3.887 \(\int \frac{x^{10}}{\sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{1}{9} \sqrt{1-x^4} x^7-\frac{7}{45} \sqrt{1-x^4} x^3-\frac{7}{15} F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac{7}{15} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

(-7*x^3*Sqrt[1 - x^4])/45 - (x^7*Sqrt[1 - x^4])/9 + (7*EllipticE[ArcSin[x], -1])
/15 - (7*EllipticF[ArcSin[x], -1])/15

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Rubi [A]  time = 0.0776288, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{9} \sqrt{1-x^4} x^7-\frac{7}{45} \sqrt{1-x^4} x^3-\frac{7}{15} F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac{7}{15} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^10/Sqrt[1 - x^4],x]

[Out]

(-7*x^3*Sqrt[1 - x^4])/45 - (x^7*Sqrt[1 - x^4])/9 + (7*EllipticE[ArcSin[x], -1])
/15 - (7*EllipticF[ArcSin[x], -1])/15

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Rubi in Sympy [A]  time = 11.8215, size = 48, normalized size = 0.91 \[ - \frac{x^{7} \sqrt{- x^{4} + 1}}{9} - \frac{7 x^{3} \sqrt{- x^{4} + 1}}{45} + \frac{7 E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{15} - \frac{7 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(-x**4+1)**(1/2),x)

[Out]

-x**7*sqrt(-x**4 + 1)/9 - 7*x**3*sqrt(-x**4 + 1)/45 + 7*elliptic_e(asin(x), -1)/
15 - 7*elliptic_f(asin(x), -1)/15

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Mathematica [A]  time = 0.0812226, size = 44, normalized size = 0.83 \[ \frac{1}{45} \left (\frac{\left (5 x^8+2 x^4-7\right ) x^3}{\sqrt{1-x^4}}-21 F\left (\left .\sin ^{-1}(x)\right |-1\right )+21 E\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/Sqrt[1 - x^4],x]

[Out]

((x^3*(-7 + 2*x^4 + 5*x^8))/Sqrt[1 - x^4] + 21*EllipticE[ArcSin[x], -1] - 21*Ell
ipticF[ArcSin[x], -1])/45

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Maple [A]  time = 0.012, size = 68, normalized size = 1.3 \[ -{\frac{{x}^{7}}{9}\sqrt{-{x}^{4}+1}}-{\frac{7\,{x}^{3}}{45}\sqrt{-{x}^{4}+1}}-{\frac{7\,{\it EllipticF} \left ( x,i \right ) -7\,{\it EllipticE} \left ( x,i \right ) }{15}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(-x^4+1)^(1/2),x)

[Out]

-1/9*x^7*(-x^4+1)^(1/2)-7/45*x^3*(-x^4+1)^(1/2)-7/15*(-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. \[ \int \frac{x^{10}}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(-x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^10/sqrt(-x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{10}}{\sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(-x^4 + 1),x, algorithm="fricas")

[Out]

integral(x^10/sqrt(-x^4 + 1), x)

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Sympy [A]  time = 3.62997, size = 31, normalized size = 0.58 \[ \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10/(-x**4+1)**(1/2),x)

[Out]

x**11*gamma(11/4)*hyper((1/2, 11/4), (15/4,), x**4*exp_polar(2*I*pi))/(4*gamma(1
5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(-x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^10/sqrt(-x^4 + 1), x)