3.998 \(\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{4-x^2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{1}{3} \sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |6\right )-\frac{1}{3} \sqrt{2} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |6\right ) \]

[Out]

-(Sqrt[2]*EllipticE[ArcSin[x/2], 6])/3 + (Sqrt[2]*EllipticF[ArcSin[x/2], 6])/3

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Rubi [A]  time = 0.110379, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{1}{3} \sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |6\right )-\frac{1}{3} \sqrt{2} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |6\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[2 - 3*x^2]*Sqrt[4 - x^2]),x]

[Out]

-(Sqrt[2]*EllipticE[ArcSin[x/2], 6])/3 + (Sqrt[2]*EllipticF[ArcSin[x/2], 6])/3

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Rubi in Sympy [A]  time = 16.5586, size = 26, normalized size = 0.74 \[ - \frac{\sqrt{2} E\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | 6\right )}{3} + \frac{\sqrt{2} F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | 6\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*elliptic_e(asin(x/2), 6)/3 + sqrt(2)*elliptic_f(asin(x/2), 6)/3

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Mathematica [A]  time = 0.0465028, size = 38, normalized size = 1.09 \[ -\frac{2 \left (E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{1}{6}\right )-F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{1}{6}\right )\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(Sqrt[2 - 3*x^2]*Sqrt[4 - x^2]),x]

[Out]

(-2*(EllipticE[ArcSin[Sqrt[3/2]*x], 1/6] - EllipticF[ArcSin[Sqrt[3/2]*x], 1/6]))
/Sqrt[3]

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Maple [A]  time = 0.031, size = 45, normalized size = 1.3 \[{\frac{2\,\sqrt{3}}{3} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}\sqrt{2}}{2}},{\frac{\sqrt{3}\sqrt{2}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{3}\sqrt{2}}{2}},{\frac{\sqrt{3}\sqrt{2}}{6}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-3*x^2+2)^(1/2)/(-x^2+4)^(1/2),x)

[Out]

2/3*3^(1/2)*(EllipticF(1/2*x*3^(1/2)*2^(1/2),1/6*3^(1/2)*2^(1/2))-EllipticE(1/2*
x*3^(1/2)*2^(1/2),1/6*3^(1/2)*2^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-x^{2} + 4} \sqrt{-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(sqrt(-x^2 + 4)*sqrt(-3*x^2 + 2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{-x^{2} + 4} \sqrt{-3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/(sqrt(-x^2 + 4)*sqrt(-3*x^2 + 2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{- \left (x - 2\right ) \left (x + 2\right )} \sqrt{- 3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/(sqrt(-(x - 2)*(x + 2))*sqrt(-3*x**2 + 2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-x^{2} + 4} \sqrt{-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(sqrt(-x^2 + 4)*sqrt(-3*x^2 + 2)), x)