3.1001 \(\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=42 \[ \frac{E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.100675, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 + x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.6391, size = 42, normalized size = 1. \[ \frac{\sqrt{3} E\left (\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | - \frac{2}{3}\right )}{3} - \frac{\sqrt{3} F\left (\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | - \frac{2}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.042918, size = 37, normalized size = 0.88 \[ \frac{E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )-F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 + x^2]),x]

[Out]

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

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Maple [A]  time = 0.02, size = 47, normalized size = 1.1 \[ -{\frac{\sqrt{3}}{3} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}\sqrt{2}}{2}},{\frac{i}{3}}\sqrt{3}\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{3}\sqrt{2}}{2}},{\frac{i}{3}}\sqrt{3}\sqrt{2} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{x^{2} + 1} \sqrt{-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(x^2 + 1)*sqrt(-3*x^2 + 2)),x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(x^2 + 1)*sqrt(-3*x^2 + 2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{x^{2} + 1} \sqrt{-3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(x^2 + 1)*sqrt(-3*x^2 + 2)),x, algorithm="fricas")

[Out]

integral(x^2/(sqrt(x^2 + 1)*sqrt(-3*x^2 + 2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{- 3 x^{2} + 2} \sqrt{x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/(sqrt(-3*x**2 + 2)*sqrt(x**2 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{x^{2} + 1} \sqrt{-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(x^2 + 1)*sqrt(-3*x^2 + 2)),x, algorithm="giac")

[Out]

integrate(x^2/(sqrt(x^2 + 1)*sqrt(-3*x^2 + 2)), x)