3.909 \(\int \frac{x^{10}}{\left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{x^7}{2 \sqrt{1-x^4}}+\frac{7}{10} \sqrt{1-x^4} x^3+\frac{21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

x^7/(2*Sqrt[1 - x^4]) + (7*x^3*Sqrt[1 - x^4])/10 - (21*EllipticE[ArcSin[x], -1])
/10 + (21*EllipticF[ArcSin[x], -1])/10

_______________________________________________________________________________________

Rubi [A]  time = 0.0760235, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{x^7}{2 \sqrt{1-x^4}}+\frac{7}{10} \sqrt{1-x^4} x^3+\frac{21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^10/(1 - x^4)^(3/2),x]

[Out]

x^7/(2*Sqrt[1 - x^4]) + (7*x^3*Sqrt[1 - x^4])/10 - (21*EllipticE[ArcSin[x], -1])
/10 + (21*EllipticF[ArcSin[x], -1])/10

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.3158, size = 48, normalized size = 0.91 \[ \frac{x^{7}}{2 \sqrt{- x^{4} + 1}} + \frac{7 x^{3} \sqrt{- x^{4} + 1}}{10} - \frac{21 E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{10} + \frac{21 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(-x**4+1)**(3/2),x)

[Out]

x**7/(2*sqrt(-x**4 + 1)) + 7*x**3*sqrt(-x**4 + 1)/10 - 21*elliptic_e(asin(x), -1
)/10 + 21*elliptic_f(asin(x), -1)/10

_______________________________________________________________________________________

Mathematica [A]  time = 0.0639899, size = 49, normalized size = 0.92 \[ \frac{1}{10} \left (-\frac{2 x^7}{\sqrt{1-x^4}}+\frac{7 x^3}{\sqrt{1-x^4}}+21 F\left (\left .\sin ^{-1}(x)\right |-1\right )-21 E\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/(1 - x^4)^(3/2),x]

[Out]

((7*x^3)/Sqrt[1 - x^4] - (2*x^7)/Sqrt[1 - x^4] - 21*EllipticE[ArcSin[x], -1] + 2
1*EllipticF[ArcSin[x], -1])/10

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 68, normalized size = 1.3 \[{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{{x}^{3}}{5}\sqrt{-{x}^{4}+1}}+{\frac{21\,{\it EllipticF} \left ( x,i \right ) -21\,{\it EllipticE} \left ( x,i \right ) }{10}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(-x^4+1)^(3/2),x)

[Out]

1/2*x^3/(-x^4+1)^(1/2)+1/5*x^3*(-x^4+1)^(1/2)+21/10*(-x^2+1)^(1/2)*(x^2+1)^(1/2)
/(-x^4+1)^(1/2)*(EllipticF(x,I)-EllipticE(x,I))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^10/(-x^4 + 1)^(3/2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

integral(-x^10/((x^4 - 1)*sqrt(-x^4 + 1)), x)

_______________________________________________________________________________________

Sympy [A]  time = 4.22732, size = 31, normalized size = 0.58 \[ \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10/(-x**4+1)**(3/2),x)

[Out]

x**11*gamma(11/4)*hyper((3/2, 11/4), (15/4,), x**4*exp_polar(2*I*pi))/(4*gamma(1
5/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^10/(-x^4 + 1)^(3/2), x)