3.908 \(\int \frac{x^{14}}{\left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ \frac{x^{11}}{2 \sqrt{1-x^4}}+\frac{11}{18} \sqrt{1-x^4} x^7+\frac{77}{90} \sqrt{1-x^4} x^3+\frac{77}{30} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{77}{30} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

x^11/(2*Sqrt[1 - x^4]) + (77*x^3*Sqrt[1 - x^4])/90 + (11*x^7*Sqrt[1 - x^4])/18 -
(77*EllipticE[ArcSin[x], -1])/30 + (77*EllipticF[ArcSin[x], -1])/30

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Rubi [A]  time = 0.0916057, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{x^{11}}{2 \sqrt{1-x^4}}+\frac{11}{18} \sqrt{1-x^4} x^7+\frac{77}{90} \sqrt{1-x^4} x^3+\frac{77}{30} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{77}{30} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^14/(1 - x^4)^(3/2),x]

[Out]

x^11/(2*Sqrt[1 - x^4]) + (77*x^3*Sqrt[1 - x^4])/90 + (11*x^7*Sqrt[1 - x^4])/18 -
(77*EllipticE[ArcSin[x], -1])/30 + (77*EllipticF[ArcSin[x], -1])/30

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Rubi in Sympy [A]  time = 13.4833, size = 63, normalized size = 0.89 \[ \frac{x^{11}}{2 \sqrt{- x^{4} + 1}} + \frac{11 x^{7} \sqrt{- x^{4} + 1}}{18} + \frac{77 x^{3} \sqrt{- x^{4} + 1}}{90} - \frac{77 E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{30} + \frac{77 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{30} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(-x**4+1)**(3/2),x)

[Out]

x**11/(2*sqrt(-x**4 + 1)) + 11*x**7*sqrt(-x**4 + 1)/18 + 77*x**3*sqrt(-x**4 + 1)
/90 - 77*elliptic_e(asin(x), -1)/30 + 77*elliptic_f(asin(x), -1)/30

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Mathematica [A]  time = 0.0844563, size = 44, normalized size = 0.62 \[ \frac{1}{90} \left (\frac{\left (-10 x^8-22 x^4+77\right ) x^3}{\sqrt{1-x^4}}+231 F\left (\left .\sin ^{-1}(x)\right |-1\right )-231 E\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/(1 - x^4)^(3/2),x]

[Out]

((x^3*(77 - 22*x^4 - 10*x^8))/Sqrt[1 - x^4] - 231*EllipticE[ArcSin[x], -1] + 231
*EllipticF[ArcSin[x], -1])/90

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Maple [A]  time = 0.016, size = 82, normalized size = 1.2 \[{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{{x}^{7}}{9}\sqrt{-{x}^{4}+1}}+{\frac{16\,{x}^{3}}{45}\sqrt{-{x}^{4}+1}}+{\frac{77\,{\it EllipticF} \left ( x,i \right ) -77\,{\it EllipticE} \left ( x,i \right ) }{30}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(-x^4+1)^(3/2),x)

[Out]

1/2*x^3/(-x^4+1)^(1/2)+1/9*x^7*(-x^4+1)^(1/2)+16/45*x^3*(-x^4+1)^(1/2)+77/30*(-x
^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*(EllipticF(x,I)-EllipticE(x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^14/(-x^4 + 1)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x^{14}}{{\left (x^{4} - 1\right )} \sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

integral(-x^14/((x^4 - 1)*sqrt(-x^4 + 1)), x)

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Sympy [A]  time = 8.63156, size = 31, normalized size = 0.44 \[ \frac{x^{15} \Gamma \left (\frac{15}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{15}{4} \\ \frac{19}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{19}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(-x**4+1)**(3/2),x)

[Out]

x**15*gamma(15/4)*hyper((3/2, 15/4), (19/4,), x**4*exp_polar(2*I*pi))/(4*gamma(1
9/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^14/(-x^4 + 1)^(3/2), x)