3.968 \(\int \frac{x^2}{\sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=21 \[ 2 E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-2 F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]

[Out]

2*EllipticE[ArcSin[x/2], -1] - 2*EllipticF[ArcSin[x/2], -1]

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Rubi [A]  time = 0.0518187, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ 2 E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-2 F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[16 - x^4],x]

[Out]

2*EllipticE[ArcSin[x/2], -1] - 2*EllipticF[ArcSin[x/2], -1]

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Rubi in Sympy [A]  time = 10.2646, size = 19, normalized size = 0.9 \[ 2 E\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -1\right ) - 2 F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*elliptic_e(asin(x/2), -1) - 2*elliptic_f(asin(x/2), -1)

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Mathematica [A]  time = 0.0295316, size = 21, normalized size = 1. \[ 2 \left (E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[16 - x^4],x]

[Out]

2*(EllipticE[ArcSin[x/2], -1] - EllipticF[ArcSin[x/2], -1])

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Maple [B]  time = 0.01, size = 43, normalized size = 2.1 \[ -2\,{\frac{\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4} \left ({\it EllipticF} \left ( x/2,i \right ) -{\it EllipticE} \left ( x/2,i \right ) \right ) }{\sqrt{-{x}^{4}+16}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-x^4+16)^(1/2),x)

[Out]

-2*(-x^2+4)^(1/2)*(x^2+4)^(1/2)/(-x^4+16)^(1/2)*(EllipticF(1/2*x,I)-EllipticE(1/
2*x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-x^{4} + 16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-x^4 + 16),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt(-x^4 + 16), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{-x^{4} + 16}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-x^4 + 16),x, algorithm="fricas")

[Out]

integral(x^2/sqrt(-x^4 + 16), x)

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Sympy [A]  time = 1.84712, size = 32, normalized size = 1.52 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), x**4*exp_polar(2*I*pi)/16)/(16*gamma(7
/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-x^{4} + 16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-x^4 + 16),x, algorithm="giac")

[Out]

integrate(x^2/sqrt(-x^4 + 16), x)