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

Optimal. Leaf size=29 \[ \frac{8}{3} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-\frac{1}{3} x \sqrt{16-x^4} \]

[Out]

-(x*Sqrt[16 - x^4])/3 + (8*EllipticF[ArcSin[x/2], -1])/3

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Rubi [A]  time = 0.0217364, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{8}{3} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-\frac{1}{3} x \sqrt{16-x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[16 - x^4],x]

[Out]

-(x*Sqrt[16 - x^4])/3 + (8*EllipticF[ArcSin[x/2], -1])/3

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Rubi in Sympy [A]  time = 3.0727, size = 22, normalized size = 0.76 \[ - \frac{x \sqrt{- x^{4} + 16}}{3} + \frac{8 F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -1\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*sqrt(-x**4 + 16)/3 + 8*elliptic_f(asin(x/2), -1)/3

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Mathematica [A]  time = 0.0335157, size = 43, normalized size = 1.48 \[ \frac{x^5+8 \sqrt{16-x^4} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-16 x}{3 \sqrt{16-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[16 - x^4],x]

[Out]

(-16*x + x^5 + 8*Sqrt[16 - x^4]*EllipticF[ArcSin[x/2], -1])/(3*Sqrt[16 - x^4])

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Maple [B]  time = 0.01, size = 47, normalized size = 1.6 \[ -{\frac{x}{3}\sqrt{-{x}^{4}+16}}+{\frac{8}{3}\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4}{\it EllipticF} \left ({\frac{x}{2}},i \right ){\frac{1}{\sqrt{-{x}^{4}+16}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-x^4+16)^(1/2),x)

[Out]

-1/3*x*(-x^4+16)^(1/2)+8/3*(-x^2+4)^(1/2)*(x^2+4)^(1/2)/(-x^4+16)^(1/2)*Elliptic
F(1/2*x,I)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/sqrt(-x^4 + 16), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/sqrt(-x^4 + 16), x)

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Sympy [A]  time = 1.99242, size = 32, normalized size = 1.1 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), x**4*exp_polar(2*I*pi)/16)/(16*gamma(9
/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/sqrt(-x^4 + 16), x)