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

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

[Out]

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

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Rubi [A]  time = 0.105872, 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 \[ \frac{1}{3} \sqrt{2} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right )-\frac{1}{3} \sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[4 - x^2]*Sqrt[2 + 3*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 = 17.268, size = 29, normalized size = 0.83 \[ \frac{\sqrt{2} E\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -6\right )}{3} - \frac{\sqrt{2} F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -6\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-x**2+4)**(1/2)/(3*x**2+2)**(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 = 0.0427926, size = 28, normalized size = 0.8 \[ \frac{1}{3} \sqrt{2} \left (E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right )-F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.031, size = 35, normalized size = 1. \[ -{\frac{\sqrt{2}}{3} \left ({\it EllipticF} \left ({\frac{x}{2}},i\sqrt{3}\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{x}{2}},i\sqrt{3}\sqrt{2} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/(sqrt(3*x^2 + 2)*sqrt(-x^2 + 4)), 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/(-x**2+4)**(1/2)/(3*x**2+2)**(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. \[ \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{-x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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