cons954, cons981, cons971) rule1721 = ReplacementRule(pattern1721, replacement1721) pattern1722 = Pattern(Integral((c_*x_)**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons5, cons669, cons954, cons981) rule1722 = ReplacementRule(pattern1722, replacement1722) pattern1723 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons5, cons149, cons954, cons968) rule1723 = ReplacementRule(pattern1723, replacement1723) pattern1724 = Pattern(Integral(u_**WC('m', S(1))*(v_**WC('j', S(1))*WC('a', S(1)) + v_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons798, cons19, cons4, cons5, cons556) rule1724 = ReplacementRule(pattern1724, replacement1724) pattern1725 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(x_**j_*WC('a', S(1)) + x_**WC('k', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons798, cons799, cons19, cons4, cons5, cons52, cons149, cons983, cons969, cons984, cons502, cons975) rule1725 = ReplacementRule(pattern1725, replacement1725) pattern1726 = Pattern(Integral((e_*x_)**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*(x_**j_*WC('a', S(1)) + x_**WC('k', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons798, cons799, cons19, cons4, cons5, cons52, cons149, cons983, cons969, cons984, cons502, cons975) rule1726 = ReplacementRule(pattern1726, replacement1726) pattern1727 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))*(x_**WC('jn', S(1))*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons798, cons19, cons4, cons5, cons985, cons149, cons73, cons986, cons987, cons973) rule1727 = ReplacementRule(pattern1727, replacement1727) pattern1728 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))*(x_**WC('jn', S(1))*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons798, cons19, cons4, cons985, cons149, cons73, cons988, cons139, cons989, cons990) rule1728 = ReplacementRule(pattern1728, replacement1728) pattern1729 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))*(x_**WC('jn', S(1))*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons798, cons5, cons985, cons149, cons73, cons95, cons90, cons991, cons992, cons973, cons993) rule1729 = ReplacementRule(pattern1729, replacement1729) pattern1730 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))*(x_**WC('jn', S(1))*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons798, cons19, cons4, cons5, cons985, cons149, cons73, cons994, cons990) rule1730 = ReplacementRule(pattern1730, replacement1730) pattern1731 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*(x_**j_*WC('a', S(1)) + x_**WC('k', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons798, cons799, cons19, cons4, cons5, cons52, cons149, cons983, cons969, cons984, cons68, cons543, cons25) rule1731 = ReplacementRule(pattern1731, replacement1731) pattern1732 = Pattern(Integral((e_*x_)**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*(x_**j_*WC('a', S(1)) + x_**WC('k', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons798, cons799, cons19, cons4, cons5, cons52, cons149, cons983, cons969, cons984, cons68, cons543, cons25) rule1732 = ReplacementRule(pattern1732, replacement1732) pattern1733 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*(x_**WC('jn', S(1))*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons798, cons19, cons4, cons5, cons52, cons985, cons149, cons73, cons995) rule1733 = ReplacementRule(pattern1733, replacement1733) pattern1734 = Pattern(Integral(Pq_*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons798, cons4, cons5, cons860, cons149, cons954, cons966, cons969, cons996) rule1734 = ReplacementRule(pattern1734, With1734) pattern1735 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons798, cons19, cons4, cons5, cons860, cons149, cons954, cons969, cons502) rule1735 = ReplacementRule(pattern1735, replacement1735) pattern1736 = Pattern(Integral(Pq_*(c_*x_)**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons4, cons5, cons860, cons149, cons954, cons969, cons502, cons33, cons997) rule1736 = ReplacementRule(pattern1736, replacement1736) pattern1737 = Pattern(Integral(Pq_*(c_*x_)**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons5, cons860, cons149, cons954, cons969, cons502) rule1737 = ReplacementRule(pattern1737, replacement1737) pattern1738 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons5, cons860, cons149, cons998, cons20, CustomConstraint(With1738)) rule1738 = ReplacementRule(pattern1738, replacement1738) pattern1739 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons19, cons5, cons66, cons149, cons999, cons1000, CustomConstraint(With1739)) rule1739 = ReplacementRule(pattern1739, replacement1739) pattern1740 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons798, cons19, cons4, cons5, cons860, cons149, cons954, cons969, cons543, cons25) rule1740 = ReplacementRule(pattern1740, replacement1740) pattern1741 = Pattern(Integral(Pq_*(c_*x_)**m_*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons4, cons5, cons860, cons149, cons954, cons969, cons543, cons25, cons33, cons997) rule1741 = ReplacementRule(pattern1741, replacement1741) pattern1742 = Pattern(Integral(Pq_*(c_*x_)**m_*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons5, cons860, cons149, cons954, cons969, cons543, cons25) rule1742 = ReplacementRule(pattern1742, replacement1742) pattern1743 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons8, cons798, cons19, cons4, cons5, cons920, cons149, cons954) rule1743 = ReplacementRule(pattern1743, replacement1743) pattern1744 = Pattern(Integral(Pq_*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons798, cons4, cons5, cons920, cons149, cons954) rule1744 = ReplacementRule(pattern1744, replacement1744) pattern1745 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons40, cons1001) rule1745 = ReplacementRule(pattern1745, replacement1745) pattern1746 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons130, cons1002) rule1746 = ReplacementRule(pattern1746, replacement1746) pattern1747 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons65, cons1002, CustomConstraint(With1747)) rule1747 = ReplacementRule(pattern1747, replacement1747) pattern1748 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons65, cons1002) rule1748 = ReplacementRule(pattern1748, With1748) pattern1749 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons5, cons149, cons1001) rule1749 = ReplacementRule(pattern1749, replacement1749) pattern1750 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons5, cons149, cons1002, CustomConstraint(With1750)) rule1750 = ReplacementRule(pattern1750, replacement1750) pattern1751 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons5, cons149, cons1002) rule1751 = ReplacementRule(pattern1751, With1751) pattern1752 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons40, cons1001) rule1752 = ReplacementRule(pattern1752, replacement1752) pattern1753 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons130, cons1002) rule1753 = ReplacementRule(pattern1753, replacement1753) pattern1754 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons65, cons1002, CustomConstraint(With1754)) rule1754 = ReplacementRule(pattern1754, replacement1754) pattern1755 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons65, cons1002) rule1755 = ReplacementRule(pattern1755, With1755) pattern1756 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons5, cons149, cons1001) rule1756 = ReplacementRule(pattern1756, replacement1756) pattern1757 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons5, cons149, cons1002, CustomConstraint(With1757)) rule1757 = ReplacementRule(pattern1757, replacement1757) pattern1758 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons5, cons149, cons1002) rule1758 = ReplacementRule(pattern1758, With1758) pattern1759 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons40, cons1003) rule1759 = ReplacementRule(pattern1759, replacement1759) pattern1760 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons130, cons1004) rule1760 = ReplacementRule(pattern1760, replacement1760) pattern1761 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons65, cons1004, CustomConstraint(With1761)) rule1761 = ReplacementRule(pattern1761, replacement1761) pattern1762 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons65, cons1004) rule1762 = ReplacementRule(pattern1762, With1762) pattern1763 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons5, cons149, cons1003) rule1763 = ReplacementRule(pattern1763, replacement1763) pattern1764 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons5, cons149, cons1004, CustomConstraint(With1764)) rule1764 = ReplacementRule(pattern1764, replacement1764) pattern1765 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons5, cons149, cons1004) rule1765 = ReplacementRule(pattern1765, With1765) pattern1766 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons50, cons127, cons19, cons40, cons1003) rule1766 = ReplacementRule(pattern1766, replacement1766) pattern1767 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons50, cons127, cons19, cons130, cons1004) rule1767 = ReplacementRule(pattern1767, replacement1767) pattern1768 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons50, cons127, cons19, cons65, cons1004, CustomConstraint(With1768)) rule1768 = ReplacementRule(pattern1768, replacement1768) pattern1769 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons50, cons127, cons19, cons65, cons1004) rule1769 = ReplacementRule(pattern1769, With1769) pattern1770 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons50, cons127, cons19, cons5, cons149, cons1003) rule1770 = ReplacementRule(pattern1770, replacement1770) pattern1771 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons50, cons127, cons19, cons5, cons149, cons1004, CustomConstraint(With1771)) rule1771 = ReplacementRule(pattern1771, replacement1771) pattern1772 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons8, cons29, cons50, cons127, cons19, cons5, cons149, cons1004) rule1772 = ReplacementRule(pattern1772, With1772) pattern1773 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons40, cons1005, cons1006) rule1773 = ReplacementRule(pattern1773, replacement1773) pattern1774 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons40, cons1005, cons1007) rule1774 = ReplacementRule(pattern1774, replacement1774) pattern1775 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons40, cons1008, cons1006) rule1775 = ReplacementRule(pattern1775, With1775) pattern1776 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons130, cons1008, cons1007) rule1776 = ReplacementRule(pattern1776, replacement1776) pattern1777 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons65, cons1008, cons1007, CustomConstraint(With1777)) rule1777 = ReplacementRule(pattern1777, replacement1777) pattern1778 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons65, cons1008, cons1007) rule1778 = ReplacementRule(pattern1778, replacement1778) pattern1779 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons149, cons1005, cons1006) rule1779 = ReplacementRule(pattern1779, replacement1779) pattern1780 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons149, cons1005, cons1007) rule1780 = ReplacementRule(pattern1780, With1780) pattern1781 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons149, cons1008, cons1006) rule1781 = ReplacementRule(pattern1781, With1781) pattern1782 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons149, cons1008, cons1007, CustomConstraint(With1782)) rule1782 = ReplacementRule(pattern1782, replacement1782) pattern1783 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons149, cons1008, cons1007) rule1783 = ReplacementRule(pattern1783, With1783) pattern1784 = Pattern(Integral(u_**p_, x_), cons5, cons1009, cons1010) rule1784 = ReplacementRule(pattern1784, replacement1784) pattern1785 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons40, cons1005, cons1006) rule1785 = ReplacementRule(pattern1785, replacement1785) pattern1786 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons40, cons1005, cons1007) rule1786 = ReplacementRule(pattern1786, With1786) pattern1787 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons40, cons1008, cons1006) rule1787 = ReplacementRule(pattern1787, With1787) pattern1788 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons130, cons1008, cons1007) rule1788 = ReplacementRule(pattern1788, replacement1788) pattern1789 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons65, cons1008, cons1007, CustomConstraint(With1789)) rule1789 = ReplacementRule(pattern1789, replacement1789) pattern1790 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons65, cons1008, cons1007) rule1790 = ReplacementRule(pattern1790, replacement1790) pattern1791 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons5, cons149, cons1005, cons1006) rule1791 = ReplacementRule(pattern1791, replacement1791) pattern1792 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons5, cons149, cons1005, cons1007) rule1792 = ReplacementRule(pattern1792, With1792) pattern1793 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons5, cons149, cons1008, cons1006) rule1793 = ReplacementRule(pattern1793, With1793) pattern1794 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons5, cons149, cons1008, cons1007, CustomConstraint(With1794)) rule1794 = ReplacementRule(pattern1794, replacement1794) pattern1795 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons5, cons149, cons1008, cons1007) rule1795 = ReplacementRule(pattern1795, With1795) pattern1796 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('p', S(1)), x_), cons19, cons5, cons70, cons1011, cons1012) rule1796 = ReplacementRule(pattern1796, replacement1796) pattern1797 = Pattern(Integral((f_ + x_**S(2)*WC('g', S(1)))/((d_ + x_**S(2)*WC('d', S(1)) + x_*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('a', S(1)) + x_**S(3)*WC('b', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons385, cons1013, cons1014) rule1797 = ReplacementRule(pattern1797, replacement1797) pattern1798 = Pattern(Integral((f_ + x_**S(2)*WC('g', S(1)))/((d_ + x_**S(2)*WC('d', S(1)) + x_*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('a', S(1)) + x_**S(3)*WC('b', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons385, cons1013, cons1015) rule1798 = ReplacementRule(pattern1798, replacement1798) pattern1799 = Pattern(Integral((x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons5, cons1016, cons1017, cons1018) rule1799 = ReplacementRule(pattern1799, replacement1799) pattern1800 = Pattern(Integral(v_**p_, x_), cons5, cons1019, cons1020, cons1017, cons1018, CustomConstraint(With1800)) rule1800 = ReplacementRule(pattern1800, replacement1800) pattern1801 = Pattern(Integral(u_*(x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons5, cons806, cons1016, cons359) rule1801 = ReplacementRule(pattern1801, replacement1801) pattern1802 = Pattern(Integral(u_*v_**p_, x_), cons5, cons806, cons1019, cons1020, cons359, CustomConstraint(With1802)) rule1802 = ReplacementRule(pattern1802, replacement1802) pattern1803 = Pattern(Integral((a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons1021, cons248) rule1803 = ReplacementRule(pattern1803, replacement1803) pattern1804 = Pattern(Integral(v_**p_, x_), cons5, cons1019, cons1020, cons248, CustomConstraint(With1804)) rule1804 = ReplacementRule(pattern1804, replacement1804) pattern1805 = Pattern(Integral((x_**S(3)*WC('D', S(1)) + x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons8, cons36, cons37, cons38, cons1025, cons1022, cons1023, cons1024) rule1805 = ReplacementRule(pattern1805, With1805) pattern1806 = Pattern(Integral((x_**S(3)*WC('D', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons8, cons36, cons37, cons1025, cons1022, cons1023, cons1024) rule1806 = ReplacementRule(pattern1806, With1806) pattern1807 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(3)*WC('D', S(1)) + x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons8, cons36, cons37, cons38, cons1025, cons19, cons1022, cons1023, cons1024) rule1807 = ReplacementRule(pattern1807, With1807) pattern1808 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(3)*WC('D', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons8, cons36, cons37, cons1025, cons19, cons1022, cons1023, cons1024) rule1808 = ReplacementRule(pattern1808, With1808) pattern1809 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons1026, cons1027, cons1028) rule1809 = ReplacementRule(pattern1809, With1809) pattern1810 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons1029, cons1030, cons1031) rule1810 = ReplacementRule(pattern1810, With1810) pattern1811 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons1026, cons1027, cons1032) rule1811 = ReplacementRule(pattern1811, With1811) pattern1812 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons1029, cons1030, cons1033) rule1812 = ReplacementRule(pattern1812, With1812) pattern1813 = Pattern(Integral((x_**S(3)*WC('D', S(1)) + x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons1025, cons1034, cons1035) rule1813 = ReplacementRule(pattern1813, replacement1813) pattern1814 = Pattern(Integral((x_**S(3)*WC('D', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons1025, cons1036, cons1037) rule1814 = ReplacementRule(pattern1814, replacement1814) pattern1815 = Pattern(Integral(u_/(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1)) + sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1038) rule1815 = ReplacementRule(pattern1815, replacement1815) pattern1816 = Pattern(Integral(u_/(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1)) + sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1039) rule1816 = ReplacementRule(pattern1816, replacement1816) pattern1817 = Pattern(Integral(u_/(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1)) + sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1040, cons1041) rule1817 = ReplacementRule(pattern1817, replacement1817) pattern1818 = Pattern(Integral(WC('u', S(1))/(x_**WC('n', S(1))*WC('d', S(1)) + sqrt(x_**WC('p', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))), x_), cons2, cons3, cons8, cons29, cons4, cons1042, cons1043) rule1818 = ReplacementRule(pattern1818, replacement1818) pattern1819 = Pattern(Integral(x_**WC('m', S(1))/(x_**WC('n', S(1))*WC('d', S(1)) + sqrt(x_**WC('p', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons1042, cons1044) rule1819 = ReplacementRule(pattern1819, replacement1819) pattern1820 = Pattern(Integral(S(1)/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons470) rule1820 = ReplacementRule(pattern1820, With1820) pattern1821 = Pattern(Integral(S(1)/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons29, cons127, cons470) rule1821 = ReplacementRule(pattern1821, With1821) pattern1822 = Pattern(Integral(S(1)/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons471) rule1822 = ReplacementRule(pattern1822, With1822) pattern1823 = Pattern(Integral(S(1)/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons29, cons127, cons471) rule1823 = ReplacementRule(pattern1823, With1823) pattern1824 = Pattern(Integral(S(1)/((d_ + x_*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons1045) rule1824 = ReplacementRule(pattern1824, replacement1824) pattern1825 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_*WC('e', S(1)))), x_), cons2, cons8, cons29, cons50, cons299) rule1825 = ReplacementRule(pattern1825, replacement1825) pattern1826 = Pattern(Integral(S(1)/((d_ + x_*WC('e', S(1)))**S(2)*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons1046, cons1047) rule1826 = ReplacementRule(pattern1826, replacement1826) pattern1827 = Pattern(Integral(S(1)/((d_ + x_*WC('e', S(1)))**S(2)*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons1046, cons1048) rule1827 = ReplacementRule(pattern1827, replacement1827) pattern1828 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_*WC('e', S(1)))**S(2)), x_), cons2, cons8, cons29, cons50, cons1049) rule1828 = ReplacementRule(pattern1828, replacement1828) pattern1829 = Pattern(Integral((A_ + x_**S(2)*WC('B', S(1)))/((d_ + x_**S(2)*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons1050, cons707) rule1829 = ReplacementRule(pattern1829, replacement1829) pattern1830 = Pattern(Integral((A_ + x_**S(2)*WC('B', S(1)))/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons8, cons29, cons50, cons36, cons37, cons1050, cons707) rule1830 = ReplacementRule(pattern1830, replacement1830) pattern1831 = Pattern(Integral((A_ + x_**S(4)*WC('B', S(1)))/(sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))*(d_ + x_**S(4)*WC('f', S(1)) + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons384, cons1051) rule1831 = ReplacementRule(pattern1831, replacement1831) pattern1832 = Pattern(Integral((A_ + x_**S(4)*WC('B', S(1)))/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(4)*WC('f', S(1)) + x_**S(2)*WC('e', S(1)))), x_), cons2, cons8, cons29, cons50, cons127, cons36, cons37, cons384, cons1051) rule1832 = ReplacementRule(pattern1832, replacement1832) pattern1833 = Pattern(Integral((A_ + x_**S(4)*WC('B', S(1)))/((d_ + x_**S(4)*WC('f', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons8, cons29, cons127, cons36, cons37, cons384, cons1051) rule1833 = ReplacementRule(pattern1833, replacement1833) pattern1834 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))/(d_ + x_**S(4)*WC('e', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons1052, cons699) rule1834 = ReplacementRule(pattern1834, replacement1834) pattern1835 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))/(d_ + x_**S(4)*WC('e', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons1052, cons711) rule1835 = ReplacementRule(pattern1835, With1835) pattern1836 = Pattern(Integral(S(1)/((a_ + x_*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons155) rule1836 = ReplacementRule(pattern1836, replacement1836) pattern1837 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))*sqrt(x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons211, cons1053, cons1054) rule1837 = ReplacementRule(pattern1837, replacement1837) pattern1838 = Pattern(Integral((u_ + (sqrt(v_)*WC('k', S(1)) + WC('j', S(0)))*WC('f', S(1)))**WC('n', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons127, cons210, cons211, cons798, cons799, cons19, cons4, cons70, cons820, cons1055, cons1056) rule1838 = ReplacementRule(pattern1838, replacement1838) pattern1839 = Pattern(Integral(((x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0)))**n_*WC('h', S(1)) + WC('g', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons211, cons4, cons1057, cons40) rule1839 = ReplacementRule(pattern1839, replacement1839) pattern1840 = Pattern(Integral(((x_*WC('e', S(1)) + sqrt(a_ + x_**S(2)*WC('c', S(1)))*WC('f', S(1)) + WC('d', S(0)))**n_*WC('h', S(1)) + WC('g', S(0)))**WC('p', S(1)), x_), cons2, cons8, cons29, cons50, cons127, cons210, cons211, cons4, cons1057, cons40) rule1840 = ReplacementRule(pattern1840, replacement1840) pattern1841 = Pattern(Integral(((u_ + sqrt(v_)*WC('f', S(1)))**n_*WC('h', S(1)) + WC('g', S(0)))**WC('p', S(1)), x_), cons127, cons210, cons211, cons4, cons70, cons820, cons821, cons1058, cons40) rule1841 = ReplacementRule(pattern1841, replacement1841) pattern1842 = Pattern(Integral((x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + WC('a', S(0)))*WC('f', S(1)))**WC('n', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons2, cons8, cons50, cons127, cons210, cons211, cons4, cons1057, cons20) rule1842 = ReplacementRule(pattern1842, replacement1842) pattern1843 = Pattern(Integral(x_**WC('p', S(1))*(g_ + x_**S(2)*WC('i', S(1)))**WC('m', S(1))*(x_*WC('e', S(1)) + sqrt(a_ + x_**S(2)*WC('c', S(1)))*WC('f', S(1)))**WC('n', S(1)), x_), cons2, cons8, cons50, cons127, cons210, cons226, cons4, cons1057, cons1059, cons1060, cons1061) rule1843 = ReplacementRule(pattern1843, replacement1843) pattern1844 = Pattern(Integral((x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1))*(x_**S(2)*WC('i', S(1)) + x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons211, cons226, cons4, cons1057, cons1059, cons1062, cons517, cons1061) rule1844 = ReplacementRule(pattern1844, replacement1844) pattern1845 = Pattern(Integral((g_ + x_**S(2)*WC('i', S(1)))**WC('m', S(1))*(x_*WC('e', S(1)) + sqrt(a_ + x_**S(2)*WC('c', S(1)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons2, cons8, cons29, cons50, cons127, cons210, cons226, cons4, cons1057, cons1059, cons517, cons1061) rule1845 = ReplacementRule(pattern1845, replacement1845) pattern1846 = Pattern(Integral((x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1))*(x_**S(2)*WC('i', S(1)) + x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons211, cons226, cons4, cons1057, cons1059, cons1062, cons75, cons1063) rule1846 = ReplacementRule(pattern1846, replacement1846) pattern1847 = Pattern(Integral((g_ + x_**S(2)*WC('i', S(1)))**WC('m', S(1))*(x_*WC('e', S(1)) + sqrt(a_ + x_**S(2)*WC('c', S(1)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons2, cons8, cons29, cons50, cons127, cons210, cons226, cons4, cons1057, cons1059, cons75, cons1063) rule1847 = ReplacementRule(pattern1847, replacement1847) pattern1848 = Pattern(Integral((x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1))*(x_**S(2)*WC('i', S(1)) + x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons211, cons226, cons4, cons1057, cons1059, cons1062, cons953, cons1063) rule1848 = ReplacementRule(pattern1848, replacement1848) pattern1849 = Pattern(Integral((g_ + x_**S(2)*WC('i', S(1)))**WC('m', S(1))*(x_*WC('e', S(1)) + sqrt(a_ + x_**S(2)*WC('c', S(1)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons2, cons8, cons29, cons50, cons127, cons210, cons226, cons4, cons1057, cons1059, cons953, cons1063) rule1849 = ReplacementRule(pattern1849, replacement1849) pattern1850 = Pattern(Integral(w_**WC('m', S(1))*(u_ + (sqrt(v_)*WC('k', S(1)) + WC('j', S(0)))*WC('f', S(1)))**WC('n', S(1)), x_), cons127, cons798, cons799, cons19, cons4, cons70, cons1064, cons1065, cons1066) rule1850 = ReplacementRule(pattern1850, replacement1850) pattern1851 = Pattern(Integral(S(1)/((a_ + x_**WC('n', S(1))*WC('b', S(1)))*sqrt(x_**S(2)*WC('c', S(1)) + (a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons4, cons1067) rule1851 = ReplacementRule(pattern1851, replacement1851) pattern1852 = Pattern(Integral(sqrt(a_ + sqrt(c_ + x_**S(2)*WC('d', S(1)))*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons1068) rule1852 = ReplacementRule(pattern1852, replacement1852) pattern1853 = Pattern(Integral(sqrt(x_**S(2)*WC('a', S(1)) + x_*sqrt(c_ + x_**S(2)*WC('d', S(1)))*WC('b', S(1)))/(x_*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons1069, cons1070) rule1853 = ReplacementRule(pattern1853, replacement1853) pattern1854 = Pattern(Integral(sqrt(x_*(x_*WC('a', S(1)) + sqrt(c_ + x_**S(2)*WC('d', S(1)))*WC('b', S(1)))*WC('e', S(1)))/(x_*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons1069, cons1071) rule1854 = ReplacementRule(pattern1854, replacement1854) pattern1855 = Pattern(Integral(sqrt(x_**S(2)*WC('c', S(1)) + sqrt(a_ + x_**S(4)*WC('b', S(1)))*WC('d', S(1)))/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons8, cons29, cons1072) rule1855 = ReplacementRule(pattern1855, replacement1855) pattern1856 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sqrt(x_**S(2)*WC('b', S(1)) + sqrt(a_ + x_**S(4)*WC('e', S(1))))/sqrt(a_ + x_**S(4)*WC('e', S(1))), x_), cons2, cons3, cons8, cons29, cons19, cons1073, cons45) rule1856 = ReplacementRule(pattern1856, replacement1856) pattern1857 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons483, cons481) rule1857 = ReplacementRule(pattern1857, With1857) pattern1858 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons483, cons482) rule1858 = ReplacementRule(pattern1858, With1858) pattern1859 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons484, cons481) rule1859 = ReplacementRule(pattern1859, With1859) pattern1860 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons484, cons482) rule1860 = ReplacementRule(pattern1860, With1860) pattern1861 = Pattern(Integral((e_ + x_*WC('f', S(1)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons483, cons481, CustomConstraint(With1861)) rule1861 = ReplacementRule(pattern1861, replacement1861) pattern1862 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons483, cons481, CustomConstraint(With1862)) rule1862 = ReplacementRule(pattern1862, replacement1862) pattern1863 = Pattern(Integral((e_ + x_*WC('f', S(1)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons483, cons482, CustomConstraint(With1863)) rule1863 = ReplacementRule(pattern1863, replacement1863) pattern1864 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons483, cons482, CustomConstraint(With1864)) rule1864 = ReplacementRule(pattern1864, replacement1864) pattern1865 = Pattern(Integral((e_ + x_*WC('f', S(1)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons484, cons481, CustomConstraint(With1865)) rule1865 = ReplacementRule(pattern1865, replacement1865) pattern1866 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons484, cons481, CustomConstraint(With1866)) rule1866 = ReplacementRule(pattern1866, replacement1866) pattern1867 = Pattern(Integral((e_ + x_*WC('f', S(1)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons484, cons482, CustomConstraint(With1867)) rule1867 = ReplacementRule(pattern1867, replacement1867) pattern1868 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons484, cons482, CustomConstraint(With1868)) rule1868 = ReplacementRule(pattern1868, replacement1868) pattern1869 = Pattern(Integral(x_**WC('m', S(1))/(c_ + x_**n_*WC('d', S(1)) + sqrt(a_ + x_**n_*WC('b', S(1)))*WC('e', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons1074, cons502) rule1869 = ReplacementRule(pattern1869, replacement1869) pattern1870 = Pattern(Integral(WC('u', S(1))/(c_ + x_**n_*WC('d', S(1)) + sqrt(a_ + x_**n_*WC('b', S(1)))*WC('e', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons1074) rule1870 = ReplacementRule(pattern1870, replacement1870) pattern1871 = Pattern(Integral((A_ + x_**n_*WC('B', S(1)))/(a_ + x_**S(2)*WC('b', S(1)) + x_**n2_*WC('d', S(1)) + x_**n_*WC('c', S(1))), x_), cons2, cons3, cons8, cons29, cons36, cons37, cons4, cons48, cons965, cons1075, cons1076) rule1871 = ReplacementRule(pattern1871, replacement1871) pattern1872 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('n', S(1))*WC('B', S(1)))/(a_ + x_**n2_*WC('d', S(1)) + x_**WC('k', S(1))*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1))), x_), cons2, cons3, cons8, cons29, cons36, cons37, cons19, cons4, cons48, cons1077, cons1078, cons1079) rule1872 = ReplacementRule(pattern1872, replacement1872) pattern1873 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_*(d_ + g_*x_**n3_ + x_**n2_*WC('f', S(1)) + x_**n_*WC('e', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons48, cons933, cons228, cons704) rule1873 = ReplacementRule(pattern1873, replacement1873) pattern1874 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_*(d_ + x_**n2_*WC('f', S(1)) + x_**n_*WC('e', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons48, cons228, cons704) rule1874 = ReplacementRule(pattern1874, replacement1874) pattern1875 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_*(d_ + g_*x_**n3_ + x_**n_*WC('e', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons210, cons4, cons48, cons933, cons228, cons704) rule1875 = ReplacementRule(pattern1875, replacement1875) pattern1876 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_*(d_ + g_*x_**n3_ + x_**n2_*WC('f', S(1))), x_), cons2, cons3, cons8, cons29, cons127, cons210, cons4, cons48, cons933, cons228, cons704) rule1876 = ReplacementRule(pattern1876, replacement1876) pattern1877 = Pattern(Integral((d_ + x_**n2_*WC('f', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons127, cons4, cons48, cons228, cons704) rule1877 = ReplacementRule(pattern1877, replacement1877) pattern1878 = Pattern(Integral((d_ + g_*x_**n3_)*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons210, cons4, cons48, cons933, cons228, cons704) rule1878 = ReplacementRule(pattern1878, replacement1878) pattern1879 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + g_*x_**n3_ + x_**n2_*WC('f', S(1)) + x_**n_*WC('e', S(1))), x_), cons2, cons8, cons29, cons50, cons127, cons210, cons4, cons48, cons933, cons704) rule1879 = ReplacementRule(pattern1879, replacement1879) pattern1880 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n2_*WC('f', S(1)) + x_**n_*WC('e', S(1))), x_), cons2, cons8, cons29, cons50, cons127, cons4, cons48, cons704) rule1880 = ReplacementRule(pattern1880, replacement1880) pattern1881 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + g_*x_**n3_ + x_**n_*WC('e', S(1))), x_), cons2, cons8, cons29, cons50, cons210, cons4, cons48, cons933, cons704) rule1881 = ReplacementRule(pattern1881, replacement1881) pattern1882 = Pattern(Integral((x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)) + WC('a', S(0)))/(d_ + x_**S(6)*WC('g', S(1)) + x_**S(4)*WC('f', S(1)) + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons1080, cons1081, cons1082, cons1083, cons1084, cons1085) rule1882 = ReplacementRule(pattern1882, With1882) pattern1883 = Pattern(Integral((x_**S(4)*WC('c', S(1)) + WC('a', S(0)))/(d_ + x_**S(6)*WC('g', S(1)) + x_**S(4)*WC('f', S(1)) + x_**S(2)*WC('e', S(1))), x_), cons2, cons8, cons29, cons50, cons127, cons210, cons1086, cons1087, cons1082, cons1088) rule1883 = ReplacementRule(pattern1883, With1883) pattern1884 = Pattern(Integral(u_*v_**p_, x_), cons13, cons139, cons806, cons1019, cons1089, cons1090, cons1091, CustomConstraint(With1884)) rule1884 = ReplacementRule(pattern1884, replacement1884) return [rule1476, rule1477, rule1478, rule1479, rule1480, rule1481, rule1482, rule1483, rule1484, rule1485, rule1486, rule1487, rule1488, rule1489, rule1490, rule1491, rule1492, rule1493, rule1494, rule1495, rule1496, rule1497, rule1498, rule1499, rule1500, rule1501, rule1502, rule1503, rule1504, rule1505, rule1506, rule1507, rule1508, rule1509, rule1510, rule1511, rule1512, rule1513, rule1514, rule1515, rule1516, rule1517, rule1518, rule1519, rule1520, rule1521, rule1522, rule1523, rule1524, rule1525, rule1526, rule1527, rule1528, rule1529, rule1530, rule1531, rule1532, rule1533, rule1534, rule1535, rule1536, rule1537, rule1538, rule1539, rule1540, rule1541, rule1542, rule1543, rule1544, rule1545, rule1546, rule1547, rule1548, rule1549, rule1550, rule1551, rule1552, rule1553, rule1554, rule1555, rule1556, rule1557, rule1558, rule1559, rule1560, rule1561, rule1562, rule1563, rule1564, rule1565, rule1566, rule1567, rule1568, rule1569, rule1570, rule1571, rule1572, rule1573, rule1574, rule1575, rule1576, rule1577, rule1578, rule1579, rule1580, rule1581, rule1582, rule1583, rule1584, rule1585, rule1586, rule1587, rule1588, rule1589, rule1590, rule1591, rule1592, rule1593, rule1594, rule1595, rule1596, rule1597, rule1598, rule1599, rule1600, rule1601, rule1602, rule1603, rule1604, rule1605, rule1606, rule1607, rule1608, rule1609, rule1610, rule1611, rule1612, rule1613, rule1614, rule1615, rule1616, rule1617, rule1618, rule1619, rule1620, rule1621, rule1622, rule1623, rule1624, rule1625, rule1626, rule1627, rule1628, rule1629, rule1630, rule1631, rule1632, rule1633, rule1634, rule1635, rule1636, rule1637, rule1638, rule1639, rule1640, rule1641, rule1642, rule1643, rule1644, rule1645, rule1646, rule1647, rule1648, rule1649, rule1650, rule1651, rule1652, rule1653, rule1654, rule1655, rule1656, rule1657, rule1658, rule1659, rule1660, rule1661, rule1662, rule1663, rule1664, rule1665, rule1666, rule1667, rule1668, rule1669, rule1670, rule1671, rule1672, rule1673, rule1674, rule1675, rule1676, rule1677, rule1678, rule1679, rule1680, rule1681, rule1682, rule1683, rule1684, rule1685, rule1686, rule1687, rule1688, rule1689, rule1690, rule1691, rule1692, rule1693, rule1694, rule1695, rule1696, rule1697, rule1698, rule1699, rule1700, rule1701, rule1702, rule1703, rule1704, rule1705, rule1706, rule1707, rule1708, rule1709, rule1710, rule1711, rule1712, rule1713, rule1714, rule1715, rule1716, rule1717, rule1718, rule1719, rule1720, rule1721, rule1722, rule1723, rule1724, rule1725, rule1726, rule1727, rule1728, rule1729, rule1730, rule1731, rule1732, rule1733, rule1734, rule1735, rule1736, rule1737, rule1738, rule1739, rule1740, rule1741, rule1742, rule1743, rule1744, rule1745, rule1746, rule1747, rule1748, rule1749, rule1750, rule1751, rule1752, rule1753, rule1754, rule1755, rule1756, rule1757, rule1758, rule1759, rule1760, rule1761, rule1762, rule1763, rule1764, rule1765, rule1766, rule1767, rule1768, rule1769, rule1770, rule1771, rule1772, rule1773, rule1774, rule1775, rule1776, rule1777, rule1778, rule1779, rule1780, rule1781, rule1782, rule1783, rule1784, rule1785, rule1786, rule1787, rule1788, rule1789, rule1790, rule1791, rule1792, rule1793, rule1794, rule1795, rule1796, rule1797, rule1798, rule1799, rule1800, rule1801, rule1802, rule1803, rule1804, rule1805, rule1806, rule1807, rule1808, rule1809, rule1810, rule1811, rule1812, rule1813, rule1814, rule1815, rule1816, rule1817, rule1818, rule1819, rule1820, rule1821, rule1822, rule1823, rule1824, rule1825, rule1826, rule1827, rule1828, rule1829, rule1830, rule1831, rule1832, rule1833, rule1834, rule1835, rule1836, rule1837, rule1838, rule1839, rule1840, rule1841, rule1842, rule1843, rule1844, rule1845, rule1846, rule1847, rule1848, rule1849, rule1850, rule1851, rule1852, rule1853, rule1854, rule1855, rule1856, rule1857, rule1858, rule1859, rule1860, rule1861, rule1862, rule1863, rule1864, rule1865, rule1866, rule1867, rule1868, rule1869, rule1870, rule1871, rule1872, rule1873, rule1874, rule1875, rule1876, rule1877, rule1878, rule1879, rule1880, rule1881, rule1882, rule1883, rule1884, ] def replacement1476(a, b, c, n, p, q, x): return Dist(x*(c*x**n)**(-S(1)/n), Subst(Int((a + b*x**(n*q))**p, x), x, (c*x**n)**(S(1)/n)), x) def replacement1477(a, b, c, m, n, p, q, x): return Dist(x**(m + S(1))*(c*x**n)**(-(m + S(1))/n), Subst(Int(x**m*(a + b*x**(n*q))**p, x), x, (c*x**n)**(S(1)/n)), x) def replacement1478(a, b, c, d, e, f, m, n, p, q, r, s, x): return Dist((e*(a + b*x**n)**r)**p*(f*(c + d*x**n)**s)**q*(a + b*x**n)**(-p*r)*(c + d*x**n)**(-q*s), Int(x**m*(a + b*x**n)**(p*r)*(c + d*x**n)**(q*s), x), x) def replacement1479(a, b, c, d, e, n, p, u, x): return Dist((b*e/d)**p, Int(u, x), x) def replacement1480(a, b, c, d, e, n, p, u, x): return Int(u*(e*(a + b*x**n))**p*(c + d*x**n)**(-p), x) def With1481(a, b, c, d, e, n, p, x): q = Denominator(p) return Dist(e*q*(-a*d + b*c)/n, Subst(Int(x**(q*(p + S(1)) + S(-1))*(-a*e + c*x**q)**(S(-1) + S(1)/n)*(b*e - d*x**q)**(S(-1) - S(1)/n), x), x, (e*(a + b*x**n)/(c + d*x**n))**(S(1)/q)), x) def With1482(a, b, c, d, e, m, n, p, x): q = Denominator(p) return Dist(e*q*(-a*d + b*c)/n, Subst(Int(x**(q*(p + S(1)) + S(-1))*(-a*e + c*x**q)**(S(-1) + (m + S(1))/n)*(b*e - d*x**q)**(S(-1) - (m + S(1))/n), x), x, (e*(a + b*x**n)/(c + d*x**n))**(S(1)/q)), x) def With1483(a, b, c, d, e, n, p, r, u, x): q = Denominator(p) return Dist(e*q*(-a*d + b*c)/n, Subst(Int(SimplifyIntegrand(x**(q*(p + S(1)) + S(-1))*(-a*e + c*x**q)**(S(-1) + S(1)/n)*(b*e - d*x**q)**(S(-1) - S(1)/n)*ReplaceAll(u, Rule(x, (-a*e + c*x**q)**(S(1)/n)*(b*e - d*x**q)**(-S(1)/n)))**r, x), x), x, (e*(a + b*x**n)/(c + d*x**n))**(S(1)/q)), x) def With1484(a, b, c, d, e, m, n, p, r, u, x): q = Denominator(p) return Dist(e*q*(-a*d + b*c)/n, Subst(Int(SimplifyIntegrand(x**(q*(p + S(1)) + S(-1))*(-a*e + c*x**q)**(S(-1) + (m + S(1))/n)*(b*e - d*x**q)**(S(-1) - (m + S(1))/n)*ReplaceAll(u, Rule(x, (-a*e + c*x**q)**(S(1)/n)*(b*e - d*x**q)**(-S(1)/n)))**r, x), x), x, (e*(a + b*x**n)/(c + d*x**n))**(S(1)/q)), x) def replacement1485(a, b, c, n, p, x): return -Dist(c, Subst(Int((a + b*x**n)**p/x**S(2), x), x, c/x), x) def replacement1486(a, b, c, m, n, p, x): return -Dist(c**(m + S(1)), Subst(Int(x**(-m + S(-2))*(a + b*x**n)**p, x), x, c/x), x) def replacement1487(a, b, c, d, m, n, p, x): return -Dist(c*(c/x)**m*(d*x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**n)**p, x), x, c/x), x) def replacement1488(a, b, c, d, n, n2, p, x): return -Dist(d, Subst(Int((a + b*x**n + c*x**(S(2)*n))**p/x**S(2), x), x, d/x), x) def replacement1489(a, b, c, d, m, n, n2, p, x): return -Dist(d**(m + S(1)), Subst(Int(x**(-m + S(-2))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, d/x), x) def replacement1490(a, b, c, d, e, m, n, n2, p, x): return -Dist(d*(d/x)**m*(e*x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, d/x), x) def replacement1491(a, b, c, d, n, n2, p, x): return -Dist(d, Subst(Int((a + b*x**n + c*d**(-S(2)*n)*x**(S(2)*n))**p/x**S(2), x), x, d/x), x) def replacement1492(a, b, c, d, m, n, n2, p, x): return -Dist(d**(m + S(1)), Subst(Int(x**(-m + S(-2))*(a + b*x**n + c*d**(-S(2)*n)*x**(S(2)*n))**p, x), x, d/x), x) def replacement1493(a, b, c, d, e, m, n, n2, p, x): return -Dist(d*(d/x)**m*(e*x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**n + c*d**(-S(2)*n)*x**(S(2)*n))**p, x), x, d/x), x) def replacement1494(m, u, x): return Int(ExpandToSum(u, x)**m, x) def replacement1495(m, n, u, v, x): return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**n, x) def replacement1496(m, n, p, u, v, w, x): return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**n*ExpandToSum(w, x)**p, x) def replacement1497(m, n, p, q, u, v, w, x, z): return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**n*ExpandToSum(w, x)**p*ExpandToSum(z, x)**q, x) def replacement1498(p, u, x): return Int(ExpandToSum(u, x)**p, x) def replacement1499(m, p, u, v, x): return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**p, x) def replacement1500(m, n, p, u, v, w, x): return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**n*ExpandToSum(w, x)**p, x) def replacement1501(p, q, u, v, x): return Int(ExpandToSum(u, x)**p*ExpandToSum(v, x)**q, x) def replacement1502(p, u, x): return Int(ExpandToSum(u, x)**p, x) def replacement1503(c, m, p, u, x): return Int((c*x)**m*ExpandToSum(u, x)**p, x) def replacement1504(p, q, u, v, x): return Int(ExpandToSum(u, x)**p*ExpandToSum(v, x)**q, x) def replacement1505(m, p, q, u, v, x): return Int(x**m*ExpandToSum(u, x)**p*ExpandToSum(v, x)**q, x) def replacement1506(m, p, q, u, v, w, x): return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**p*ExpandToSum(w, x)**q, x) def replacement1507(m, p, q, r, u, v, x, z): return Int(x**m*ExpandToSum(u, x)**p*ExpandToSum(v, x)**q*ExpandToSum(z, x)**r, x) def replacement1508(p, u, x): return Int(ExpandToSum(u, x)**p, x) def replacement1509(m, p, u, x): return Int(x**m*ExpandToSum(u, x)**p, x) def replacement1510(p, u, x): return Int(ExpandToSum(u, x)**p, x) def replacement1511(d, m, p, u, x): return Int((d*x)**m*ExpandToSum(u, x)**p, x) def replacement1512(p, q, u, v, x): return Int(ExpandToSum(u, x)**q*ExpandToSum(v, x)**p, x) def replacement1513(p, q, u, v, x): return Int(ExpandToSum(u, x)**q*ExpandToSum(v, x)**p, x) def replacement1514(m, p, q, u, x, z): return Int(x**m*ExpandToSum(u, x)**p*ExpandToSum(z, x)**q, x) def replacement1515(m, p, q, u, x, z): return Int(x**m*ExpandToSum(u, x)**p*ExpandToSum(z, x)**q, x) def replacement1516(p, u, x): return Int(ExpandToSum(u, x)**p, x) def replacement1517(m, p, u, x): return Int(x**m*ExpandToSum(u, x)**p, x) def replacement1518(p, u, x, z): return Int(ExpandToSum(u, x)**p*ExpandToSum(z, x), x) def replacement1519(m, p, u, x, z): return Int(x**m*ExpandToSum(u, x)**p*ExpandToSum(z, x), x) def replacement1520(a, c, e, f, g, h, m, n, q, r, x): return -Simp((S(2)*a*g + S(4)*a*h*x**(n/S(4)) - S(2)*c*f*x**(n/S(2)))/(a*c*n*sqrt(a + c*x**n)), x) def replacement1521(a, c, d, e, f, g, h, m, n, q, r, x): return Dist(x**(-m)*(d*x)**m, Int(x**m*(e + f*x**(n/S(4)) + g*x**(S(3)*n/S(4)) + h*x**n)/(a + c*x**n)**(S(3)/2), x), x) def With1522(Pq, a, b, c, m, p, x): n = Denominator(p) return Dist(n/b, Subst(Int(x**(n*p + n + S(-1))*(-a*c/b + c*x**n/b)**m*ReplaceAll(Pq, Rule(x, -a/b + x**n/b)), x), x, (a + b*x)**(S(1)/n)), x) def replacement1523(Pq, a, b, m, n, p, x): return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))))**p*SubstFor(x**(m + S(1)), Pq, x), x), x, x**(m + S(1))), x) def replacement1524(Pq, a, b, n, p, x): return Int((a + b*x**n)**p*ExpandToSum(Pq - x**(n + S(-1))*Coeff(Pq, x, n + S(-1)), x), x) + Simp((a + b*x**n)**(p + S(1))*Coeff(Pq, x, n + S(-1))/(b*n*(p + S(1))), x) def replacement1525(Pq, a, b, c, m, n, p, x): return Int(ExpandIntegrand(Pq*(c*x)**m*(a + b*x**n)**p, x), x) def replacement1526(Pq, a, b, n, p, x): return Int(ExpandIntegrand(Pq*(a + b*x**n)**p, x), x) def replacement1527(Pq, a, b, m, n, p, x): return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x)**p*SubstFor(x**n, Pq, x), x), x, x**n), x) def replacement1528(Pq, a, b, c, m, n, p, x): return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(Pq*x**m*(a + b*x**n)**p, x), x) def replacement1529(Pq, a, b, m, n, p, x): return -Dist(S(1)/(b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*D(Pq, x), x), x) + Simp(Pq*(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x) def replacement1530(Pq, a, b, d, m, n, p, x): return Dist(S(1)/d, Int((d*x)**(m + S(1))*(a + b*x**n)**p*ExpandToSum(Pq/x, x), x), x) def replacement1531(Pq, a, b, n, p, x): return Int(x*(a + b*x**n)**p*ExpandToSum(Pq/x, x), x) def With1532(Pq, a, b, m, n, p, x): u = IntHide(Pq*x**m, x) return -Dist(b*n*p, Int(x**(m + n)*(a + b*x**n)**(p + S(-1))*ExpandToSum(u*x**(-m + S(-1)), x), x), x) + Simp(u*(a + b*x**n)**p, x) def With1533(Pq, a, b, c, m, n, p, x): q = Expon(Pq, x) i = Symbol('i') return Dist(a*n*p, Int((c*x)**m*(a + b*x**n)**(p + S(-1))*Sum_doit(x**i*Coeff(Pq, x, i)/(i + m + n*p + S(1)), List(i, S(0), q)), x), x) + Simp((c*x)**m*(a + b*x**n)**p*Sum_doit(x**(i + S(1))*Coeff(Pq, x, i)/(i + m + n*p + S(1)), List(i, S(0), q)), x) def With1534(Pq, a, b, n, p, x): q = Expon(Pq, x) i = Symbol('i') return Dist(a*n*p, Int((a + b*x**n)**(p + S(-1))*Sum_doit(x**i*Coeff(Pq, x, i)/(i + n*p + S(1)), List(i, S(0), q)), x), x) + Simp((a + b*x**n)**p*Sum_doit(x**(i + S(1))*Coeff(Pq, x, i)/(i + n*p + S(1)), List(i, S(0), q)), x) def With1535(Pq, a, b, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) i = Symbol('i') if Equal(q, n + S(-1)): return True return False def replacement1535(Pq, a, b, n, p, x): q = Expon(Pq, x) i = Symbol('i') return Dist(S(1)/(a*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*Sum_doit(x**i*(i + n*(p + S(1)) + S(1))*Coeff(Pq, x, i), List(i, S(0), q + S(-1))), x), x) + Simp((a + b*x**n)**(p + S(1))*(a*Coeff(Pq, x, q) - b*x*ExpandToSum(Pq - x**q*Coeff(Pq, x, q), x))/(a*b*n*(p + S(1))), x) def replacement1536(Pq, a, b, n, p, x): return Dist(S(1)/(a*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*ExpandToSum(Pq*n*(p + S(1)) + D(Pq*x, x), x), x), x) - Simp(Pq*x*(a + b*x**n)**(p + S(1))/(a*n*(p + S(1))), x) def replacement1537(a, b, d, e, f, g, x): return -Simp((S(2)*a*f + S(4)*a*g*x - S(2)*b*e*x**S(2))/(S(4)*a*b*sqrt(a + b*x**S(4))), x) def replacement1538(a, b, d, f, g, x): return -Simp((f + S(2)*g*x)/(S(2)*b*sqrt(a + b*x**S(4))), x) def replacement1539(a, b, d, e, g, x): return -Simp(x*(S(2)*a*g - b*e*x)/(S(2)*a*b*sqrt(a + b*x**S(4))), x) def replacement1540(a, b, e, f, h, x): return -Simp((f - S(2)*h*x**S(3))/(S(2)*b*sqrt(a + b*x**S(4))), x) def replacement1541(a, b, e, h, x): return Simp(h*x**S(3)/(b*sqrt(a + b*x**S(4))), x) def replacement1542(a, b, d, e, f, g, h, x): return -Simp((a*f - S(2)*a*h*x**S(3) - S(2)*b*d*x)/(S(2)*a*b*sqrt(a + b*x**S(4))), x) def replacement1543(a, b, d, e, g, h, x): return Simp(x*(a*h*x**S(2) + b*d)/(a*b*sqrt(a + b*x**S(4))), x) def With1544(Pq, a, b, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Q = PolynomialQuotient(Pq*b**(Floor((q + S(-1))/n) + S(1)), a + b*x**n, x) R = PolynomialRemainder(Pq*b**(Floor((q + S(-1))/n) + S(1)), a + b*x**n, x) if GreaterEqual(q, n): return True return False def replacement1544(Pq, a, b, n, p, x): q = Expon(Pq, x) Q = PolynomialQuotient(Pq*b**(Floor((q + S(-1))/n) + S(1)), a + b*x**n, x) R = PolynomialRemainder(Pq*b**(Floor((q + S(-1))/n) + S(1)), a + b*x**n, x) return Dist(b**(-Floor((q - 1)/n) - 1)/(a*n*(p + 1)), Int((a + b*x**n)**(p + 1)*ExpandToSum(Q*a*n*(p + 1) + R*n*(p + 1) + D(R*x, x), x), x), x) - Simp(R*b**(-Floor((q - 1)/n) - 1)*x*(a + b*x**n)**(p + 1)/(a*n*(p + 1)), x) def With1545(Pq, a, b, m, n, p, x): q = Expon(Pq, x) Q = PolynomialQuotient(Pq*a*b**(Floor((q + S(-1))/n) + S(1))*x**m, a + b*x**n, x) R = PolynomialRemainder(Pq*a*b**(Floor((q + S(-1))/n) + S(1))*x**m, a + b*x**n, x) return Dist(b**(-Floor((q - 1)/n) - 1)/(a*n*(p + 1)), Int(x**m*(a + b*x**n)**(p + 1)*ExpandToSum(Q*n*x**(-m)*(p + 1) + Sum_doit(x**(i - m)*(i + n*(p + 1) + 1)*Coeff(R, x, i)/a, List(i, 0, n - 1)), x), x), x) - Simp(R*b**(-Floor((q - 1)/n) - 1)*x*(a + b*x**n)**(p + 1)/(a**2*n*(p + 1)), x) def With1546(Pq, a, b, m, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False g = GCD(m + S(1), n) if Unequal(g, S(1)): return True return False def replacement1546(Pq, a, b, m, n, p, x): g = GCD(m + S(1), n) return Dist(S(1)/g, Subst(Int(x**(S(-1) + (m + S(1))/g)*(a + b*x**(n/g))**p*ReplaceAll(Pq, Rule(x, x**(S(1)/g))), x), x, x**g), x) def replacement1547(A, B, a, b, x): return Dist(B**S(3)/b, Int(S(1)/(A**S(2) - A*B*x + B**S(2)*x**S(2)), x), x) def With1548(A, B, a, b, x): r = Numerator(Rt(a/b, S(3))) s = Denominator(Rt(a/b, S(3))) return Dist(r/(S(3)*a*s), Int((r*(S(2)*A*s + B*r) + s*x*(-A*s + B*r))/(r**S(2) - r*s*x + s**S(2)*x**S(2)), x), x) - Dist(r*(-A*s + B*r)/(S(3)*a*s), Int(S(1)/(r + s*x), x), x) def With1549(A, B, a, b, x): r = Numerator(Rt(-a/b, S(3))) s = Denominator(Rt(-a/b, S(3))) return -Dist(r/(S(3)*a*s), Int((r*(-S(2)*A*s + B*r) - s*x*(A*s + B*r))/(r**S(2) + r*s*x + s**S(2)*x**S(2)), x), x) + Dist(r*(A*s + B*r)/(S(3)*a*s), Int(S(1)/(r - s*x), x), x) def replacement1550(A, B, C, a, b, x): return -Dist(C**S(2)/b, Int(S(1)/(B - C*x), x), x) def With1551(A, B, C, a, b, x): q = a**(S(1)/3)/b**(S(1)/3) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1552(B, C, a, b, x): q = a**(S(1)/3)/b**(S(1)/3) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1553(A, C, a, b, x): q = a**(S(1)/3)/b**(S(1)/3) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist(C*q/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1554(A, B, C, a, b, x): q = (-a)**(S(1)/3)/(-b)**(S(1)/3) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1555(B, C, a, b, x): q = (-a)**(S(1)/3)/(-b)**(S(1)/3) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1556(A, C, a, b, x): q = (-a)**(S(1)/3)/(-b)**(S(1)/3) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist(C*q/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1557(A, B, C, a, b, x): q = (-a)**(S(1)/3)/b**(S(1)/3) return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1558(B, C, a, b, x): q = (-a)**(S(1)/3)/b**(S(1)/3) return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1559(A, C, a, b, x): q = (-a)**(S(1)/3)/b**(S(1)/3) return -Dist(C/b, Int(S(1)/(q - x), x), x) - Dist(C*q/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1560(A, B, C, a, b, x): q = a**(S(1)/3)/(-b)**(S(1)/3) return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1561(B, C, a, b, x): q = a**(S(1)/3)/(-b)**(S(1)/3) return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1562(A, C, a, b, x): q = a**(S(1)/3)/(-b)**(S(1)/3) return -Dist(C/b, Int(S(1)/(q - x), x), x) - Dist(C*q/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1563(A, B, C, a, b, x): q = (a/b)**(S(1)/3) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1564(B, C, a, b, x): q = (a/b)**(S(1)/3) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1565(A, C, a, b, x): q = (a/b)**(S(1)/3) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist(C*q/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1566(A, B, C, a, b, x): q = Rt(a/b, S(3)) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1567(B, C, a, b, x): q = Rt(a/b, S(3)) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1568(A, C, a, b, x): q = Rt(a/b, S(3)) return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist(C*q/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x) def With1569(A, B, C, a, b, x): q = (-a/b)**(S(1)/3) return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1570(B, C, a, b, x): q = (-a/b)**(S(1)/3) return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1571(A, C, a, b, x): q = (-a/b)**(S(1)/3) return -Dist(C/b, Int(S(1)/(q - x), x), x) - Dist(C*q/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1572(A, B, C, a, b, x): q = Rt(-a/b, S(3)) return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1573(B, C, a, b, x): q = Rt(-a/b, S(3)) return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def With1574(A, C, a, b, x): q = Rt(-a/b, S(3)) return -Dist(C/b, Int(S(1)/(q - x), x), x) - Dist(C*q/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x) def replacement1575(A, B, C, a, b, x): return Dist(C, Int(x**S(2)/(a + b*x**S(3)), x), x) + Int((A + B*x)/(a + b*x**S(3)), x) def replacement1576(B, C, a, b, x): return Dist(B, Int(x/(a + b*x**S(3)), x), x) + Dist(C, Int(x**S(2)/(a + b*x**S(3)), x), x) def replacement1577(A, C, a, b, x): return Dist(A, Int(S(1)/(a + b*x**S(3)), x), x) + Dist(C, Int(x**S(2)/(a + b*x**S(3)), x), x) def With1578(A, B, C, a, b, x): q = (a/b)**(S(1)/3) return Dist(q**S(2)/a, Int((A + C*q*x)/(q**S(2) - q*x + x**S(2)), x), x) def With1579(B, C, a, b, x): q = (a/b)**(S(1)/3) return Dist(C*q**S(3)/a, Int(x/(q**S(2) - q*x + x**S(2)), x), x) def With1580(A, C, a, b, x): q = (a/b)**(S(1)/3) return Dist(q**S(2)/a, Int((A + C*q*x)/(q**S(2) - q*x + x**S(2)), x), x) def With1581(A, B, C, a, b, x): q = (-a/b)**(S(1)/3) return Dist(q/a, Int((A*q + x*(A + B*q))/(q**S(2) + q*x + x**S(2)), x), x) def With1582(B, C, a, b, x): q = (-a/b)**(S(1)/3) return Dist(B*q**S(2)/a, Int(x/(q**S(2) + q*x + x**S(2)), x), x) def With1583(A, C, a, b, x): q = (-a/b)**(S(1)/3) return Dist(A*q/a, Int((q + x)/(q**S(2) + q*x + x**S(2)), x), x) def With1584(A, B, C, a, b, x): if isinstance(x, (int, Integer, float, Float)): return False q = (a/b)**(S(1)/3) if NonzeroQ(A - B*q + C*q**S(2)): return True return False def replacement1584(A, B, C, a, b, x): q = (a/b)**(S(1)/3) return Dist(q/(S(3)*a), Int((q*(S(2)*A + B*q - C*q**S(2)) - x*(A - B*q - S(2)*C*q**S(2)))/(q**S(2) - q*x + x**S(2)), x), x) + Dist(q*(A - B*q + C*q**S(2))/(S(3)*a), Int(S(1)/(q + x), x), x) def With1585(B, C, a, b, x): if isinstance(x, (int, Integer, float, Float)): return False q = (a/b)**(S(1)/3) if NonzeroQ(B*q - C*q**S(2)): return True return False def replacement1585(B, C, a, b, x): q = (a/b)**(S(1)/3) return Dist(q/(S(3)*a), Int((q*(B*q - C*q**S(2)) + x*(B*q + S(2)*C*q**S(2)))/(q**S(2) - q*x + x**S(2)), x), x) - Dist(q*(B*q - C*q**S(2))/(S(3)*a), Int(S(1)/(q + x), x), x) def With1586(A, C, a, b, x): if isinstance(x, (int, Integer, float, Float)): return False q = (a/b)**(S(1)/3) if NonzeroQ(A + C*q**S(2)): return True return False def replacement1586(A, C, a, b, x): q = (a/b)**(S(1)/3) return Dist(q/(S(3)*a), Int((q*(S(2)*A - C*q**S(2)) - x*(A - S(2)*C*q**S(2)))/(q**S(2) - q*x + x**S(2)), x), x) + Dist(q*(A + C*q**S(2))/(S(3)*a), Int(S(1)/(q + x), x), x) def With1587(A, B, C, a, b, x): if isinstance(x, (int, Integer, float, Float)): return False q = (-a/b)**(S(1)/3) if NonzeroQ(A + B*q + C*q**S(2)): return True return False def replacement1587(A, B, C, a, b, x): q = (-a/b)**(S(1)/3) return Dist(q/(S(3)*a), Int((q*(S(2)*A - B*q - C*q**S(2)) + x*(A + B*q - S(2)*C*q**S(2)))/(q**S(2) + q*x + x**S(2)), x), x) + Dist(q*(A + B*q + C*q**S(2))/(S(3)*a), Int(S(1)/(q - x), x), x) def With1588(B, C, a, b, x): if isinstance(x, (int, Integer, float, Float)): return False q = (-a/b)**(S(1)/3) if NonzeroQ(B*q + C*q**S(2)): return True return False def replacement1588(B, C, a, b, x): q = (-a/b)**(S(1)/3) return Dist(q/(S(3)*a), Int((-q*(B*q + C*q**S(2)) + x*(B*q - S(2)*C*q**S(2)))/(q**S(2) + q*x + x**S(2)), x), x) + Dist(q*(B*q + C*q**S(2))/(S(3)*a), Int(S(1)/(q - x), x), x) def With1589(A, C, a, b, x): if isinstance(x, (int, Integer, float, Float)): return False q = (-a/b)**(S(1)/3) if NonzeroQ(A + C*q**S(2)): return True return False def replacement1589(A, C, a, b, x): q = (-a/b)**(S(1)/3) return Dist(q/(S(3)*a), Int((q*(S(2)*A - C*q**S(2)) + x*(A - S(2)*C*q**S(2)))/(q**S(2) + q*x + x**S(2)), x), x) + Dist(q*(A + C*q**S(2))/(S(3)*a), Int(S(1)/(q - x), x), x) def With1590(Pq, a, b, c, m, n, x): if isinstance(x, (int, Integer, float, Float)): return False v = Sum_doit(c**(-ii)*(c*x)**(ii + m)*(x**(n/S(2))*Coeff(Pq, x, ii + n/S(2)) + Coeff(Pq, x, ii))/(a + b*x**n), List(ii, S(0), n/S(2) + S(-1))) if SumQ(v): return True return False def replacement1590(Pq, a, b, c, m, n, x): v = Sum_doit(c**(-ii)*(c*x)**(ii + m)*(x**(n/S(2))*Coeff(Pq, x, ii + n/S(2)) + Coeff(Pq, x, ii))/(a + b*x**n), List(ii, S(0), n/S(2) + S(-1))) return Int(v, x) def With1591(Pq, a, b, n, x): if isinstance(x, (int, Integer, float, Float)): return False v = Sum_doit(x**ii*(x**(n/S(2))*Coeff(Pq, x, ii + n/S(2)) + Coeff(Pq, x, ii))/(a + b*x**n), List(ii, S(0), n/S(2) + S(-1))) if SumQ(v): return True return False def replacement1591(Pq, a, b, n, x): v = Sum_doit(x**ii*(x**(n/S(2))*Coeff(Pq, x, ii + n/S(2)) + Coeff(Pq, x, ii))/(a + b*x**n), List(ii, S(0), n/S(2) + S(-1))) return Int(v, x) def With1592(a, b, c, d, x): r = Numer(Rt(b/a, S(3))) s = Denom(Rt(b/a, S(3))) return Simp(S(2)*d*s**S(3)*sqrt(a + b*x**S(3))/(a*r**S(2)*(r*x + s*(S(1) + sqrt(S(3))))), x) - Simp(S(3)**(S(1)/4)*d*s*sqrt((r**S(2)*x**S(2) - r*s*x + s**S(2))/(r*x + s*(S(1) + sqrt(S(3))))**S(2))*sqrt(S(2) - sqrt(S(3)))*(r*x + s)*EllipticE(asin((r*x + s*(S(1) - sqrt(S(3))))/(r*x + s*(S(1) + sqrt(S(3))))), S(-7) - S(4)*sqrt(S(3)))/(r**S(2)*sqrt(s*(r*x + s)/(r*x + s*(S(1) + sqrt(S(3))))**S(2))*sqrt(a + b*x**S(3))), x) def With1593(a, b, c, d, x): r = Numer(Rt(b/a, S(3))) s = Denom(Rt(b/a, S(3))) return Dist(d/r, Int((r*x + s*(S(1) - sqrt(S(3))))/sqrt(a + b*x**S(3)), x), x) + Dist((c*r - d*s*(S(1) - sqrt(S(3))))/r, Int(S(1)/sqrt(a + b*x**S(3)), x), x) def With1594(a, b, c, d, x): r = Numer(Rt(b/a, S(3))) s = Denom(Rt(b/a, S(3))) return Simp(S(2)*d*s**S(3)*sqrt(a + b*x**S(3))/(a*r**S(2)*(r*x + s*(S(1) - sqrt(S(3))))), x) + Simp(S(3)**(S(1)/4)*d*s*sqrt((r**S(2)*x**S(2) - r*s*x + s**S(2))/(r*x + s*(S(1) - sqrt(S(3))))**S(2))*sqrt(sqrt(S(3)) + S(2))*(r*x + s)*EllipticE(asin((r*x + s*(S(1) + sqrt(S(3))))/(r*x + s*(S(1) - sqrt(S(3))))), S(-7) + S(4)*sqrt(S(3)))/(r**S(2)*sqrt(-s*(r*x + s)/(r*x + s*(S(1) - sqrt(S(3))))**S(2))*sqrt(a + b*x**S(3))), x) def With1595(a, b, c, d, x): r = Numer(Rt(b/a, S(3))) s = Denom(Rt(b/a, S(3))) return Dist(d/r, Int((r*x + s*(S(1) + sqrt(S(3))))/sqrt(a + b*x**S(3)), x), x) + Dist((c*r - d*s*(S(1) + sqrt(S(3))))/r, Int(S(1)/sqrt(a + b*x**S(3)), x), x) def With1596(a, b, c, d, x): r = Numer(Rt(b/a, S(3))) s = Denom(Rt(b/a, S(3))) return Simp(d*s**S(3)*x*(S(1) + sqrt(S(3)))*sqrt(a + b*x**S(6))/(S(2)*a*r**S(2)*(r*x**S(2)*(S(1) + sqrt(S(3))) + s)), x) - Simp(S(3)**(S(1)/4)*d*s*x*sqrt((r**S(2)*x**S(4) - r*s*x**S(2) + s**S(2))/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)**S(2))*(r*x**S(2) + s)*EllipticE(acos((r*x**S(2)*(S(1) - sqrt(S(3))) + s)/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)), sqrt(S(3))/S(4) + S(1)/2)/(S(2)*r**S(2)*sqrt(r*x**S(2)*(r*x**S(2) + s)/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)**S(2))*sqrt(a + b*x**S(6))), x) def With1597(a, b, c, d, x): q = Rt(b/a, S(3)) return Dist(d/(S(2)*q**S(2)), Int((S(2)*q**S(2)*x**S(4) - sqrt(S(3)) + S(1))/sqrt(a + b*x**S(6)), x), x) + Dist((S(2)*c*q**S(2) - d*(S(1) - sqrt(S(3))))/(S(2)*q**S(2)), Int(S(1)/sqrt(a + b*x**S(6)), x), x) def replacement1598(a, b, c, d, x): return -Simp(c*d*x**S(3)*sqrt(-(c - d*x**S(2))**S(2)/(c*d*x**S(2)))*sqrt(-d**S(2)*(a + b*x**S(8))/(b*c**S(2)*x**S(4)))*EllipticF(asin(sqrt((sqrt(S(2))*c**S(2) + S(2)*c*d*x**S(2) + sqrt(S(2))*d**S(2)*x**S(4))/(c*d*x**S(2)))/S(2)), S(-2) + S(2)*sqrt(S(2)))/(sqrt(sqrt(S(2)) + S(2))*sqrt(a + b*x**S(8))*(c - d*x**S(2))), x) def replacement1599(a, b, c, d, x): return -Dist((-c*Rt(b/a, S(4)) + d)/(S(2)*Rt(b/a, S(4))), Int((-x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x) + Dist((c*Rt(b/a, S(4)) + d)/(S(2)*Rt(b/a, S(4))), Int((x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x) def replacement1600(Pq, a, b, n, x): return Dist(Coeff(Pq, x, S(0)), Int(S(1)/(x*sqrt(a + b*x**n)), x), x) + Int(ExpandToSum((Pq - Coeff(Pq, x, S(0)))/x, x)/sqrt(a + b*x**n), x) def With1601(Pq, a, b, c, m, n, p, x): q = Expon(Pq, x) j = Symbol('j') k = Symbol('k') return Int(Sum_doit(c**(-j)*(c*x)**(j + m)*(a + b*x**n)**p*Sum_doit(x**(k*n/S(2))*Coeff(Pq, x, j + k*n/S(2)), List(k, S(0), S(1) + S(2)*(-j + q)/n)), List(j, S(0), n/S(2) + S(-1))), x) def With1602(Pq, a, b, n, p, x): q = Expon(Pq, x) j = Symbol('j') k = Symbol('k') return Int(Sum_doit(x**j*(a + b*x**n)**p*Sum_doit(x**(k*n/S(2))*Coeff(Pq, x, j + k*n/S(2)), List(k, S(0), S(1) + S(2)*(-j + q)/n)), List(j, S(0), n/S(2) + S(-1))), x) def replacement1603(Pq, a, b, n, p, x): return Dist(Coeff(Pq, x, n + S(-1)), Int(x**(n + S(-1))*(a + b*x**n)**p, x), x) + Int((a + b*x**n)**p*ExpandToSum(Pq - x**(n + S(-1))*Coeff(Pq, x, n + S(-1)), x), x) def replacement1604(Pq, a, b, c, m, n, x): return Int(ExpandIntegrand(Pq*(c*x)**m/(a + b*x**n), x), x) def replacement1605(Pq, a, b, n, x): return Int(ExpandIntegrand(Pq/(a + b*x**n), x), x) def With1606(Pq, a, b, c, m, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False Pq0 = Coeff(Pq, x, S(0)) if NonzeroQ(Pq0): return True return False def replacement1606(Pq, a, b, c, m, n, p, x): Pq0 = Coeff(Pq, x, S(0)) return Dist(S(1)/(S(2)*a*c*(m + S(1))), Int((c*x)**(m + S(1))*(a + b*x**n)**p*ExpandToSum(-S(2)*Pq0*b*x**(n + S(-1))*(m + n*(p + S(1)) + S(1)) + S(2)*a*(Pq - Pq0)*(m + S(1))/x, x), x), x) + Simp(Pq0*(c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*(m + S(1))), x) def With1607(Pq, a, b, c, m, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) if And(NonzeroQ(m + n*p + q + S(1)), GreaterEqual(-n + q, S(0)), Or(IntegerQ(S(2)*p), IntegerQ(p + (q + S(1))/(S(2)*n)))): return True return False def replacement1607(Pq, a, b, c, m, n, p, x): q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) return Dist(1/(b*(m + n*p + q + 1)), Int((c*x)**m*(a + b*x**n)**p*ExpandToSum(-Pqq*a*x**(-n + q)*(m - n + q + 1) + b*(Pq - Pqq*x**q)*(m + n*p + q + 1), x), x), x) + Simp(Pqq*c**(n - q - 1)*(c*x)**(m - n + q + 1)*(a + b*x**n)**(p + 1)/(b*(m + n*p + q + 1)), x) def With1608(Pq, a, b, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) if And(NonzeroQ(n*p + q + S(1)), GreaterEqual(-n + q, S(0)), Or(IntegerQ(S(2)*p), IntegerQ(p + (q + S(1))/(S(2)*n)))): return True return False def replacement1608(Pq, a, b, n, p, x): q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) return Dist(1/(b*(n*p + q + 1)), Int((a + b*x**n)**p*ExpandToSum(-Pqq*a*x**(-n + q)*(-n + q + 1) + b*(Pq - Pqq*x**q)*(n*p + q + 1), x), x), x) + Simp(Pqq*x**(-n + q + 1)*(a + b*x**n)**(p + 1)/(b*(n*p + q + 1)), x) def With1609(Pq, a, b, m, n, p, x): q = Expon(Pq, x) return -Subst(Int(x**(-m - q + S(-2))*(a + b*x**(-n))**p*ExpandToSum(x**q*ReplaceAll(Pq, Rule(x, S(1)/x)), x), x), x, S(1)/x) def With1610(Pq, a, b, c, m, n, p, x): g = Denominator(m) q = Expon(Pq, x) return -Dist(g/c, Subst(Int(x**(-g*(m + q + S(1)) + S(-1))*(a + b*c**(-n)*x**(-g*n))**p*ExpandToSum(x**(g*q)*ReplaceAll(Pq, Rule(x, x**(-g)/c)), x), x), x, (c*x)**(-S(1)/g)), x) def With1611(Pq, a, b, c, m, n, p, x): q = Expon(Pq, x) return -Dist((c*x)**m*(S(1)/x)**m, Subst(Int(x**(-m - q + S(-2))*(a + b*x**(-n))**p*ExpandToSum(x**q*ReplaceAll(Pq, Rule(x, S(1)/x)), x), x), x, S(1)/x), x) def With1612(Pq, a, b, m, n, p, x): g = Denominator(n) return Dist(g, Subst(Int(x**(g*(m + S(1)) + S(-1))*(a + b*x**(g*n))**p*ReplaceAll(Pq, Rule(x, x**g)), x), x, x**(S(1)/g)), x) def With1613(Pq, a, b, n, p, x): g = Denominator(n) return Dist(g, Subst(Int(x**(g + S(-1))*(a + b*x**(g*n))**p*ReplaceAll(Pq, Rule(x, x**g)), x), x, x**(S(1)/g)), x) def replacement1614(Pq, a, b, c, m, n, p, x): return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(Pq*x**m*(a + b*x**n)**p, x), x) def replacement1615(Pq, a, b, m, n, p, x): return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))))**p*ReplaceAll(SubstFor(x**n, Pq, x), Rule(x, x**(n/(m + S(1))))), x), x, x**(m + S(1))), x) def replacement1616(Pq, a, b, c, m, n, p, x): return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(Pq*x**m*(a + b*x**n)**p, x), x) def replacement1617(A, B, a, b, m, n, p, x): return Dist(A, Int((a + b*x**n)**p, x), x) + Dist(B, Int(x**m*(a + b*x**n)**p, x), x) def replacement1618(Pq, a, b, c, m, n, p, x): return Int(ExpandIntegrand(Pq*(c*x)**m*(a + b*x**n)**p, x), x) def replacement1619(Pq, a, b, n, p, x): return Int(ExpandIntegrand(Pq*(a + b*x**n)**p, x), x) def replacement1620(Pq, a, b, m, n, p, u, v, x): return Dist(u**m*v**(-m)/Coeff(v, x, S(1)), Subst(Int(x**m*(a + b*x**n)**p*SubstFor(v, Pq, x), x), x, v), x) def replacement1621(Pq, a, b, n, p, v, x): return Dist(S(1)/Coeff(v, x, S(1)), Subst(Int((a + b*x**n)**p*SubstFor(v, Pq, x), x), x, v), x) def replacement1622(Pq, a1, a2, b1, b2, c, m, n, p, x): return Int(Pq*(c*x)**m*(a1*a2 + b1*b2*x**(S(2)*n))**p, x) def replacement1623(Pq, a1, a2, b1, b2, n, p, x): return Int(Pq*(a1*a2 + b1*b2*x**(S(2)*n))**p, x) def replacement1624(Pq, a1, a2, b1, b2, c, m, n, p, x): return Dist((a1 + b1*x**n)**FracPart(p)*(a2 + b2*x**n)**FracPart(p)*(a1*a2 + b1*b2*x**(S(2)*n))**(-FracPart(p)), Int(Pq*(c*x)**m*(a1*a2 + b1*b2*x**(S(2)*n))**p, x), x) def replacement1625(Pq, a1, a2, b1, b2, n, p, x): return Dist((a1 + b1*x**n)**FracPart(p)*(a2 + b2*x**n)**FracPart(p)*(a1*a2 + b1*b2*x**(S(2)*n))**(-FracPart(p)), Int(Pq*(a1*a2 + b1*b2*x**(S(2)*n))**p, x), x) def replacement1626(a, b, c, d, e, f, g, n, n2, p, x): return Simp(e*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(p + S(1))/(a*c), x) def replacement1627(a, b, c, d, e, g, n, n2, p, x): return Simp(e*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(p + S(1))/(a*c), x) def replacement1628(a, b, c, d, e, f, g, h, m, n, n2, p, x): return Simp(e*(h*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(p + S(1))/(a*c*h*(m + S(1))), x) def replacement1629(a, b, c, d, e, g, h, m, n, n2, p, x): return Simp(e*(h*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(p + S(1))/(a*c*h*(m + S(1))), x) def replacement1630(A, B, a, b, c, d, m, n, p, q, x): return Dist(A, Int((a + b*x**n)**p*(c + d*x**n)**q, x), x) + Dist(B, Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x) def With1631(Px, a, b, c, d, n, p, q, x): k = Denominator(n) return Dist(k/d, Subst(Int(SimplifyIntegrand(x**(k + S(-1))*(a + b*x**(k*n))**p*ReplaceAll(Px, Rule(x, -c/d + x**k/d))**q, x), x), x, (c + d*x)**(S(1)/k)), x) def replacement1632(Pq, a, b, c, m, n, n2, p, x): return Dist(S(1)/n, Subst(Int((a + b*x + c*x**S(2))**p*SubstFor(x**n, Pq, x), x), x, x**n), x) def replacement1633(Pq, a, b, c, d, m, n, n2, p, x): return Int(ExpandIntegrand(Pq*(d*x)**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x) def replacement1634(Pq, a, b, c, n, n2, p, x): return Int(ExpandIntegrand(Pq*(a + b*x**n + c*x**(S(2)*n))**p, x), x) def replacement1635(a, b, c, d, e, f, n, n2, p, x): return Simp(d*x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/a, x) def replacement1636(a, b, c, d, f, n, n2, p, x): return Simp(d*x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/a, x) def replacement1637(a, b, c, d, e, f, g, m, n, n2, p, x): return Simp(d*(g*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(a*g*(m + S(1))), x) def replacement1638(a, b, c, d, f, g, m, n, n2, p, x): return Simp(d*(g*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(a*g*(m + S(1))), x) def replacement1639(Pq, a, b, c, d, m, n, n2, p, x): return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int(Pq*(d*x)**m*(b + S(2)*c*x**n)**(S(2)*p), x), x) def replacement1640(Pq, a, b, c, n, n2, p, x): return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int(Pq*(b + S(2)*c*x**n)**(S(2)*p), x), x) def replacement1641(Pq, a, b, c, m, n, n2, p, x): return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x + c*x**S(2))**p*SubstFor(x**n, Pq, x), x), x, x**n), x) def replacement1642(Pq, a, b, c, d, m, n, n2, p, x): return Dist(x**(-m)*(d*x)**m, Int(Pq*x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x) def replacement1643(Pq, a, b, c, d, m, n, n2, p, x): return Dist(S(1)/d, Int((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**p*ExpandToSum(Pq/x, x), x), x) def replacement1644(Pq, a, b, c, n, n2, p, x): return Int(x*(a + b*x**n + c*x**(S(2)*n))**p*ExpandToSum(Pq/x, x), x) def replacement1645(a, b, c, d, e, f, g, n, n2, n3, p, x): return Simp(x*(S(3)*a*d - x**S(2)*(-a*e + S(2)*b*d*p + S(3)*b*d))*(a + b*x**S(2) + c*x**S(4))**(p + S(1))/(S(3)*a**S(2)), x) def replacement1646(a, b, c, d, f, g, n, n2, n3, p, x): return Simp(x*(S(3)*a*d - x**S(2)*(S(2)*b*d*p + S(3)*b*d))*(a + b*x**S(2) + c*x**S(4))**(p + S(1))/(S(3)*a**S(2)), x) def replacement1647(a, b, c, d, e, g, n, n2, n3, p, x): return Simp(x*(S(3)*a*d - x**S(2)*(-a*e + S(2)*b*d*p + S(3)*b*d))*(a + b*x**S(2) + c*x**S(4))**(p + S(1))/(S(3)*a**S(2)), x) def replacement1648(a, b, c, d, g, n, n2, n3, p, x): return Simp(x*(S(3)*a*d - x**S(2)*(S(2)*b*d*p + S(3)*b*d))*(a + b*x**S(2) + c*x**S(4))**(p + S(1))/(S(3)*a**S(2)), x) def replacement1649(a, b, c, e, f, g, h, m, n, n2, q, r, s, x): return -Simp((S(2)*c*x**n*(-b*g + S(2)*c*f) + S(2)*c*(-S(2)*a*g + b*f) + S(2)*h*x**(n/S(2))*(-S(4)*a*c + b**S(2)))/(c*n*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**n + c*x**(S(2)*n))), x) def replacement1650(a, b, c, d, e, f, g, h, m, n, n2, q, r, s, x): return Dist(x**(-m)*(d*x)**m, Int(x**m*(e + f*x**(n/S(2)) + g*x**(S(3)*n/S(2)) + h*x**(S(2)*n))/(a + b*x**n + c*x**(S(2)*n))**(S(3)/2), x), x) def With1651(Pq, a, b, c, n, n2, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) i = Symbol('i') if Less(q, S(2)*n): return True return False def replacement1651(Pq, a, b, c, n, n2, p, x): q = Expon(Pq, x) i = Symbol('i') return Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Sum_doit(c*x**(i + n)*(-S(2)*a*Coeff(Pq, x, i + n) + b*Coeff(Pq, x, i))*(i + n*(S(2)*p + S(3)) + S(1)) + x**i*(-a*b*(i + S(1))*Coeff(Pq, x, i + n) + (-S(2)*a*c*(i + S(2)*n*(p + S(1)) + S(1)) + b**S(2)*(i + n*(p + S(1)) + S(1)))*Coeff(Pq, x, i)), List(i, S(0), n + S(-1))), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Sum_doit(c*x**(i + n)*(-S(2)*a*Coeff(Pq, x, i + n) + b*Coeff(Pq, x, i)) + x**i*(-a*b*Coeff(Pq, x, i + n) + (-S(2)*a*c + b**S(2))*Coeff(Pq, x, i)), List(i, S(0), n + S(-1)))/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x) def replacement1652(a, b, c, d, e, f, g, x): return -Simp((c*x**S(2)*(-b*f + S(2)*c*e) + c*(-S(2)*a*f + b*e) + g*x*(-S(4)*a*c + b**S(2)))/(c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x) def replacement1653(a, b, c, d, f, g, x): return Simp((S(2)*a*c*f + b*c*f*x**S(2) - g*x*(-S(4)*a*c + b**S(2)))/(c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x) def replacement1654(a, b, c, d, e, g, x): return -Simp((b*c*e + S(2)*c**S(2)*e*x**S(2) + g*x*(-S(4)*a*c + b**S(2)))/(c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x) def replacement1655(a, b, c, e, f, g, h, x): return Simp((S(2)*a**S(2)*c*f + a*b*c*f*x**S(2) + a*h*x**S(3)*(-S(4)*a*c + b**S(2)))/(a*c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x) def replacement1656(a, b, c, e, g, h, x): return Simp(h*x**S(3)/(c*sqrt(a + b*x**S(2) + c*x**S(4))), x) def replacement1657(a, b, c, d, e, f, g, h, x): return Simp((S(2)*a**S(2)*c*f + a*b*c*f*x**S(2) + a*h*x**S(3)*(-S(4)*a*c + b**S(2)) + c*d*x*(-S(4)*a*c + b**S(2)))/(a*c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x) def replacement1658(a, b, c, d, e, f, h, x): return Simp((S(2)*a**S(2)*c*f + a*b*c*f*x**S(2) + a*h*x**S(3)*(-S(4)*a*c + b**S(2)) + c*d*x*(-S(4)*a*c + b**S(2)))/(a*c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x) def With1659(Pq, a, b, c, n, n2, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Q = PolynomialQuotient(Pq*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x) R = PolynomialRemainder(Pq*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x) if GreaterEqual(q, S(2)*n): return True return False def replacement1659(Pq, a, b, c, n, n2, p, x): q = Expon(Pq, x) Q = PolynomialQuotient(Pq*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x) R = PolynomialRemainder(Pq*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x) return Dist((b*c)**(-Floor((q - 1)/n) - 1)/(a*n*(p + 1)*(-4*a*c + b**2)), Int((a + b*x**n + c*x**(2*n))**(p + 1)*ExpandToSum(Q*a*n*(p + 1)*(-4*a*c + b**2) + Sum_doit(c*x**(i + n)*(-2*a*Coeff(R, x, i + n) + b*Coeff(R, x, i))*(i + n*(2*p + 3) + 1) + x**i*(-a*b*(i + 1)*Coeff(R, x, i + n) + (-2*a*c*(i + 2*n*(p + 1) + 1) + b**2*(i + n*(p + 1) + 1))*Coeff(R, x, i)), List(i, 0, n - 1)), x), x), x) - Simp(x*(b*c)**(-Floor((q - 1)/n) - 1)*(a + b*x**n + c*x**(2*n))**(p + 1)*Sum_doit(c*x**(i + n)*(-2*a*Coeff(R, x, i + n) + b*Coeff(R, x, i)) + x**i*(-a*b*Coeff(R, x, i + n) + (-2*a*c + b**2)*Coeff(R, x, i)), List(i, 0, n - 1))/(a*n*(p + 1)*(-4*a*c + b**2)), x) def With1660(Pq, a, b, c, m, n, n2, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Q = PolynomialQuotient(Pq*a*x**m*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x) R = PolynomialRemainder(Pq*a*x**m*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x) if GreaterEqual(q, S(2)*n): return True return False def replacement1660(Pq, a, b, c, m, n, n2, p, x): q = Expon(Pq, x) Q = PolynomialQuotient(Pq*a*x**m*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x) R = PolynomialRemainder(Pq*a*x**m*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x) return Dist((b*c)**(-Floor((q - 1)/n) - 1)/(a*n*(p + 1)*(-4*a*c + b**2)), Int(x**m*(a + b*x**n + c*x**(2*n))**(p + 1)*ExpandToSum(Q*n*x**(-m)*(p + 1)*(-4*a*c + b**2) + Sum_doit(c*x**(i - m + n)*(-2*Coeff(R, x, i + n) + b*Coeff(R, x, i)/a)*(i + n*(2*p + 3) + 1) + x**(i - m)*(-b*(i + 1)*Coeff(R, x, i + n) + (-2*c*(i + 2*n*(p + 1) + 1) + b**2*(i + n*(p + 1) + 1)/a)*Coeff(R, x, i)), List(i, 0, n - 1)), x), x), x) - Simp(x*(b*c)**(-Floor((q - 1)/n) - 1)*(a + b*x**n + c*x**(2*n))**(p + 1)*Sum_doit(c*x**(i + n)*(-2*a*Coeff(R, x, i + n) + b*Coeff(R, x, i)) + x**i*(-a*b*Coeff(R, x, i + n) + (-2*a*c + b**2)*Coeff(R, x, i)), List(i, 0, n - 1))/(a**2*n*(p + 1)*(-4*a*c + b**2)), x) def With1661(Pq, a, b, c, m, n, n2, p, x): if isinstance(x, (int, Integer, float, Float)): return False g = GCD(m + S(1), n) if Unequal(g, S(1)): return True return False def replacement1661(Pq, a, b, c, m, n, n2, p, x): g = GCD(m + S(1), n) return Dist(S(1)/g, Subst(Int(x**(S(-1) + (m + S(1))/g)*(a + b*x**(n/g) + c*x**(S(2)*n/g))**p*ReplaceAll(Pq, Rule(x, x**(S(1)/g))), x), x, x**g), x) def replacement1662(Pq, a, b, c, d, m, n, n2, x): return Int(ExpandIntegrand(Pq*(d*x)**m/(a + b*x**n + c*x**(S(2)*n)), x), x) def replacement1663(Pq, a, b, c, n, n2, x): return Int(ExpandIntegrand(Pq/(a + b*x**n + c*x**(S(2)*n)), x), x) def With1664(Pq, a, b, c, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) if Equal(S(2)*p + q + S(1), S(0)): return True return False def replacement1664(Pq, a, b, c, p, x): q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) return Dist(1/2, Int((a + b*x + c*x**2)**p*ExpandToSum(2*Pq - Pqq*c**p*(b + 2*c*x)*(a + b*x + c*x**2)**(-p - 1), x), x), x) + Simp(Pqq*c**p*log(a + b*x + c*x**2)/2, x) def With1665(Pq, a, b, c, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) if Equal(S(2)*p + q + S(1), S(0)): return True return False def replacement1665(Pq, a, b, c, p, x): q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) return Int((a + b*x + c*x**2)**p*ExpandToSum(Pq - Pqq*c**(p + 1/2)*(a + b*x + c*x**2)**(-p - 1/2), x), x) + Simp(Pqq*c**p*atanh((b + 2*c*x)/(2*sqrt(a + b*x + c*x**2)*Rt(c, 2))), x) def With1666(Pq, a, b, c, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) if Equal(S(2)*p + q + S(1), S(0)): return True return False def replacement1666(Pq, a, b, c, p, x): q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) return Int((a + b*x + c*x**2)**p*ExpandToSum(Pq - Pqq*(-c)**(p + 1/2)*(a + b*x + c*x**2)**(-p - 1/2), x), x) - Simp(Pqq*(-c)**p*ArcTan((b + 2*c*x)/(2*sqrt(a + b*x + c*x**2)*Rt(-c, 2))), x) def With1667(Pq, a, b, c, d, m, n, n2, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) if And(GreaterEqual(q, S(2)*n), Unequal(m + S(2)*n*p + q + S(1), S(0)), Or(IntegerQ(S(2)*p), And(Equal(n, S(1)), IntegerQ(S(4)*p)), IntegerQ(p + (q + S(1))/(S(2)*n)))): return True return False def replacement1667(Pq, a, b, c, d, m, n, n2, p, x): q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) return Int((d*x)**m*(a + b*x**n + c*x**(2*n))**p*ExpandToSum(Pq - Pqq*x**q - Pqq*(a*x**(-2*n + q)*(m - 2*n + q + 1) + b*x**(-n + q)*(m + n*(p - 1) + q + 1))/(c*(m + 2*n*p + q + 1)), x), x) + Simp(Pqq*d**(2*n - q - 1)*(d*x)**(m - 2*n + q + 1)*(a + b*x**n + c*x**(2*n))**(p + 1)/(c*(m + 2*n*p + q + 1)), x) def With1668(Pq, a, b, c, n, n2, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) if And(GreaterEqual(q, S(2)*n), Unequal(S(2)*n*p + q + S(1), S(0)), Or(IntegerQ(S(2)*p), And(Equal(n, S(1)), IntegerQ(S(4)*p)), IntegerQ(p + (q + S(1))/(S(2)*n)))): return True return False def replacement1668(Pq, a, b, c, n, n2, p, x): q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) return Int((a + b*x**n + c*x**(2*n))**p*ExpandToSum(Pq - Pqq*x**q - Pqq*(a*x**(-2*n + q)*(-2*n + q + 1) + b*x**(-n + q)*(n*(p - 1) + q + 1))/(c*(2*n*p + q + 1)), x), x) + Simp(Pqq*x**(-2*n + q + 1)*(a + b*x**n + c*x**(2*n))**(p + 1)/(c*(2*n*p + q + 1)), x) def With1669(Pq, a, b, c, d, m, n, n2, p, x): q = Expon(Pq, x) j = Symbol('j') k = Symbol('k') return Int(Sum_doit(d**(-j)*(d*x)**(j + m)*(a + b*x**n + c*x**(S(2)*n))**p*Sum_doit(x**(k*n)*Coeff(Pq, x, j + k*n), List(k, S(0), S(1) + (-j + q)/n)), List(j, S(0), n + S(-1))), x) def With1670(Pq, a, b, c, n, n2, p, x): q = Expon(Pq, x) j = Symbol('j') k = Symbol('k') return Int(Sum_doit(x**j*(a + b*x**n + c*x**(S(2)*n))**p*Sum_doit(x**(k*n)*Coeff(Pq, x, j + k*n), List(k, S(0), S(1) + (-j + q)/n)), List(j, S(0), n + S(-1))), x) def replacement1671(Pq, a, b, c, d, m, n, n2, x): return Int(RationalFunctionExpand(Pq*(d*x)**m/(a + b*x**n + c*x**(S(2)*n)), x), x) def replacement1672(Pq, a, b, c, n, n2, x): return Int(RationalFunctionExpand(Pq/(a + b*x**n + c*x**(S(2)*n)), x), x) def With1673(Pq, a, b, c, m, n, n2, p, x): q = Expon(Pq, x) return -Subst(Int(x**(-m - q + S(-2))*(a + b*x**(-n) + c*x**(-S(2)*n))**p*ExpandToSum(x**q*ReplaceAll(Pq, Rule(x, S(1)/x)), x), x), x, S(1)/x) def With1674(Pq, a, b, c, d, m, n, n2, p, x): g = Denominator(m) q = Expon(Pq, x) return -Dist(g/d, Subst(Int(x**(-g*(m + q + S(1)) + S(-1))*(a + b*d**(-n)*x**(-g*n) + c*d**(-S(2)*n)*x**(-S(2)*g*n))**p*ExpandToSum(x**(g*q)*ReplaceAll(Pq, Rule(x, x**(-g)/d)), x), x), x, (d*x)**(-S(1)/g)), x) def With1675(Pq, a, b, c, d, m, n, n2, p, x): q = Expon(Pq, x) return -Dist((d*x)**m*(S(1)/x)**m, Subst(Int(x**(-m - q + S(-2))*(a + b*x**(-n) + c*x**(-S(2)*n))**p*ExpandToSum(x**q*ReplaceAll(Pq, Rule(x, S(1)/x)), x), x), x, S(1)/x), x) def With1676(Pq, a, b, c, m, n, n2, p, x): g = Denominator(n) return Dist(g, Subst(Int(x**(g*(m + S(1)) + S(-1))*(a + b*x**(g*n) + c*x**(S(2)*g*n))**p*ReplaceAll(Pq, Rule(x, x**g)), x), x, x**(S(1)/g)), x) def With1677(Pq, a, b, c, n, n2, p, x): g = Denominator(n) return Dist(g, Subst(Int(x**(g + S(-1))*(a + b*x**(g*n) + c*x**(S(2)*g*n))**p*ReplaceAll(Pq, Rule(x, x**g)), x), x, x**(S(1)/g)), x) def replacement1678(Pq, a, b, c, d, m, n, n2, p, x): return Dist(d**(m + S(-1)/2)*sqrt(d*x)/sqrt(x), Int(Pq*x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x) def replacement1679(Pq, a, b, c, d, m, n, n2, p, x): return Dist(d**(m + S(1)/2)*sqrt(x)/sqrt(d*x), Int(Pq*x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x) def replacement1680(Pq, a, b, c, d, m, n, n2, p, x): return Dist(x**(-m)*(d*x)**m, Int(Pq*x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x) def replacement1681(Pq, a, b, c, m, n, n2, p, x): return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))) + c*x**(S(2)*n/(m + S(1))))**p*ReplaceAll(SubstFor(x**n, Pq, x), Rule(x, x**(n/(m + S(1))))), x), x, x**(m + S(1))), x) def replacement1682(Pq, a, b, c, d, m, n, n2, p, x): return Dist(x**(-m)*(d*x)**m, Int(Pq*x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x) def With1683(Pq, a, b, c, d, m, n, n2, x): q = Rt(-S(4)*a*c + b**S(2), S(2)) return Dist(S(2)*c/q, Int(Pq*(d*x)**m/(b + S(2)*c*x**n - q), x), x) - Dist(S(2)*c/q, Int(Pq*(d*x)**m/(b + S(2)*c*x**n + q), x), x) def With1684(Pq, a, b, c, n, n2, x): q = Rt(-S(4)*a*c + b**S(2), S(2)) return Dist(S(2)*c/q, Int(Pq/(b + S(2)*c*x**n - q), x), x) - Dist(S(2)*c/q, Int(Pq/(b + S(2)*c*x**n + q), x), x) def replacement1685(Pq, a, b, c, d, m, n, n2, p, x): return Int(ExpandIntegrand(Pq*(d*x)**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x) def replacement1686(Pq, a, b, c, n, n2, p, x): return Int(ExpandIntegrand(Pq*(a + b*x**n + c*x**(S(2)*n))**p, x), x) def replacement1687(Pq, a, b, c, d, m, n, n2, p, x): return Int(Pq*(d*x)**m*(a + b*x**n + c*x**(S(2)*n))**p, x) def replacement1688(Pq, a, b, c, n, n2, p, x): return Int(Pq*(a + b*x**n + c*x**(S(2)*n))**p, x) def replacement1689(Pq, a, b, c, m, n, n2, p, u, v, x): return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a + b*x**n + c*x**(S(2)*n))**p*SubstFor(v, Pq, x), x), x, v), x) def replacement1690(Pq, a, b, c, n, n2, p, v, x): return Dist(S(1)/Coefficient(v, x, S(1)), Subst(Int((a + b*x**n + c*x**(S(2)*n))**p*SubstFor(v, Pq, x), x), x, v), x) def replacement1691(a, b, j, n, p, x): return Simp(x**(S(1) - n)*(a*x**j + b*x**n)**(p + S(1))/(b*(-j + n)*(p + S(1))), x) def replacement1692(a, b, j, n, p, x): return Dist((-j + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int(x**(-j)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(x**(S(1) - j)*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x) def replacement1693(a, b, j, n, p, x): return -Dist(b*(-j + n*p + n + S(1))/(a*(j*p + S(1))), Int(x**(-j + n)*(a*x**j + b*x**n)**p, x), x) + Simp(x**(S(1) - j)*(a*x**j + b*x**n)**(p + S(1))/(a*(j*p + S(1))), x) def replacement1694(a, b, j, n, p, x): return -Dist(b*p*(-j + n)/(j*p + S(1)), Int(x**n*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp(x*(a*x**j + b*x**n)**p/(j*p + S(1)), x) def replacement1695(a, b, j, n, p, x): return Dist(a*p*(-j + n)/(n*p + S(1)), Int(x**j*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp(x*(a*x**j + b*x**n)**p/(n*p + S(1)), x) def replacement1696(a, b, j, n, p, x): return -Dist((j*p + j - n + S(1))/(b*(-j + n)*(p + S(1))), Int(x**(-n)*(a*x**j + b*x**n)**(p + S(1)), x), x) + Simp(x**(S(1) - n)*(a*x**j + b*x**n)**(p + S(1))/(b*(-j + n)*(p + S(1))), x) def replacement1697(a, b, j, n, p, x): return Dist((-j + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int(x**(-j)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(x**(S(1) - j)*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x) def replacement1698(a, b, j, n, p, x): return Dist(a, Int(x**j*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp(x*(a*x**j + b*x**n)**p/(p*(-j + n)), x) def replacement1699(a, b, n, x): return Dist(S(2)/(S(2) - n), Subst(Int(S(1)/(-a*x**S(2) + S(1)), x), x, x/sqrt(a*x**S(2) + b*x**n)), x) def replacement1700(a, b, j, n, p, x): return Dist((-j + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int(x**(-j)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(x**(S(1) - j)*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x) def replacement1701(a, b, j, n, x): return -Dist(a*(-j + S(2)*n + S(-2))/(b*(n + S(-2))), Int(x**(j - n)/sqrt(a*x**j + b*x**n), x), x) + Simp(-S(2)*x**(S(1) - n)*sqrt(a*x**j + b*x**n)/(b*(n + S(-2))), x) def replacement1702(a, b, j, n, p, x): return Dist(x**(-j*FracPart(p))*(a + b*x**(-j + n))**(-FracPart(p))*(a*x**j + b*x**n)**FracPart(p), Int(x**(j*p)*(a + b*x**(-j + n))**p, x), x) def replacement1703(a, b, j, n, p, u, x): return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a*x**j + b*x**n)**p, x), x, u), x) def replacement1704(a, b, j, m, n, p, x): return Dist(S(1)/n, Subst(Int((a*x**(j/n) + b*x)**p, x), x, x**n), x) def replacement1705(a, b, c, j, m, n, p, x): return -Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x) def replacement1706(a, b, c, j, m, n, p, x): return Dist(c**j*(-j + m + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int((c*x)**(-j + m)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x) def replacement1707(a, b, c, j, m, n, p, x): return -Dist(b*c**(j - n)*(-j + m + n*p + n + S(1))/(a*(j*p + m + S(1))), Int((c*x)**(-j + m + n)*(a*x**j + b*x**n)**p, x), x) + Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(j*p + m + S(1))), x) def replacement1708(a, b, c, j, m, n, p, x): return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a*x**j + b*x**n)**p, x), x) def replacement1709(a, b, j, m, n, p, x): return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a*x**(j/n) + b*x)**p, x), x, x**n), x) def replacement1710(a, b, c, j, m, n, p, x): return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a*x**j + b*x**n)**p, x), x) def replacement1711(a, b, c, j, m, n, p, x): return -Dist(b*c**(-n)*p*(-j + n)/(j*p + m + S(1)), Int((c*x)**(m + n)*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a*x**j + b*x**n)**p/(c*(j*p + m + S(1))), x) def replacement1712(a, b, c, j, m, n, p, x): return Dist(a*c**(-j)*p*(-j + n)/(m + n*p + S(1)), Int((c*x)**(j + m)*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a*x**j + b*x**n)**p/(c*(m + n*p + S(1))), x) def replacement1713(a, b, c, j, m, n, p, x): return -Dist(c**n*(j*p + j + m - n + S(1))/(b*(-j + n)*(p + S(1))), Int((c*x)**(m - n)*(a*x**j + b*x**n)**(p + S(1)), x), x) + Simp(c**(n + S(-1))*(c*x)**(m - n + S(1))*(a*x**j + b*x**n)**(p + S(1))/(b*(-j + n)*(p + S(1))), x) def replacement1714(a, b, c, j, m, n, p, x): return Dist(c**j*(-j + m + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int((c*x)**(-j + m)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x) def replacement1715(a, b, c, j, m, n, p, x): return -Dist(a*c**(-j + n)*(j*p + j + m - n + S(1))/(b*(m + n*p + S(1))), Int((c*x)**(j + m - n)*(a*x**j + b*x**n)**p, x), x) + Simp(c**(n + S(-1))*(c*x)**(m - n + S(1))*(a*x**j + b*x**n)**(p + S(1))/(b*(m + n*p + S(1))), x) def replacement1716(a, b, c, j, m, n, p, x): return -Dist(b*c**(j - n)*(-j + m + n*p + n + S(1))/(a*(j*p + m + S(1))), Int((c*x)**(-j + m + n)*(a*x**j + b*x**n)**p, x), x) + Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(j*p + m + S(1))), x) def replacement1717(a, b, j, m, n, p, x): return Dist(S(1)/(m + S(1)), Subst(Int((a*x**(j/(m + S(1))) + b*x**(n/(m + S(1))))**p, x), x, x**(m + S(1))), x) def replacement1718(a, b, c, j, m, n, p, x): return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a*x**j + b*x**n)**p, x), x) def replacement1719(a, b, c, j, m, n, p, x): return Dist(a*c**(-j), Int((c*x)**(j + m)*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a*x**j + b*x**n)**p/(c*p*(-j + n)), x) def replacement1720(a, b, j, m, n, x): return Dist(-S(2)/(-j + n), Subst(Int(S(1)/(-a*x**S(2) + S(1)), x), x, x**(j/S(2))/sqrt(a*x**j + b*x**n)), x) def replacement1721(a, b, c, j, m, n, p, x): return Dist(c**j*(-j + m + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int((c*x)**(-j + m)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x) def replacement1722(a, b, c, j, m, n, p, x): return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a*x**j + b*x**n)**p, x), x) def replacement1723(a, b, c, j, m, n, p, x): return Dist(c**IntPart(m)*x**(-j*FracPart(p) - FracPart(m))*(c*x)**FracPart(m)*(a + b*x**(-j + n))**(-FracPart(p))*(a*x**j + b*x**n)**FracPart(p), Int(x**(j*p + m)*(a + b*x**(-j + n))**p, x), x) def replacement1724(a, b, j, m, n, p, u, v, x): return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a*x**j + b*x**n)**p, x), x, v), x) def replacement1725(a, b, c, d, j, k, m, n, p, q, x): return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(c + d*x)**q*(a*x**(j/n) + b*x**(k/n))**p, x), x, x**n), x) def replacement1726(a, b, c, d, e, j, k, m, n, p, q, x): return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(c + d*x**n)**q*(a*x**j + b*x**k)**p, x), x) def replacement1727(jn, a, b, c, d, e, j, m, n, p, x): return Simp(c*e**(j + S(-1))*(e*x)**(-j + m + S(1))*(a*x**j + b*x**(j + n))**(p + S(1))/(a*(j*p + m + S(1))), x) def replacement1728(jn, a, b, c, d, e, j, m, n, p, x): return -Dist(e**j*(a*d*(j*p + m + S(1)) - b*c*(m + n + p*(j + n) + S(1)))/(a*b*n*(p + S(1))), Int((e*x)**(-j + m)*(a*x**j + b*x**(j + n))**(p + S(1)), x), x) - Simp(e**(j + S(-1))*(e*x)**(-j + m + S(1))*(-a*d + b*c)*(a*x**j + b*x**(j + n))**(p + S(1))/(a*b*n*(p + S(1))), x) def replacement1729(jn, a, b, c, d, e, j, m, n, p, x): return Dist(e**(-n)*(a*d*(j*p + m + S(1)) - b*c*(m + n + p*(j + n) + S(1)))/(a*(j*p + m + S(1))), Int((e*x)**(m + n)*(a*x**j + b*x**(j + n))**p, x), x) + Simp(c*e**(j + S(-1))*(e*x)**(-j + m + S(1))*(a*x**j + b*x**(j + n))**(p + S(1))/(a*(j*p + m + S(1))), x) def replacement1730(jn, a, b, c, d, e, j, m, n, p, x): return -Dist((a*d*(j*p + m + S(1)) - b*c*(m + n + p*(j + n) + S(1)))/(b*(m + n + p*(j + n) + S(1))), Int((e*x)**m*(a*x**j + b*x**(j + n))**p, x), x) + Simp(d*e**(j + S(-1))*(e*x)**(-j + m + S(1))*(a*x**j + b*x**(j + n))**(p + S(1))/(b*(m + n + p*(j + n) + S(1))), x) def replacement1731(a, b, c, d, j, k, m, n, p, q, x): return Dist(S(1)/(m + S(1)), Subst(Int((c + d*x**(n/(m + S(1))))**q*(a*x**(j/(m + S(1))) + b*x**(k/(m + S(1))))**p, x), x, x**(m + S(1))), x) def replacement1732(a, b, c, d, e, j, k, m, n, p, q, x): return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(c + d*x**n)**q*(a*x**j + b*x**k)**p, x), x) def replacement1733(jn, a, b, c, d, e, j, m, n, p, q, x): return Dist(e**IntPart(m)*x**(-j*FracPart(p) - FracPart(m))*(e*x)**FracPart(m)*(a + b*x**n)**(-FracPart(p))*(a*x**j + b*x**(j + n))**FracPart(p), Int(x**(j*p + m)*(a + b*x**n)**p*(c + d*x**n)**q, x), x) def With1734(Pq, a, b, j, n, p, x): d = Denominator(n) return Dist(d, Subst(Int(x**(d + S(-1))*(a*x**(d*j) + b*x**(d*n))**p*ReplaceAll(SubstFor(x**n, Pq, x), Rule(x, x**(d*n))), x), x, x**(S(1)/d)), x) def replacement1735(Pq, a, b, j, m, n, p, x): return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a*x**(j/n) + b*x)**p*SubstFor(x**n, Pq, x), x), x, x**n), x) def replacement1736(Pq, a, b, c, j, m, n, p, x): return Dist(c**(Quotient(m, sign(m))*sign(m))*x**(-Mod(m, sign(m)))*(c*x)**Mod(m, sign(m)), Int(Pq*x**m*(a*x**j + b*x**n)**p, x), x) def replacement1737(Pq, a, b, c, j, m, n, p, x): return Dist(x**(-m)*(c*x)**m, Int(Pq*x**m*(a*x**j + b*x**n)**p, x), x) def With1738(Pq, a, b, j, m, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False g = GCD(m + S(1), n) if Unequal(g, S(1)): return True return False def replacement1738(Pq, a, b, j, m, n, p, x): g = GCD(m + S(1), n) return Dist(S(1)/g, Subst(Int(x**(S(-1) + (m + S(1))/g)*(a*x**(j/g) + b*x**(n/g))**p*ReplaceAll(Pq, Rule(x, x**(S(1)/g))), x), x, x**g), x) def With1739(Pq, a, b, c, j, m, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) if And(Greater(q, n + S(-1)), Unequal(m + n*p + q + S(1), S(0)), Or(IntegerQ(S(2)*p), IntegerQ(p + (q + S(1))/(S(2)*n)))): return True return False def replacement1739(Pq, a, b, c, j, m, n, p, x): q = Expon(Pq, x) Pqq = Coeff(Pq, x, q) return Int((c*x)**m*(a*x**j + b*x**n)**p*ExpandToSum(Pq - Pqq*a*x**(-n + q)*(m - n + q + 1)/(b*(m + n*p + q + 1)) - Pqq*x**q, x), x) + Simp(Pqq*c**(n - q - 1)*(c*x)**(m - n + q + 1)*(a*x**j + b*x**n)**(p + 1)/(b*(m + n*p + q + 1)), x) def replacement1740(Pq, a, b, j, m, n, p, x): return Dist(S(1)/(m + S(1)), Subst(Int((a*x**(j/(m + S(1))) + b*x**(n/(m + S(1))))**p*ReplaceAll(SubstFor(x**n, Pq, x), Rule(x, x**(n/(m + S(1))))), x), x, x**(m + S(1))), x) def replacement1741(Pq, a, b, c, j, m, n, p, x): return Dist(c**(Quotient(m, sign(m))*sign(m))*x**(-Mod(m, sign(m)))*(c*x)**Mod(m, sign(m)), Int(Pq*x**m*(a*x**j + b*x**n)**p, x), x) def replacement1742(Pq, a, b, c, j, m, n, p, x): return Dist(x**(-m)*(c*x)**m, Int(Pq*x**m*(a*x**j + b*x**n)**p, x), x) def replacement1743(Pq, a, b, c, j, m, n, p, x): return Int(ExpandIntegrand(Pq*(c*x)**m*(a*x**j + b*x**n)**p, x), x) def replacement1744(Pq, a, b, j, n, p, x): return Int(ExpandIntegrand(Pq*(a*x**j + b*x**n)**p, x), x) def replacement1745(a, b, d, p, x): return Dist(S(3)**(-S(3)*p)*a**(-S(2)*p), Int((S(3)*a - b*x)**p*(S(3)*a + S(2)*b*x)**(S(2)*p), x), x) def replacement1746(a, b, d, p, x): return Int(ExpandToSum((a + b*x + d*x**S(3))**p, x), x) def With1747(a, b, d, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = Factor(a + b*x + d*x**S(3)) if ProductQ(NonfreeFactors(u, x)): return True return False def replacement1747(a, b, d, p, x): u = Factor(a + b*x + d*x**S(3)) return Dist(FreeFactors(u, x)**p, Int(DistributeDegree(NonfreeFactors(u, x), p), x), x) def With1748(a, b, d, p, x): r = Rt(-S(27)*a*d**S(2) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*b**S(3)*d), S(3)) return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p, x), x) def replacement1749(a, b, d, p, x): return Dist((S(3)*a - b*x)**(-p)*(S(3)*a + S(2)*b*x)**(-S(2)*p)*(a + b*x + d*x**S(3))**p, Int((S(3)*a - b*x)**p*(S(3)*a + S(2)*b*x)**(S(2)*p), x), x) def With1750(a, b, d, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = NonfreeFactors(Factor(a + b*x + d*x**S(3)), x) if ProductQ(u): return True return False def replacement1750(a, b, d, p, x): u = NonfreeFactors(Factor(a + b*x + d*x**S(3)), x) return Dist((a + b*x + d*x**S(3))**p/DistributeDegree(u, p), Int(DistributeDegree(u, p), x), x) def With1751(a, b, d, p, x): r = Rt(-S(27)*a*d**S(2) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*b**S(3)*d), S(3)) return Dist((-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**(-p)*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**(-p)*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(a + b*x + d*x**S(3))**p, Int((-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p, x), x) def replacement1752(a, b, d, e, f, m, p, x): return Dist(S(3)**(-S(3)*p)*a**(-S(2)*p), Int((S(3)*a - b*x)**p*(S(3)*a + S(2)*b*x)**(S(2)*p)*(e + f*x)**m, x), x) def replacement1753(a, b, d, e, f, m, p, x): return Int(ExpandIntegrand((e + f*x)**m*(a + b*x + d*x**S(3))**p, x), x) def With1754(a, b, d, e, f, m, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = Factor(a + b*x + d*x**S(3)) if ProductQ(NonfreeFactors(u, x)): return True return False def replacement1754(a, b, d, e, f, m, p, x): u = Factor(a + b*x + d*x**S(3)) return Dist(FreeFactors(u, x)**p, Int((e + f*x)**m*DistributeDegree(NonfreeFactors(u, x), p), x), x) def With1755(a, b, d, e, f, m, p, x): r = Rt(-S(27)*a*d**S(2) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*b**S(3)*d), S(3)) return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((e + f*x)**m*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p, x), x) def replacement1756(a, b, d, e, f, m, p, x): return Dist((S(3)*a - b*x)**(-p)*(S(3)*a + S(2)*b*x)**(-S(2)*p)*(a + b*x + d*x**S(3))**p, Int((S(3)*a - b*x)**p*(S(3)*a + S(2)*b*x)**(S(2)*p)*(e + f*x)**m, x), x) def With1757(a, b, d, e, f, m, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = NonfreeFactors(Factor(a + b*x + d*x**S(3)), x) if ProductQ(u): return True return False def replacement1757(a, b, d, e, f, m, p, x): u = NonfreeFactors(Factor(a + b*x + d*x**S(3)), x) return Dist((a + b*x + d*x**S(3))**p/DistributeDegree(u, p), Int((e + f*x)**m*DistributeDegree(u, p), x), x) def With1758(a, b, d, e, f, m, p, x): r = Rt(-S(27)*a*d**S(2) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*b**S(3)*d), S(3)) return Dist((-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**(-p)*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**(-p)*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(a + b*x + d*x**S(3))**p, Int((e + f*x)**m*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p, x), x) def replacement1759(a, c, d, p, x): return -Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((c - S(3)*d*x)**p*(S(2)*c + S(3)*d*x)**(S(2)*p), x), x) def replacement1760(a, c, d, p, x): return Int(ExpandToSum((a + c*x**S(2) + d*x**S(3))**p, x), x) def With1761(a, c, d, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = Factor(a + c*x**S(2) + d*x**S(3)) if ProductQ(NonfreeFactors(u, x)): return True return False def replacement1761(a, c, d, p, x): u = Factor(a + c*x**S(2) + d*x**S(3)) return Dist(FreeFactors(u, x)**p, Int(DistributeDegree(NonfreeFactors(u, x), p), x), x) def With1762(a, c, d, p, x): r = Rt(-S(27)*a*d**S(2) - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*a*c**S(3)), S(3)) return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p, x), x) def replacement1763(a, c, d, p, x): return Dist((c - S(3)*d*x)**(-p)*(S(2)*c + S(3)*d*x)**(-S(2)*p)*(a + c*x**S(2) + d*x**S(3))**p, Int((c - S(3)*d*x)**p*(S(2)*c + S(3)*d*x)**(S(2)*p), x), x) def With1764(a, c, d, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = NonfreeFactors(Factor(a + c*x**S(2) + d*x**S(3)), x) if ProductQ(u): return True return False def replacement1764(a, c, d, p, x): u = NonfreeFactors(Factor(a + c*x**S(2) + d*x**S(3)), x) return Dist((a + c*x**S(2) + d*x**S(3))**p/DistributeDegree(u, p), Int(DistributeDegree(u, p), x), x) def With1765(a, c, d, p, x): r = Rt(-S(27)*a*d**S(2) - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*a*c**S(3)), S(3)) return Dist((a + c*x**S(2) + d*x**S(3))**p*(c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**(-p), Int((c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p, x), x) def replacement1766(a, c, d, e, f, m, p, x): return -Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((c - S(3)*d*x)**p*(S(2)*c + S(3)*d*x)**(S(2)*p)*(e + f*x)**m, x), x) def replacement1767(a, c, d, e, f, m, p, x): return Int(ExpandIntegrand((e + f*x)**m*(a + c*x**S(2) + d*x**S(3))**p, x), x) def With1768(a, c, d, e, f, m, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = Factor(a + c*x**S(2) + d*x**S(3)) if ProductQ(NonfreeFactors(u, x)): return True return False def replacement1768(a, c, d, e, f, m, p, x): u = Factor(a + c*x**S(2) + d*x**S(3)) return Dist(FreeFactors(u, x)**p, Int((e + f*x)**m*DistributeDegree(NonfreeFactors(u, x), p), x), x) def With1769(a, c, d, e, f, m, p, x): r = Rt(-S(27)*a*d**S(2) - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*a*c**S(3)), S(3)) return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((e + f*x)**m*(c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p, x), x) def replacement1770(a, c, d, e, f, m, p, x): return Dist((c - S(3)*d*x)**(-p)*(S(2)*c + S(3)*d*x)**(-S(2)*p)*(a + c*x**S(2) + d*x**S(3))**p, Int((c - S(3)*d*x)**p*(S(2)*c + S(3)*d*x)**(S(2)*p)*(e + f*x)**m, x), x) def With1771(a, c, d, e, f, m, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = NonfreeFactors(Factor(a + c*x**S(2) + d*x**S(3)), x) if ProductQ(u): return True return False def replacement1771(a, c, d, e, f, m, p, x): u = NonfreeFactors(Factor(a + c*x**S(2) + d*x**S(3)), x) return Dist((a + c*x**S(2) + d*x**S(3))**p/DistributeDegree(u, p), Int((e + f*x)**m*DistributeDegree(u, p), x), x) def With1772(a, c, d, e, f, m, p, x): r = Rt(-S(27)*a*d**S(2) - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*a*c**S(3)), S(3)) return Dist((a + c*x**S(2) + d*x**S(3))**p*(c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**(-p), Int((e + f*x)**m*(c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p, x), x) def replacement1773(a, b, c, d, p, x): return Dist(S(3)**(-p)*b**(-p)*c**(-p), Int((b + c*x)**(S(3)*p), x), x) def replacement1774(a, b, c, d, p, x): return Dist(S(3)**(-p)*b**(-p)*c**(-p), Subst(Int((S(3)*a*b*c - b**S(3) + c**S(3)*x**S(3))**p, x), x, c/(S(3)*d) + x), x) def With1775(a, b, c, d, p, x): r = Rt(-S(3)*b*c*d + c**S(3), S(3)) return Dist(S(3)**(-p)*b**(-p)*c**(-p), Int((b + x*(c - r))**p*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**p*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**p, x), x) def replacement1776(a, b, c, d, p, x): return Int(ExpandToSum((a + b*x + c*x**S(2) + d*x**S(3))**p, x), x) def With1777(a, b, c, d, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = Factor(a + b*x + c*x**S(2) + d*x**S(3)) if ProductQ(NonfreeFactors(u, x)): return True return False def replacement1777(a, b, c, d, p, x): u = Factor(a + b*x + c*x**S(2) + d*x**S(3)) return Dist(FreeFactors(u, x)**p, Int(DistributeDegree(NonfreeFactors(u, x), p), x), x) def replacement1778(a, b, c, d, p, x): return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Subst(Int((S(27)*a*d**S(2) - S(9)*b*c*d + S(2)*c**S(3) + S(27)*d**S(3)*x**S(3) - S(9)*d*x*(-S(3)*b*d + c**S(2)))**p, x), x, c/(S(3)*d) + x), x) def replacement1779(a, b, c, d, p, x): return Dist((b + c*x)**(-S(3)*p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((b + c*x)**(S(3)*p), x), x) def With1780(a, b, c, d, p, x): r = Rt(-S(3)*a*b*c + b**S(3), S(3)) return Dist((b + c*x - r)**(-p)*(b + c*x + r*(S(1) - sqrt(S(3))*I)/S(2))**(-p)*(b + c*x + r*(S(1) + sqrt(S(3))*I)/S(2))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((b + c*x - r)**p*(b + c*x + r*(S(1) - sqrt(S(3))*I)/S(2))**p*(b + c*x + r*(S(1) + sqrt(S(3))*I)/S(2))**p, x), x) def With1781(a, b, c, d, p, x): r = Rt(-S(3)*b*c*d + c**S(3), S(3)) return Dist((b + x*(c - r))**(-p)*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**(-p)*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((b + x*(c - r))**p*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**p*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**p, x), x) def With1782(a, b, c, d, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = NonfreeFactors(Factor(a + b*x + c*x**S(2) + d*x**S(3)), x) if ProductQ(u): return True return False def replacement1782(a, b, c, d, p, x): u = NonfreeFactors(Factor(a + b*x + c*x**S(2) + d*x**S(3)), x) return Dist((a + b*x + c*x**S(2) + d*x**S(3))**p/DistributeDegree(u, p), Int(DistributeDegree(u, p), x), x) def With1783(a, b, c, d, p, x): r = Rt(-S(27)*a*d**S(2) + S(9)*b*c*d - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) - S(18)*a*b*c*d + S(4)*a*c**S(3) + S(4)*b**S(3)*d - b**S(2)*c**S(2)), S(3)) return Dist((c + S(3)*d*x - S(2)**(S(1)/3)*(-S(6)*b*d + S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) - sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) - I))/(S(4)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) + sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) + I))/(S(4)*r))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((c + S(3)*d*x - S(2)**(S(1)/3)*(-S(6)*b*d + S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) - sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) - I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) + sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) + I))/(S(4)*r))**p, x), x) def replacement1784(p, u, x): return Int(ExpandToSum(u, x)**p, x) def replacement1785(a, b, c, d, e, f, m, p, x): return Dist(S(3)**(-p)*b**(-p)*c**(-p), Int((b + c*x)**(S(3)*p)*(e + f*x)**m, x), x) def With1786(a, b, c, d, e, f, m, p, x): r = Rt(-S(3)*a*b*c + b**S(3), S(3)) return Dist(S(3)**(-p)*b**(-p)*c**(-p), Int((e + f*x)**m*(b + c*x - r)**p*(b + c*x + r*(S(1) - sqrt(S(3))*I)/S(2))**p*(b + c*x + r*(S(1) + sqrt(S(3))*I)/S(2))**p, x), x) def With1787(a, b, c, d, e, f, m, p, x): r = Rt(-S(3)*b*c*d + c**S(3), S(3)) return Dist(S(3)**(-p)*b**(-p)*c**(-p), Int((b + x*(c - r))**p*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**p*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**p*(e + f*x)**m, x), x) def replacement1788(a, b, c, d, e, f, m, p, x): return Int(ExpandIntegrand((e + f*x)**m*(a + b*x + c*x**S(2) + d*x**S(3))**p, x), x) def With1789(a, b, c, d, e, f, m, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = Factor(a + b*x + c*x**S(2) + d*x**S(3)) if ProductQ(NonfreeFactors(u, x)): return True return False def replacement1789(a, b, c, d, e, f, m, p, x): u = Factor(a + b*x + c*x**S(2) + d*x**S(3)) return Dist(FreeFactors(u, x)**p, Int((e + f*x)**m*DistributeDegree(NonfreeFactors(u, x), p), x), x) def replacement1790(a, b, c, d, e, f, m, p, x): return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Subst(Int((S(27)*a*d**S(2) - S(9)*b*c*d + S(2)*c**S(3) + S(27)*d**S(3)*x**S(3) - S(9)*d*x*(-S(3)*b*d + c**S(2)))**p, x), x, c/(S(3)*d) + x), x) def replacement1791(a, b, c, d, e, f, m, p, x): return Dist((b + c*x)**(-S(3)*p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((b + c*x)**(S(3)*p)*(e + f*x)**m, x), x) def With1792(a, b, c, d, e, f, m, p, x): r = Rt(-S(3)*a*b*c + b**S(3), S(3)) return Dist((b + c*x - r)**(-p)*(b + c*x + r*(S(1) - sqrt(S(3))*I)/S(2))**(-p)*(b + c*x + r*(S(1) + sqrt(S(3))*I)/S(2))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((e + f*x)**m*(b + c*x - r)**p*(b + c*x + r*(S(1) - sqrt(S(3))*I)/S(2))**p*(b + c*x + r*(S(1) + sqrt(S(3))*I)/S(2))**p, x), x) def With1793(a, b, c, d, e, f, m, p, x): r = Rt(-S(3)*b*c*d + c**S(3), S(3)) return Dist((b + x*(c - r))**(-p)*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**(-p)*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((b + x*(c - r))**p*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**p*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**p*(e + f*x)**m, x), x) def With1794(a, b, c, d, e, f, m, p, x): if isinstance(x, (int, Integer, float, Float)): return False u = NonfreeFactors(Factor(a + b*x + c*x**S(2) + d*x**S(3)), x) if ProductQ(u): return True return False def replacement1794(a, b, c, d, e, f, m, p, x): u = NonfreeFactors(Factor(a + b*x + c*x**S(2) + d*x**S(3)), x) return Dist((a + b*x + c*x**S(2) + d*x**S(3))**p/DistributeDegree(u, p), Int((e + f*x)**m*DistributeDegree(u, p), x), x) def With1795(a, b, c, d, e, f, m, p, x): r = Rt(-S(27)*a*d**S(2) + S(9)*b*c*d - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) - S(18)*a*b*c*d + S(4)*a*c**S(3) + S(4)*b**S(3)*d - b**S(2)*c**S(2)), S(3)) return Dist((c + S(3)*d*x - S(2)**(S(1)/3)*(-S(6)*b*d + S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) - sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) - I))/(S(4)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) + sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) + I))/(S(4)*r))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((e + f*x)**m*(c + S(3)*d*x - S(2)**(S(1)/3)*(-S(6)*b*d + S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) - sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) - I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) + sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) + I))/(S(4)*r))**p, x), x) def replacement1796(m, p, u, v, x): return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**p, x) def replacement1797(a, b, c, d, e, f, g, x): return Simp(a*f*ArcTan((a*b*x**S(2) + a*b + x*(S(4)*a**S(2) - S(2)*a*c + b**S(2)))/(S(2)*sqrt(a*x**S(4) + a + b*x**S(3) + b*x + c*x**S(2))*Rt(a**S(2)*(S(2)*a - c), S(2))))/(d*Rt(a**S(2)*(S(2)*a - c), S(2))), x) def replacement1798(a, b, c, d, e, f, g, x): return -Simp(a*f*atanh((a*b*x**S(2) + a*b + x*(S(4)*a**S(2) - S(2)*a*c + b**S(2)))/(S(2)*sqrt(a*x**S(4) + a + b*x**S(3) + b*x + c*x**S(2))*Rt(-a**S(2)*(S(2)*a - c), S(2))))/(d*Rt(-a**S(2)*(S(2)*a - c), S(2))), x) def replacement1799(a, b, c, d, e, p, x): return Subst(Int(SimplifyIntegrand((a - b*d/(S(8)*e) + d**S(4)/(S(256)*e**S(3)) + e*x**S(4) + x**S(2)*(c - S(3)*d**S(2)/(S(8)*e)))**p, x), x), x, d/(S(4)*e) + x) def With1800(p, v, x): if isinstance(x, (int, Integer, float, Float)): return False a = Coefficient(v, x, S(0)) b = Coefficient(v, x, S(1)) c = Coefficient(v, x, S(2)) d = Coefficient(v, x, S(3)) e = Coefficient(v, x, S(4)) if And(ZeroQ(S(8)*b*e**S(2) - S(4)*c*d*e + d**S(3)), NonzeroQ(d)): return True return False def replacement1800(p, v, x): a = Coefficient(v, x, S(0)) b = Coefficient(v, x, S(1)) c = Coefficient(v, x, S(2)) d = Coefficient(v, x, S(3)) e = Coefficient(v, x, S(4)) return Subst(Int(SimplifyIntegrand((a - b*d/(S(8)*e) + d**S(4)/(S(256)*e**S(3)) + e*x**S(4) + x**S(2)*(c - S(3)*d**S(2)/(S(8)*e)))**p, x), x), x, d/(S(4)*e) + x) def replacement1801(a, b, c, d, e, p, u, x): return Subst(Int(SimplifyIntegrand((a - b*d/(S(8)*e) + d**S(4)/(S(256)*e**S(3)) + e*x**S(4) + x**S(2)*(c - S(3)*d**S(2)/(S(8)*e)))**p*ReplaceAll(u, Rule(x, -d/(S(4)*e) + x)), x), x), x, d/(S(4)*e) + x) def With1802(p, u, v, x): if isinstance(x, (int, Integer, float, Float)): return False a = Coefficient(v, x, S(0)) b = Coefficient(v, x, S(1)) c = Coefficient(v, x, S(2)) d = Coefficient(v, x, S(3)) e = Coefficient(v, x, S(4)) if And(ZeroQ(S(8)*b*e**S(2) - S(4)*c*d*e + d**S(3)), NonzeroQ(d)): return True return False def replacement1802(p, u, v, x): a = Coefficient(v, x, S(0)) b = Coefficient(v, x, S(1)) c = Coefficient(v, x, S(2)) d = Coefficient(v, x, S(3)) e = Coefficient(v, x, S(4)) return Subst(Int(SimplifyIntegrand((a - b*d/(S(8)*e) + d**S(4)/(S(256)*e**S(3)) + e*x**S(4) + x**S(2)*(c - S(3)*d**S(2)/(S(8)*e)))**p*ReplaceAll(u, Rule(x, -d/(S(4)*e) + x)), x), x), x, d/(S(4)*e) + x) def replacement1803(a, b, c, d, e, p, x): return Dist(-S(16)*a**S(2), Subst(Int((a*(S(256)*a**S(4)*x**S(4) + S(256)*a**S(3)*e - S(64)*a**S(2)*b*d - S(32)*a**S(2)*x**S(2)*(-S(8)*a*c + S(3)*b**S(2)) + S(16)*a*b**S(2)*c - S(3)*b**S(4))/(-S(4)*a*x + b)**S(4))**p/(-S(4)*a*x + b)**S(2), x), x, S(1)/x + b/(S(4)*a)), x) def With1804(p, v, x): if isinstance(x, (int, Integer, float, Float)): return False a = Coefficient(v, x, S(0)) b = Coefficient(v, x, S(1)) c = Coefficient(v, x, S(2)) d = Coefficient(v, x, S(3)) e = Coefficient(v, x, S(4)) if And(NonzeroQ(a), NonzeroQ(b), ZeroQ(S(8)*a**S(2)*d - S(4)*a*b*c + b**S(3))): return True return False def replacement1804(p, v, x): a = Coefficient(v, x, S(0)) b = Coefficient(v, x, S(1)) c = Coefficient(v, x, S(2)) d = Coefficient(v, x, S(3)) e = Coefficient(v, x, S(4)) return Dist(-S(16)*a**S(2), Subst(Int((a*(S(256)*a**S(4)*x**S(4) + S(256)*a**S(3)*e - S(64)*a**S(2)*b*d - S(32)*a**S(2)*x**S(2)*(-S(8)*a*c + S(3)*b**S(2)) + S(16)*a*b**S(2)*c - S(3)*b**S(4))/(-S(4)*a*x + b)**S(4))**p/(-S(4)*a*x + b)**S(2), x), x, S(1)/x + b/(S(4)*a)), x) def With1805(A, B, C, D, a, b, c, d, e, x): q = sqrt(S(8)*a**S(2) - S(4)*a*c + b**S(2)) return -Dist(S(1)/q, Int((A*b - A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a - S(2)*C*a + D*b - D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b - q)), x), x) + Dist(S(1)/q, Int((A*b + A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a - S(2)*C*a + D*b + D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b + q)), x), x) def With1806(A, B, D, a, b, c, d, e, x): q = sqrt(S(8)*a**S(2) - S(4)*a*c + b**S(2)) return -Dist(S(1)/q, Int((A*b - A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a + D*b - D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b - q)), x), x) + Dist(S(1)/q, Int((A*b + A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a + D*b + D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b + q)), x), x) def With1807(A, B, C, D, a, b, c, d, e, m, x): q = sqrt(S(8)*a**S(2) - S(4)*a*c + b**S(2)) return -Dist(S(1)/q, Int(x**m*(A*b - A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a - S(2)*C*a + D*b - D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b - q)), x), x) + Dist(S(1)/q, Int(x**m*(A*b + A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a - S(2)*C*a + D*b + D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b + q)), x), x) def With1808(A, B, D, a, b, c, d, e, m, x): q = sqrt(S(8)*a**S(2) - S(4)*a*c + b**S(2)) return -Dist(S(1)/q, Int(x**m*(A*b - A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a + D*b - D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b - q)), x), x) + Dist(S(1)/q, Int(x**m*(A*b + A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a + D*b + D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b + q)), x), x) def With1809(A, B, C, a, b, c, d, e, x): q = Rt(C*(C*(-S(4)*c*e + d**S(2)) + S(2)*e*(-S(4)*A*e + B*d)), S(2)) return Simp(-S(2)*C**S(2)*atanh((-B*e + C*d + S(2)*C*e*x)/q)/q, x) + Simp(S(2)*C**S(2)*atanh(C*(S(12)*A*B*e - S(4)*A*C*d - S(3)*B**S(2)*d + S(4)*B*C*c + S(8)*C**S(2)*e*x**S(3) + S(4)*C*x**S(2)*(-B*e + S(2)*C*d) + S(4)*C*x*(S(2)*A*e - B*d + S(2)*C*c))/(q*(-S(4)*A*C + B**S(2))))/q, x) def With1810(A, C, a, b, c, d, e, x): q = Rt(C*(-S(8)*A*e**S(2) + C*(-S(4)*c*e + d**S(2))), S(2)) return Simp(-S(2)*C**S(2)*atanh(C*(d + S(2)*e*x)/q)/q, x) + Simp(S(2)*C**S(2)*atanh(C*(A*d - S(2)*C*d*x**S(2) - S(2)*C*e*x**S(3) - S(2)*x*(A*e + C*c))/(A*q))/q, x) def With1811(A, B, C, a, b, c, d, e, x): q = Rt(-C*(C*(-S(4)*c*e + d**S(2)) + S(2)*e*(-S(4)*A*e + B*d)), S(2)) return Simp(S(2)*C**S(2)*ArcTan((-B*e + C*d + S(2)*C*e*x)/q)/q, x) - Simp(S(2)*C**S(2)*ArcTan(C*(S(12)*A*B*e - S(4)*A*C*d - S(3)*B**S(2)*d + S(4)*B*C*c + S(8)*C**S(2)*e*x**S(3) + S(4)*C*x**S(2)*(-B*e + S(2)*C*d) + S(4)*C*x*(S(2)*A*e - B*d + S(2)*C*c))/(q*(-S(4)*A*C + B**S(2))))/q, x) def With1812(A, C, a, b, c, d, e, x): q = Rt(-C*(-S(8)*A*e**S(2) + C*(-S(4)*c*e + d**S(2))), S(2)) return Simp(S(2)*C**S(2)*ArcTan((C*d + S(2)*C*e*x)/q)/q, x) - Simp(S(2)*C**S(2)*ArcTan(-C*(-A*d + S(2)*C*d*x**S(2) + S(2)*C*e*x**S(3) + S(2)*x*(A*e + C*c))/(A*q))/q, x) def replacement1813(A, B, C, D, a, b, c, d, e, x): return -Dist(S(1)/(S(4)*e), Int((-S(4)*A*e + D*b + x**S(2)*(-S(4)*C*e + S(3)*D*d) + S(2)*x*(-S(2)*B*e + D*c))/(a + b*x + c*x**S(2) + d*x**S(3) + e*x**S(4)), x), x) + Simp(D*log(a + b*x + c*x**S(2) + d*x**S(3) + e*x**S(4))/(S(4)*e), x) def replacement1814(A, B, D, a, b, c, d, e, x): return -Dist(S(1)/(S(4)*e), Int((-S(4)*A*e + D*b + S(3)*D*d*x**S(2) + S(2)*x*(-S(2)*B*e + D*c))/(a + b*x + c*x**S(2) + d*x**S(3) + e*x**S(4)), x), x) + Simp(D*log(a + b*x + c*x**S(2) + d*x**S(3) + e*x**S(4))/(S(4)*e), x) def replacement1815(a, b, c, d, e, f, u, x): return -Dist(a/(f*(-a*d + b*c)), Int(u*sqrt(c + d*x)/x, x), x) + Dist(c/(e*(-a*d + b*c)), Int(u*sqrt(a + b*x)/x, x), x) def replacement1816(a, b, c, d, e, f, u, x): return Dist(b/(f*(-a*d + b*c)), Int(u*sqrt(c + d*x), x), x) - Dist(d/(e*(-a*d + b*c)), Int(u*sqrt(a + b*x), x), x) def replacement1817(a, b, c, d, e, f, u, x): return Dist(e, Int(u*sqrt(a + b*x)/(a*e**S(2) - c*f**S(2) + x*(b*e**S(2) - d*f**S(2))), x), x) - Dist(f, Int(u*sqrt(c + d*x)/(a*e**S(2) - c*f**S(2) + x*(b*e**S(2) - d*f**S(2))), x), x) def replacement1818(a, b, c, d, n, p, u, x): return Dist(S(1)/(a*c), Int(u*sqrt(a + b*x**(S(2)*n)), x), x) - Dist(b/(a*d), Int(u*x**n, x), x) def replacement1819(a, b, c, d, m, n, p, x): return Dist(c, Int(x**m*sqrt(a + b*x**(S(2)*n))/(a*c**S(2) + x**(S(2)*n)*(b*c**S(2) - d**S(2))), x), x) - Dist(d, Int(x**(m + n)/(a*c**S(2) + x**(S(2)*n)*(b*c**S(2) - d**S(2))), x), x) def With1820(a, b, d, e, f, x): r = Numerator(Rt(a/b, S(3))) s = Denominator(Rt(a/b, S(3))) return Dist(r/(S(3)*a), Int(S(1)/((r + s*x)*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(r/(S(3)*a), Int((S(2)*r - s*x)/(sqrt(d + e*x + f*x**S(2))*(r**S(2) - r*s*x + s**S(2)*x**S(2))), x), x) def With1821(a, b, d, f, x): r = Numerator(Rt(a/b, S(3))) s = Denominator(Rt(a/b, S(3))) return Dist(r/(S(3)*a), Int(S(1)/(sqrt(d + f*x**S(2))*(r + s*x)), x), x) + Dist(r/(S(3)*a), Int((S(2)*r - s*x)/(sqrt(d + f*x**S(2))*(r**S(2) - r*s*x + s**S(2)*x**S(2))), x), x) def With1822(a, b, d, e, f, x): r = Numerator(Rt(-a/b, S(3))) s = Denominator(Rt(-a/b, S(3))) return Dist(r/(S(3)*a), Int(S(1)/((r - s*x)*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(r/(S(3)*a), Int((S(2)*r + s*x)/(sqrt(d + e*x + f*x**S(2))*(r**S(2) + r*s*x + s**S(2)*x**S(2))), x), x) def With1823(a, b, d, f, x): r = Numerator(Rt(-a/b, S(3))) s = Denominator(Rt(-a/b, S(3))) return Dist(r/(S(3)*a), Int(S(1)/(sqrt(d + f*x**S(2))*(r - s*x)), x), x) + Dist(r/(S(3)*a), Int((S(2)*r + s*x)/(sqrt(d + f*x**S(2))*(r**S(2) + r*s*x + s**S(2)*x**S(2))), x), x) def replacement1824(a, b, c, d, e, x): return Dist(d, Int(S(1)/((d**S(2) - e**S(2)*x**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x), x) - Dist(e, Int(x/((d**S(2) - e**S(2)*x**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x), x) def replacement1825(a, c, d, e, x): return Dist(d, Int(S(1)/(sqrt(a + c*x**S(4))*(d**S(2) - e**S(2)*x**S(2))), x), x) - Dist(e, Int(x/(sqrt(a + c*x**S(4))*(d**S(2) - e**S(2)*x**S(2))), x), x) def replacement1826(a, b, c, d, e, x): return -Dist(c/(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4)), Int((d**S(2) - e**S(2)*x**S(2))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) - Simp(e**S(3)*sqrt(a + b*x**S(2) + c*x**S(4))/((d + e*x)*(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4))), x) def replacement1827(a, b, c, d, e, x): return -Dist(c/(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4)), Int((d**S(2) - e**S(2)*x**S(2))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist((b*d*e**S(2) + S(2)*c*d**S(3))/(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4)), Int(S(1)/((d + e*x)*sqrt(a + b*x**S(2) + c*x**S(4))), x), x) - Simp(e**S(3)*sqrt(a + b*x**S(2) + c*x**S(4))/((d + e*x)*(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4))), x) def replacement1828(a, c, d, e, x): return -Dist(c/(a*e**S(4) + c*d**S(4)), Int((d**S(2) - e**S(2)*x**S(2))/sqrt(a + c*x**S(4)), x), x) + Dist(S(2)*c*d**S(3)/(a*e**S(4) + c*d**S(4)), Int(S(1)/(sqrt(a + c*x**S(4))*(d + e*x)), x), x) - Simp(e**S(3)*sqrt(a + c*x**S(4))/((d + e*x)*(a*e**S(4) + c*d**S(4))), x) def replacement1829(A, B, a, b, c, d, e, x): return Dist(A, Subst(Int(S(1)/(d - x**S(2)*(-S(2)*a*e + b*d)), x), x, x/sqrt(a + b*x**S(2) + c*x**S(4))), x) def replacement1830(A, B, a, c, d, e, x): return Dist(A, Subst(Int(S(1)/(S(2)*a*e*x**S(2) + d), x), x, x/sqrt(a + c*x**S(4))), x) def replacement1831(A, B, a, b, c, d, e, f, x): return Dist(A, Subst(Int(S(1)/(d - x**S(2)*(-a*e + b*d)), x), x, x/sqrt(a + b*x**S(2) + c*x**S(4))), x) def replacement1832(A, B, a, c, d, e, f, x): return Dist(A, Subst(Int(S(1)/(a*e*x**S(2) + d), x), x, x/sqrt(a + c*x**S(4))), x) def replacement1833(A, B, a, b, c, d, f, x): return Dist(A, Subst(Int(S(1)/(-b*d*x**S(2) + d), x), x, x/sqrt(a + b*x**S(2) + c*x**S(4))), x) def replacement1834(a, b, c, d, e, x): return Dist(a/d, Subst(Int(S(1)/(-S(2)*b*x**S(2) + x**S(4)*(-S(4)*a*c + b**S(2)) + S(1)), x), x, x/sqrt(a + b*x**S(2) + c*x**S(4))), x) def With1835(a, b, c, d, e, x): q = sqrt(-S(4)*a*c + b**S(2)) return Simp(sqrt(S(2))*a*sqrt(-b + q)*atanh(sqrt(S(2))*x*sqrt(-b + q)*(b + S(2)*c*x**S(2) + q)/(S(4)*sqrt(a + b*x**S(2) + c*x**S(4))*Rt(-a*c, S(2))))/(S(4)*d*Rt(-a*c, S(2))), x) - Simp(sqrt(S(2))*a*sqrt(b + q)*ArcTan(sqrt(S(2))*x*sqrt(b + q)*(b + S(2)*c*x**S(2) - q)/(S(4)*sqrt(a + b*x**S(2) + c*x**S(4))*Rt(-a*c, S(2))))/(S(4)*d*Rt(-a*c, S(2))), x) def replacement1836(a, b, c, d, e, f, x): return Dist(a, Int(S(1)/((a**S(2) - b**S(2)*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) - Dist(b, Int(x/((a**S(2) - b**S(2)*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) def replacement1837(a, b, c, d, e, f, g, h, x): return Simp(S(2)*sqrt(d + e*x + f*sqrt(a + b*x + c*x**S(2)))*(S(9)*c**S(2)*f*g*h*x**S(2) + S(3)*c**S(2)*f*h**S(2)*x**S(3) + c*f*x*(a*h**S(2) - b*g*h + S(10)*c*g**S(2)) + f*(S(2)*a*b*h**S(2) - S(3)*a*c*g*h - S(2)*b**S(2)*g*h + S(5)*b*c*g**S(2)) - (-d*h + e*g)*sqrt(a + b*x + c*x**S(2))*(-S(2)*b*h + S(5)*c*g + c*h*x))/(S(15)*c**S(2)*f*(g + h*x)), x) def replacement1838(f, g, h, j, k, m, n, u, v, x): return Int((g + h*x)**m*(f*k*sqrt(ExpandToSum(v, x)) + ExpandToSum(f*j + u, x))**n, x) def replacement1839(a, b, c, d, e, f, g, h, n, p, x): return Dist(S(2), Subst(Int((g + h*x**n)**p*(d**S(2)*e + e*x**S(2) - f**S(2)*(-a*e + b*d) - x*(-b*f**S(2) + S(2)*d*e))/(b*f**S(2) - S(2)*d*e + S(2)*e*x)**S(2), x), x, d + e*x + f*sqrt(a + b*x + c*x**S(2))), x) def replacement1840(a, c, d, e, f, g, h, n, p, x): return Dist(S(1)/(S(2)*e), Subst(Int((g + h*x**n)**p*(a*f**S(2) + d**S(2) - S(2)*d*x + x**S(2))/(d - x)**S(2), x), x, d + e*x + f*sqrt(a + c*x**S(2))), x) def replacement1841(f, g, h, n, p, u, v, x): return Int((g + h*(f*sqrt(ExpandToSum(v, x)) + ExpandToSum(u, x))**n)**p, x) def replacement1842(a, c, e, f, g, h, m, n, x): return Dist(S(2)**(-m + S(-1))*e**(-m + S(-1)), Subst(Int(x**(-m + n + S(-2))*(a*f**S(2) + x**S(2))*(-a*f**S(2)*h + S(2)*e*g*x + h*x**S(2))**m, x), x, e*x + f*sqrt(a + c*x**S(2))), x) def replacement1843(a, c, e, f, g, i, m, n, p, x): return Dist(S(2)**(-S(2)*m - p + S(-1))*e**(-p + S(-1))*f**(-S(2)*m)*(i/c)**m, Subst(Int(x**(-S(2)*m + n - p + S(-2))*(-a*f**S(2) + x**S(2))**p*(a*f**S(2) + x**S(2))**(S(2)*m + S(1)), x), x, e*x + f*sqrt(a + c*x**S(2))), x) def replacement1844(a, b, c, d, e, f, g, h, i, m, n, x): return Dist(S(2)*f**(-S(2)*m)*(i/c)**m, Subst(Int(x**n*(b*f**S(2) - S(2)*d*e + S(2)*e*x)**(-S(2)*m + S(-2))*(d**S(2)*e + e*x**S(2) - f**S(2)*(-a*e + b*d) - x*(-b*f**S(2) + S(2)*d*e))**(S(2)*m + S(1)), x), x, d + e*x + f*sqrt(a + b*x + c*x**S(2))), x) def replacement1845(a, c, d, e, f, g, i, m, n, x): return Dist(S(2)**(-S(2)*m + S(-1))*f**(-S(2)*m)*(i/c)**m/e, Subst(Int(x**n*(-d + x)**(-S(2)*m + S(-2))*(a*f**S(2) + d**S(2) - S(2)*d*x + x**S(2))**(S(2)*m + S(1)), x), x, d + e*x + f*sqrt(a + c*x**S(2))), x) def replacement1846(a, b, c, d, e, f, g, h, i, m, n, x): return Dist((i/c)**(m + S(-1)/2)*sqrt(g + h*x + i*x**S(2))/sqrt(a + b*x + c*x**S(2)), Int((a + b*x + c*x**S(2))**m*(d + e*x + f*sqrt(a + b*x + c*x**S(2)))**n, x), x) def replacement1847(a, c, d, e, f, g, i, m, n, x): return Dist((i/c)**(m + S(-1)/2)*sqrt(g + i*x**S(2))/sqrt(a + c*x**S(2)), Int((a + c*x**S(2))**m*(d + e*x + f*sqrt(a + c*x**S(2)))**n, x), x) def replacement1848(a, b, c, d, e, f, g, h, i, m, n, x): return Dist((i/c)**(m + S(1)/2)*sqrt(a + b*x + c*x**S(2))/sqrt(g + h*x + i*x**S(2)), Int((a + b*x + c*x**S(2))**m*(d + e*x + f*sqrt(a + b*x + c*x**S(2)))**n, x), x) def replacement1849(a, c, d, e, f, g, i, m, n, x): return Dist((i/c)**(m + S(1)/2)*sqrt(a + c*x**S(2))/sqrt(g + i*x**S(2)), Int((a + c*x**S(2))**m*(d + e*x + f*sqrt(a + c*x**S(2)))**n, x), x) def replacement1850(f, j, k, m, n, u, v, w, x): return Int((f*k*sqrt(ExpandToSum(v, x)) + ExpandToSum(f*j + u, x))**n*ExpandToSum(w, x)**m, x) def replacement1851(a, b, c, d, n, p, x): return Dist(S(1)/a, Subst(Int(S(1)/(-c*x**S(2) + S(1)), x), x, x/sqrt(c*x**S(2) + d*(a + b*x**n)**(S(2)/n))), x) def replacement1852(a, b, c, d, x): return Simp(S(2)*a*x/sqrt(a + b*sqrt(c + d*x**S(2))), x) + Simp(S(2)*b**S(2)*d*x**S(3)/(S(3)*(a + b*sqrt(c + d*x**S(2)))**(S(3)/2)), x) def replacement1853(a, b, c, d, x): return Dist(sqrt(S(2))*b/a, Subst(Int(S(1)/sqrt(S(1) + x**S(2)/a), x), x, a*x + b*sqrt(c + d*x**S(2))), x) def replacement1854(a, b, c, d, e, x): return Int(sqrt(a*e*x**S(2) + b*e*x*sqrt(c + d*x**S(2)))/(x*sqrt(c + d*x**S(2))), x) def replacement1855(a, b, c, d, x): return Dist(d, Subst(Int(S(1)/(-S(2)*c*x**S(2) + S(1)), x), x, x/sqrt(c*x**S(2) + d*sqrt(a + b*x**S(4)))), x) def replacement1856(a, b, c, d, e, m, x): return Dist(S(1)/2 - I/S(2), Int((c + d*x)**m/sqrt(sqrt(a) - I*b*x**S(2)), x), x) + Dist(S(1)/2 + I/S(2), Int((c + d*x)**m/sqrt(sqrt(a) + I*b*x**S(2)), x), x) def With1857(a, b, c, d, x): q = Rt(b/a, S(3)) return Dist(d/(-c*q + d*(S(1) + sqrt(S(3)))), Int((q*x + S(1) + sqrt(S(3)))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) - Dist(q/(-c*q + d*(S(1) + sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x) def With1858(a, b, c, d, x): q = Rt(-b/a, S(3)) return Dist(d/(c*q + d*(S(1) + sqrt(S(3)))), Int((-q*x + S(1) + sqrt(S(3)))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist(q/(c*q + d*(S(1) + sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x) def With1859(a, b, c, d, x): q = Rt(-b/a, S(3)) return Dist(d/(c*q + d*(S(1) - sqrt(S(3)))), Int((-q*x - sqrt(S(3)) + S(1))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist(q/(c*q + d*(S(1) - sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x) def With1860(a, b, c, d, x): q = Rt(b/a, S(3)) return Dist(d/(-c*q + d*(S(1) - sqrt(S(3)))), Int((q*x - sqrt(S(3)) + S(1))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) - Dist(q/(-c*q + d*(S(1) - sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x) def With1861(a, b, c, d, e, f, x): if isinstance(x, (int, Integer, float, Float)): return False q = Rt(b/a, S(3)) if ZeroQ(-e*q + f*(S(1) + sqrt(S(3)))): return True return False def replacement1861(a, b, c, d, e, f, x): q = Rt(b/a, S(3)) return Dist(S(4)*S(3)**(S(1)/4)*f*sqrt((q**S(2)*x**S(2) - q*x + S(1))/(q*x + S(1) + sqrt(S(3)))**S(2))*sqrt(S(2) - sqrt(S(3)))*(q*x + S(1))/(q*sqrt((q*x + S(1))/(q*x + S(1) + sqrt(S(3)))**S(2))*sqrt(a + b*x**S(3))), Subst(Int(S(1)/(sqrt(S(1) - x**S(2))*sqrt(x**S(2) - S(4)*sqrt(S(3)) + S(7))*(-c*q + d*(S(1) - sqrt(S(3))) + x*(-c*q + d*(S(1) + sqrt(S(3)))))), x), x, (-q*x + S(-1) + sqrt(S(3)))/(q*x + S(1) + sqrt(S(3)))), x) def With1862(a, b, c, d, e, f, x): if isinstance(x, (int, Integer, float, Float)): return False q = Rt(b/a, S(3)) if NonzeroQ(-e*q + f*(S(1) + sqrt(S(3)))): return True return False def replacement1862(a, b, c, d, e, f, x): q = Rt(b/a, S(3)) return Dist((-c*f + d*e)/(-c*q + d*(S(1) + sqrt(S(3)))), Int((q*x + S(1) + sqrt(S(3)))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist((-e*q + f*(S(1) + sqrt(S(3))))/(-c*q + d*(S(1) + sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x) def With1863(a, b, c, d, e, f, x): if isinstance(x, (int, Integer, float, Float)): return False q = Rt(-b/a, S(3)) if ZeroQ(e*q + f*(S(1) + sqrt(S(3)))): return True return False def replacement1863(a, b, c, d, e, f, x): q = Rt(-b/a, S(3)) return Dist(-S(4)*S(3)**(S(1)/4)*f*sqrt((q**S(2)*x**S(2) + q*x + S(1))/(-q*x + S(1) + sqrt(S(3)))**S(2))*sqrt(S(2) - sqrt(S(3)))*(-q*x + S(1))/(q*sqrt((-q*x + S(1))/(-q*x + S(1) + sqrt(S(3)))**S(2))*sqrt(a + b*x**S(3))), Subst(Int(S(1)/(sqrt(S(1) - x**S(2))*sqrt(x**S(2) - S(4)*sqrt(S(3)) + S(7))*(c*q + d*(S(1) - sqrt(S(3))) + x*(c*q + d*(S(1) + sqrt(S(3)))))), x), x, (q*x + S(-1) + sqrt(S(3)))/(-q*x + S(1) + sqrt(S(3)))), x) def With1864(a, b, c, d, e, f, x): if isinstance(x, (int, Integer, float, Float)): return False q = Rt(-b/a, S(3)) if NonzeroQ(e*q + f*(S(1) + sqrt(S(3)))): return True return False def replacement1864(a, b, c, d, e, f, x): q = Rt(-b/a, S(3)) return Dist((-c*f + d*e)/(c*q + d*(S(1) + sqrt(S(3)))), Int((-q*x + S(1) + sqrt(S(3)))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist((e*q + f*(S(1) + sqrt(S(3))))/(c*q + d*(S(1) + sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x) def With1865(a, b, c, d, e, f, x): if isinstance(x, (int, Integer, float, Float)): return False q = Rt(-b/a, S(3)) if ZeroQ(e*q + f*(S(1) - sqrt(S(3)))): return True return False def replacement1865(a, b, c, d, e, f, x): q = Rt(-b/a, S(3)) return Dist(S(4)*S(3)**(S(1)/4)*f*sqrt((q**S(2)*x**S(2) + q*x + S(1))/(-q*x - sqrt(S(3)) + S(1))**S(2))*sqrt(sqrt(S(3)) + S(2))*(-q*x + S(1))/(q*sqrt(-(-q*x + S(1))/(-q*x - sqrt(S(3)) + S(1))**S(2))*sqrt(a + b*x**S(3))), Subst(Int(S(1)/(sqrt(S(1) - x**S(2))*sqrt(x**S(2) + S(4)*sqrt(S(3)) + S(7))*(c*q + d*(S(1) + sqrt(S(3))) + x*(c*q + d*(S(1) - sqrt(S(3)))))), x), x, (-q*x + S(1) + sqrt(S(3)))/(q*x + S(-1) + sqrt(S(3)))), x) def With1866(a, b, c, d, e, f, x): if isinstance(x, (int, Integer, float, Float)): return False q = Rt(-b/a, S(3)) if NonzeroQ(e*q + f*(S(1) - sqrt(S(3)))): return True return False def replacement1866(a, b, c, d, e, f, x): q = Rt(-b/a, S(3)) return Dist((-c*f + d*e)/(c*q + d*(S(1) - sqrt(S(3)))), Int((-q*x - sqrt(S(3)) + S(1))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist((e*q + f*(S(1) - sqrt(S(3))))/(c*q + d*(S(1) - sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x) def With1867(a, b, c, d, e, f, x): if isinstance(x, (int, Integer, float, Float)): return False q = Rt(b/a, S(3)) if ZeroQ(-e*q + f*(S(1) - sqrt(S(3)))): return True return False def replacement1867(a, b, c, d, e, f, x): q = Rt(b/a, S(3)) return Dist(-S(4)*S(3)**(S(1)/4)*f*sqrt((q**S(2)*x**S(2) - q*x + S(1))/(q*x - sqrt(S(3)) + S(1))**S(2))*sqrt(sqrt(S(3)) + S(2))*(q*x + S(1))/(q*sqrt(-(q*x + S(1))/(q*x - sqrt(S(3)) + S(1))**S(2))*sqrt(a + b*x**S(3))), Subst(Int(S(1)/(sqrt(S(1) - x**S(2))*sqrt(x**S(2) + S(4)*sqrt(S(3)) + S(7))*(-c*q + d*(S(1) + sqrt(S(3))) + x*(-c*q + d*(S(1) - sqrt(S(3)))))), x), x, (q*x + S(1) + sqrt(S(3)))/(-q*x + S(-1) + sqrt(S(3)))), x) def With1868(a, b, c, d, e, f, x): if isinstance(x, (int, Integer, float, Float)): return False q = Rt(b/a, S(3)) if NonzeroQ(-e*q + f*(S(1) - sqrt(S(3)))): return True return False def replacement1868(a, b, c, d, e, f, x): q = Rt(b/a, S(3)) return Dist((-c*f + d*e)/(-c*q + d*(S(1) - sqrt(S(3)))), Int((q*x - sqrt(S(3)) + S(1))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist((-e*q + f*(S(1) - sqrt(S(3))))/(-c*q + d*(S(1) - sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x) def replacement1869(a, b, c, d, e, m, n, x): return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)/(c + d*x + e*sqrt(a + b*x)), x), x, x**n), x) def replacement1870(a, b, c, d, e, n, u, x): return Dist(c, Int(u/(-a*e**S(2) + c**S(2) + c*d*x**n), x), x) - Dist(a*e, Int(u/(sqrt(a + b*x**n)*(-a*e**S(2) + c**S(2) + c*d*x**n)), x), x) def replacement1871(A, B, a, b, c, d, n, n2, x): return Dist(A**S(2)*(n + S(-1)), Subst(Int(S(1)/(A**S(2)*b*x**S(2)*(n + S(-1))**S(2) + a), x), x, x/(A*(n + S(-1)) - B*x**n)), x) def replacement1872(A, B, a, b, c, d, k, m, n, n2, x): return Dist(A**S(2)*(m - n + S(1))/(m + S(1)), Subst(Int(S(1)/(A**S(2)*b*x**S(2)*(m - n + S(1))**S(2) + a), x), x, x**(m + S(1))/(A*(m - n + S(1)) + B*x**n*(m + S(1)))), x) def replacement1873(a, b, c, d, e, f, g, n, n2, n3, p, x): return -Dist(S(1)/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(a*b*(a*g + c*e) - S(2)*a*c*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))) - b**S(2)*c*d*(n*p + n + S(1)) + x**n*(a*b**S(2)*g*(n*(p + S(2)) + S(1)) - S(2)*a*c*(a*g*(n + S(1)) - c*e*(n*(S(2)*p + S(3)) + S(1))) - b*c*(a*f + c*d)*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a*b*(a*g + c*e) - S(2)*a*c*(-a*f + c*d) + b**S(2)*c*d + x**n*(-a*b**S(2)*g - S(2)*a*c*(-a*g + c*e) + b*c*(a*f + c*d)))/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x) def replacement1874(a, b, c, d, e, f, n, n2, p, x): return -Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(a*b*e - S(2)*a*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))) - b**S(2)*d*(n*p + n + S(1)) - x**n*(-S(2)*a*c*e*(n*(S(2)*p + S(3)) + S(1)) + b*(a*f + c*d)*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a*b*e - S(2)*a*(-a*f + c*d) + b**S(2)*d + x**n*(-S(2)*a*c*e + b*(a*f + c*d)))/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x) def replacement1875(a, b, c, d, e, g, n, n2, n3, p, x): return -Dist(S(1)/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(a*b*(a*g + c*e) + S(2)*a*c**S(2)*d*(S(2)*n*(p + S(1)) + S(1)) - b**S(2)*c*d*(n*p + n + S(1)) + x**n*(a*b**S(2)*g*(n*(p + S(2)) + S(1)) - S(2)*a*c*(a*g*(n + S(1)) - c*e*(n*(S(2)*p + S(3)) + S(1))) - b*c**S(2)*d*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a*b*(a*g + c*e) - S(2)*a*c**S(2)*d + b**S(2)*c*d + x**n*(-a*b**S(2)*g - S(2)*a*c*(-a*g + c*e) + b*c**S(2)*d))/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x) def replacement1876(a, b, c, d, f, g, n, n2, n3, p, x): return -Dist(S(1)/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(a**S(2)*b*g - S(2)*a*c*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))) - b**S(2)*c*d*(n*p + n + S(1)) + x**n*(-S(2)*a**S(2)*c*g*(n + S(1)) + a*b**S(2)*g*(n*(p + S(2)) + S(1)) - b*c*(a*f + c*d)*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a**S(2)*b*g - S(2)*a*c*(-a*f + c*d) + b**S(2)*c*d + x**n*(S(2)*a**S(2)*c*g - a*b**S(2)*g + b*c*(a*f + c*d)))/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x) def replacement1877(a, b, c, d, f, n, n2, p, x): return Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(S(2)*a*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))) + b**S(2)*d*(n*p + n + S(1)) + b*x**n*(a*f + c*d)*(n*(S(2)*p + S(3)) + S(1)), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*(-a*f + c*d) + b**S(2)*d + b*x**n*(a*f + c*d))/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x) def replacement1878(a, b, c, d, g, n, n2, n3, p, x): return -Dist(S(1)/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(a**S(2)*b*g + S(2)*a*c**S(2)*d*(S(2)*n*(p + S(1)) + S(1)) - b**S(2)*c*d*(n*p + n + S(1)) + x**n*(-S(2)*a**S(2)*c*g*(n + S(1)) + a*b**S(2)*g*(n*(p + S(2)) + S(1)) - b*c**S(2)*d*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a**S(2)*b*g - S(2)*a*c**S(2)*d + b**S(2)*c*d + x**n*(S(2)*a**S(2)*c*g - a*b**S(2)*g + b*c**S(2)*d))/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x) def replacement1879(a, c, d, e, f, g, n, n2, n3, p, x): return -Dist(-S(1)/(S(4)*a**S(2)*c**S(2)*n*(p + S(1))), Int((a + c*x**(S(2)*n))**(p + S(1))*Simp(-S(2)*a*c*x**n*(a*g*(n + S(1)) - c*e*(n*(S(2)*p + S(3)) + S(1))) - S(2)*a*c*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))), x), x), x) - Simp(-x*(a + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*c*x**n*(-a*g + c*e) - S(2)*a*c*(-a*f + c*d))/(S(4)*a**S(2)*c**S(2)*n*(p + S(1))), x) def replacement1880(a, c, d, e, f, n, n2, p, x): return -Dist(-S(1)/(S(4)*a**S(2)*c*n*(p + S(1))), Int((a + c*x**(S(2)*n))**(p + S(1))*Simp(S(2)*a*c*e*x**n*(n*(S(2)*p + S(3)) + S(1)) - S(2)*a*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))), x), x), x) - Simp(-x*(a + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*c*e*x**n - S(2)*a*(-a*f + c*d))/(S(4)*a**S(2)*c*n*(p + S(1))), x) def replacement1881(a, c, d, e, g, n, n2, n3, p, x): return -Dist(-S(1)/(S(4)*a**S(2)*c**S(2)*n*(p + S(1))), Int((a + c*x**(S(2)*n))**(p + S(1))*Simp(S(2)*a*c**S(2)*d*(S(2)*n*(p + S(1)) + S(1)) - S(2)*a*c*x**n*(a*g*(n + S(1)) - c*e*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(-x*(a + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*c**S(2)*d - S(2)*a*c*x**n*(-a*g + c*e))/(S(4)*a**S(2)*c**S(2)*n*(p + S(1))), x) def With1882(a, b, c, d, e, f, g, x): q = Rt((S(12)*a**S(2)*g**S(2) - a*c*f**S(2) + f*(-S(2)*a*b*g + S(3)*c**S(2)*d))/(c*g*(-a*f + S(3)*c*d)), S(2)) r = Rt((a*c*f**S(2) - f*(S(2)*a*b*g + S(3)*c**S(2)*d) + S(4)*g*(a**S(2)*g + b*c*d))/(c*g*(-a*f + S(3)*c*d)), S(2)) return -Simp(c*ArcTan((r - S(2)*x)/q)/(g*q), x) + Simp(c*ArcTan((r + S(2)*x)/q)/(g*q), x) - Simp(c*ArcTan(x*(-a*f + S(3)*c*d)*(S(6)*a**S(2)*b*g**S(2) - S(2)*a**S(2)*c*f*g - a*b**S(2)*f*g + b*c**S(2)*d*f + c**S(2)*g*x**S(4)*(-a*f + S(3)*c*d) + c*x**S(2)*(S(2)*a**S(2)*g**S(2) - a*c*f**S(2) - b*c*d*g + S(3)*c**S(2)*d*f))/(g*q*(-S(2)*a**S(2)*g + b*c*d)*(S(4)*a**S(2)*g - a*b*f + b*c*d)))/(g*q), x) def With1883(a, c, d, e, f, g, x): q = Rt((S(12)*a**S(2)*g**S(2) - a*c*f**S(2) + S(3)*c**S(2)*d*f)/(c*g*(-a*f + S(3)*c*d)), S(2)) r = Rt((S(4)*a**S(2)*g**S(2) + a*c*f**S(2) - S(3)*c**S(2)*d*f)/(c*g*(-a*f + S(3)*c*d)), S(2)) return -Simp(c*ArcTan((r - S(2)*x)/q)/(g*q), x) + Simp(c*ArcTan((r + S(2)*x)/q)/(g*q), x) - Simp(c*ArcTan(c*x*(-a*f + S(3)*c*d)*(S(2)*a**S(2)*f*g - c*g*x**S(4)*(-a*f + S(3)*c*d) - x**S(2)*(S(2)*a**S(2)*g**S(2) - a*c*f**S(2) + S(3)*c**S(2)*d*f))/(S(8)*a**S(4)*g**S(3)*q))/(g*q), x) def With1884(p, u, v, x): if isinstance(x, (int, Integer, float, Float)): return False try: m = Exponent(u, x) n = Exponent(v, x) c = Coefficient(u, x, m)/((m + n*p + S(1))*Coefficient(v, x, n)) c = Coefficient(u, x, m)/((m + n*p + S(1))*Coefficient(v, x, n)) w = Apart(-c*x**(m - n)*(v*(m - n + S(1)) + x*(p + S(1))*D(v, x)) + u, x) res = And(Inequality(S(1), Less, n, LessEqual, m + S(1)), Less(m + n*p, S(-1)), FalseQ(DerivativeDivides(v, u, x))) except (TypeError, AttributeError): return False if res: return True return False def replacement1884(p, u, v, x): m = Exponent(u, x) n = Exponent(v, x) c = Coefficient(u, x, m)/((m + n*p + S(1))*Coefficient(v, x, n)) c = Coefficient(u, x, m)/((m + n*p + S(1))*Coefficient(v, x, n)) w = Apart(-c*x**(m - n)*(v*(m - n + S(1)) + x*(p + S(1))*D(v, x)) + u, x) return Simp(If(ZeroQ(w), c*v**(p + 1)*x**(m - n + 1), c*v**(p + 1)*x**(m - n + 1) + Int(v**p*w, x)), x)