3.188 \(\int \left (1-x^2\right ) \left (1+b x^4\right )^p \, dx\)

Optimal. Leaf size=42 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )-\frac{1}{3} x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right ) \]

[Out]

x*Hypergeometric2F1[1/4, -p, 5/4, -(b*x^4)] - (x^3*Hypergeometric2F1[3/4, -p, 7/
4, -(b*x^4)])/3

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Rubi [A]  time = 0.047833, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )-\frac{1}{3} x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^2)*(1 + b*x^4)^p,x]

[Out]

x*Hypergeometric2F1[1/4, -p, 5/4, -(b*x^4)] - (x^3*Hypergeometric2F1[3/4, -p, 7/
4, -(b*x^4)])/3

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Rubi in Sympy [A]  time = 6.66806, size = 32, normalized size = 0.76 \[ - \frac{x^{3}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{- b x^{4}} \right )}}{3} + x{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- b x^{4}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+1)*(b*x**4+1)**p,x)

[Out]

-x**3*hyper((-p, 3/4), (7/4,), -b*x**4)/3 + x*hyper((-p, 1/4), (5/4,), -b*x**4)

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Mathematica [A]  time = 0.0163358, size = 42, normalized size = 1. \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )-\frac{1}{3} x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^2)*(1 + b*x^4)^p,x]

[Out]

x*Hypergeometric2F1[1/4, -p, 5/4, -(b*x^4)] - (x^3*Hypergeometric2F1[3/4, -p, 7/
4, -(b*x^4)])/3

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Maple [A]  time = 0.042, size = 37, normalized size = 0.9 \[ x{\mbox{$_2$F$_1$}({\frac{1}{4}},-p;\,{\frac{5}{4}};\,-b{x}^{4})}-{\frac{{x}^{3}}{3}{\mbox{$_2$F$_1$}({\frac{3}{4}},-p;\,{\frac{7}{4}};\,-b{x}^{4})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)*(b*x^4+1)^p,x)

[Out]

x*hypergeom([1/4,-p],[5/4],-b*x^4)-1/3*x^3*hypergeom([3/4,-p],[7/4],-b*x^4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int{\left (x^{2} - 1\right )}{\left (b x^{4} + 1\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)*(b*x^4 + 1)^p,x, algorithm="maxima")

[Out]

-integrate((x^2 - 1)*(b*x^4 + 1)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (x^{2} - 1\right )}{\left (b x^{4} + 1\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)*(b*x^4 + 1)^p,x, algorithm="fricas")

[Out]

integral(-(x^2 - 1)*(b*x^4 + 1)^p, x)

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Sympy [A]  time = 155.555, size = 61, normalized size = 1.45 \[ - \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, - p \\ \frac{7}{4} \end{matrix}\middle |{b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, - p \\ \frac{5}{4} \end{matrix}\middle |{b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)*(b*x**4+1)**p,x)

[Out]

-x**3*gamma(3/4)*hyper((3/4, -p), (7/4,), b*x**4*exp_polar(I*pi))/(4*gamma(7/4))
+ x*gamma(1/4)*hyper((1/4, -p), (5/4,), b*x**4*exp_polar(I*pi))/(4*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -{\left (x^{2} - 1\right )}{\left (b x^{4} + 1\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)*(b*x^4 + 1)^p,x, algorithm="giac")

[Out]

integrate(-(x^2 - 1)*(b*x^4 + 1)^p, x)