3.903 \(\int \frac{x^8}{\left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=43 \[ \frac{5}{6} \sqrt{1-x^4} x+\frac{x^5}{2 \sqrt{1-x^4}}-\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

x^5/(2*Sqrt[1 - x^4]) + (5*x*Sqrt[1 - x^4])/6 - (5*EllipticF[ArcSin[x], -1])/6

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Rubi [A]  time = 0.0360646, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5}{6} \sqrt{1-x^4} x+\frac{x^5}{2 \sqrt{1-x^4}}-\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^8/(1 - x^4)^(3/2),x]

[Out]

x^5/(2*Sqrt[1 - x^4]) + (5*x*Sqrt[1 - x^4])/6 - (5*EllipticF[ArcSin[x], -1])/6

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Rubi in Sympy [A]  time = 4.63866, size = 36, normalized size = 0.84 \[ \frac{x^{5}}{2 \sqrt{- x^{4} + 1}} + \frac{5 x \sqrt{- x^{4} + 1}}{6} - \frac{5 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(-x**4+1)**(3/2),x)

[Out]

x**5/(2*sqrt(-x**4 + 1)) + 5*x*sqrt(-x**4 + 1)/6 - 5*elliptic_f(asin(x), -1)/6

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Mathematica [A]  time = 0.0447375, size = 41, normalized size = 0.95 \[ -\frac{2 x^5+5 \sqrt{1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )-5 x}{6 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(1 - x^4)^(3/2),x]

[Out]

-(-5*x + 2*x^5 + 5*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(6*Sqrt[1 - x^4])

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Maple [A]  time = 0.015, size = 57, normalized size = 1.3 \[{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{x}{3}\sqrt{-{x}^{4}+1}}-{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{6}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(-x^4+1)^(3/2),x)

[Out]

1/2*x/(-x^4+1)^(1/2)+1/3*x*(-x^4+1)^(1/2)-5/6*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4
+1)^(1/2)*EllipticF(x,I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^8/(-x^4 + 1)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

integral(-x^8/((x^4 - 1)*sqrt(-x^4 + 1)), x)

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Sympy [A]  time = 2.94299, size = 31, normalized size = 0.72 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(-x**4+1)**(3/2),x)

[Out]

x**9*gamma(9/4)*hyper((3/2, 9/4), (13/4,), x**4*exp_polar(2*I*pi))/(4*gamma(13/4
))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^8/(-x^4 + 1)^(3/2), x)