3.189 \(\int \left (1+b x^4\right )^p \, dx\)

Optimal. Leaf size=18 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right ) \]

[Out]

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

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Rubi [A]  time = 0.010481, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + b*x^4)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 1.2766, size = 14, normalized size = 0.78 \[ x{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- b x^{4}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+1)**p,x)

[Out]

x*hyper((-p, 1/4), (5/4,), -b*x**4)

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Mathematica [A]  time = 0.00727353, size = 18, normalized size = 1. \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + b*x^4)^p,x]

[Out]

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

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Maple [A]  time = 0.03, size = 17, normalized size = 0.9 \[ x{\mbox{$_2$F$_1$}({\frac{1}{4}},-p;\,{\frac{5}{4}};\,-b{x}^{4})} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+1)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + 1)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + 1)^p, x)

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Sympy [A]  time = 32.4571, size = 29, normalized size = 1.61 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, - p \\ \frac{5}{4} \end{matrix}\middle |{b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+1)**p,x)

[Out]

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. \[ \int{\left (b x^{4} + 1\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + 1)^p, x)