3.1531 \(\int \frac{x^4}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=22 \[ \frac{1}{5} x^5 \, _2F_1\left (\frac{1}{2},\frac{5}{8};\frac{13}{8};-x^8\right ) \]

[Out]

(x^5*Hypergeometric2F1[1/2, 5/8, 13/8, -x^8])/5

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Rubi [A]  time = 0.0189804, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{1}{5} x^5 \, _2F_1\left (\frac{1}{2},\frac{5}{8};\frac{13}{8};-x^8\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[1 + x^8],x]

[Out]

(x^5*Hypergeometric2F1[1/2, 5/8, 13/8, -x^8])/5

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Rubi in Sympy [A]  time = 2.75843, size = 15, normalized size = 0.68 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{8} \\ \frac{13}{8} \end{matrix}\middle |{- x^{8}} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(x**8+1)**(1/2),x)

[Out]

x**5*hyper((1/2, 5/8), (13/8,), -x**8)/5

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Mathematica [A]  time = 0.0219217, size = 22, normalized size = 1. \[ \frac{1}{5} x^5 \, _2F_1\left (\frac{1}{2},\frac{5}{8};\frac{13}{8};-x^8\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[1 + x^8],x]

[Out]

(x^5*Hypergeometric2F1[1/2, 5/8, 13/8, -x^8])/5

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Maple [A]  time = 0.036, size = 17, normalized size = 0.8 \[{\frac{{x}^{5}}{5}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{5}{8}};\,{\frac{13}{8}};\,-{x}^{8})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^8+1)^(1/2),x)

[Out]

1/5*x^5*hypergeom([1/2,5/8],[13/8],-x^8)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{x^{8} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(x^8 + 1),x, algorithm="maxima")

[Out]

integrate(x^4/sqrt(x^8 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{\sqrt{x^{8} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(x^8 + 1),x, algorithm="fricas")

[Out]

integral(x^4/sqrt(x^8 + 1), x)

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Sympy [A]  time = 1.92328, size = 29, normalized size = 1.32 \[ \frac{x^{5} \Gamma \left (\frac{5}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{8} \\ \frac{13}{8} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{13}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(x**8+1)**(1/2),x)

[Out]

x**5*gamma(5/8)*hyper((1/2, 5/8), (13/8,), x**8*exp_polar(I*pi))/(8*gamma(13/8))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{x^{8} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(x^8 + 1),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(x^8 + 1), x)