3.966 \(\int \frac{x^6}{\sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{1}{5} \sqrt{16-x^4} x^3-\frac{96}{5} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )+\frac{96}{5} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]

[Out]

-(x^3*Sqrt[16 - x^4])/5 + (96*EllipticE[ArcSin[x/2], -1])/5 - (96*EllipticF[ArcS
in[x/2], -1])/5

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Rubi [A]  time = 0.0674675, antiderivative size = 43, 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 \[ -\frac{1}{5} \sqrt{16-x^4} x^3-\frac{96}{5} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )+\frac{96}{5} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^6/Sqrt[16 - x^4],x]

[Out]

-(x^3*Sqrt[16 - x^4])/5 + (96*EllipticE[ArcSin[x/2], -1])/5 - (96*EllipticF[ArcS
in[x/2], -1])/5

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Rubi in Sympy [A]  time = 11.5037, size = 36, normalized size = 0.84 \[ - \frac{x^{3} \sqrt{- x^{4} + 16}}{5} + \frac{96 E\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -1\right )}{5} - \frac{96 F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -1\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(-x**4+16)**(1/2),x)

[Out]

-x**3*sqrt(-x**4 + 16)/5 + 96*elliptic_e(asin(x/2), -1)/5 - 96*elliptic_f(asin(x
/2), -1)/5

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Mathematica [A]  time = 0.0712276, size = 56, normalized size = 1.3 \[ \frac{1}{5} \left (\frac{x^7}{\sqrt{16-x^4}}-\frac{16 x^3}{\sqrt{16-x^4}}-96 F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )+96 E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/Sqrt[16 - x^4],x]

[Out]

((-16*x^3)/Sqrt[16 - x^4] + x^7/Sqrt[16 - x^4] + 96*EllipticE[ArcSin[x/2], -1] -
96*EllipticF[ArcSin[x/2], -1])/5

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Maple [A]  time = 0.012, size = 58, normalized size = 1.4 \[ -{\frac{{x}^{3}}{5}\sqrt{-{x}^{4}+16}}-{\frac{96}{5}\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4} \left ({\it EllipticF} \left ({\frac{x}{2}},i \right ) -{\it EllipticE} \left ({\frac{x}{2}},i \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+16}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-x^4+16)^(1/2),x)

[Out]

-1/5*x^3*(-x^4+16)^(1/2)-96/5*(-x^2+4)^(1/2)*(x^2+4)^(1/2)/(-x^4+16)^(1/2)*(Elli
pticF(1/2*x,I)-EllipticE(1/2*x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{-x^{4} + 16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(-x^4 + 16),x, algorithm="maxima")

[Out]

integrate(x^6/sqrt(-x^4 + 16), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{\sqrt{-x^{4} + 16}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(-x^4 + 16),x, algorithm="fricas")

[Out]

integral(x^6/sqrt(-x^4 + 16), x)

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Sympy [A]  time = 2.30131, size = 32, normalized size = 0.74 \[ \frac{x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(-x**4+16)**(1/2),x)

[Out]

x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), x**4*exp_polar(2*I*pi)/16)/(16*gamma(
11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{-x^{4} + 16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(-x^4 + 16),x, algorithm="giac")

[Out]

integrate(x^6/sqrt(-x^4 + 16), x)