3.902 \(\int \frac{x^{12}}{\left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=61 \[ \frac{15}{14} \sqrt{1-x^4} x+\frac{x^9}{2 \sqrt{1-x^4}}+\frac{9}{14} \sqrt{1-x^4} x^5-\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

x^9/(2*Sqrt[1 - x^4]) + (15*x*Sqrt[1 - x^4])/14 + (9*x^5*Sqrt[1 - x^4])/14 - (15
*EllipticF[ArcSin[x], -1])/14

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Rubi [A]  time = 0.0523767, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{15}{14} \sqrt{1-x^4} x+\frac{x^9}{2 \sqrt{1-x^4}}+\frac{9}{14} \sqrt{1-x^4} x^5-\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^12/(1 - x^4)^(3/2),x]

[Out]

x^9/(2*Sqrt[1 - x^4]) + (15*x*Sqrt[1 - x^4])/14 + (9*x^5*Sqrt[1 - x^4])/14 - (15
*EllipticF[ArcSin[x], -1])/14

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Rubi in Sympy [A]  time = 5.97614, size = 51, normalized size = 0.84 \[ \frac{x^{9}}{2 \sqrt{- x^{4} + 1}} + \frac{9 x^{5} \sqrt{- x^{4} + 1}}{14} + \frac{15 x \sqrt{- x^{4} + 1}}{14} - \frac{15 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12/(-x**4+1)**(3/2),x)

[Out]

x**9/(2*sqrt(-x**4 + 1)) + 9*x**5*sqrt(-x**4 + 1)/14 + 15*x*sqrt(-x**4 + 1)/14 -
15*elliptic_f(asin(x), -1)/14

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Mathematica [A]  time = 0.0501682, size = 46, normalized size = 0.75 \[ -\frac{2 x^9+6 x^5+15 \sqrt{1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )-15 x}{14 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12/(1 - x^4)^(3/2),x]

[Out]

-(-15*x + 6*x^5 + 2*x^9 + 15*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(14*Sqrt[1
- x^4])

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Maple [A]  time = 0.014, size = 71, normalized size = 1.2 \[{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{{x}^{5}}{7}\sqrt{-{x}^{4}+1}}+{\frac{4\,x}{7}\sqrt{-{x}^{4}+1}}-{\frac{15\,{\it EllipticF} \left ( x,i \right ) }{14}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12/(-x^4+1)^(3/2),x)

[Out]

1/2*x/(-x^4+1)^(1/2)+1/7*x^5*(-x^4+1)^(1/2)+4/7*x*(-x^4+1)^(1/2)-15/14*(-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. \[ \int \frac{x^{12}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^12/(-x^4 + 1)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x^{12}}{{\left (x^{4} - 1\right )} \sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

integral(-x^12/((x^4 - 1)*sqrt(-x^4 + 1)), x)

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Sympy [A]  time = 5.85672, size = 31, normalized size = 0.51 \[ \frac{x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{17}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12/(-x**4+1)**(3/2),x)

[Out]

x**13*gamma(13/4)*hyper((3/2, 13/4), (17/4,), x**4*exp_polar(2*I*pi))/(4*gamma(1
7/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{12}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^12/(-x^4 + 1)^(3/2), x)