3.999 \(\int \frac{x^2}{\sqrt{1-4 x^2} \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{E\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right )}{3 \sqrt{2}}-\frac{F\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right )}{3 \sqrt{2}} \]

[Out]

EllipticE[ArcSin[2*x], -3/8]/(3*Sqrt[2]) - EllipticF[ArcSin[2*x], -3/8]/(3*Sqrt[
2])

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Rubi [A]  time = 0.108538, 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 \[ \frac{E\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right )}{3 \sqrt{2}}-\frac{F\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

EllipticE[ArcSin[2*x], -3/8]/(3*Sqrt[2]) - EllipticF[ArcSin[2*x], -3/8]/(3*Sqrt[
2])

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Rubi in Sympy [A]  time = 18.2143, size = 32, normalized size = 0.91 \[ \frac{\sqrt{2} E\left (\operatorname{asin}{\left (2 x \right )}\middle | - \frac{3}{8}\right )}{6} - \frac{\sqrt{2} F\left (\operatorname{asin}{\left (2 x \right )}\middle | - \frac{3}{8}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*elliptic_e(asin(2*x), -3/8)/6 - sqrt(2)*elliptic_f(asin(2*x), -3/8)/6

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Mathematica [A]  time = 0.0455505, size = 28, normalized size = 0.8 \[ \frac{E\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right )-F\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(EllipticE[ArcSin[2*x], -3/8] - EllipticF[ArcSin[2*x], -3/8])/(3*Sqrt[2])

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Maple [A]  time = 0.029, size = 35, normalized size = 1. \[ -{\frac{ \left ({\it EllipticF} \left ( 2\,x,{\frac{i}{4}}\sqrt{3}\sqrt{2} \right ) -{\it EllipticE} \left ( 2\,x,{\frac{i}{4}}\sqrt{3}\sqrt{2} \right ) \right ) \sqrt{2}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(EllipticF(2*x,1/4*I*3^(1/2)*2^(1/2))-EllipticE(2*x,1/4*I*3^(1/2)*2^(1/2)))
*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{-4 \, x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(-4*x^2 + 1)),x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(-4*x^2 + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{-4 \, x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(-4*x^2 + 1)),x, algorithm="fricas")

[Out]

integral(x^2/(sqrt(3*x^2 + 2)*sqrt(-4*x^2 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{-4 \, x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(-4*x^2 + 1)),x, algorithm="giac")

[Out]

integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(-4*x^2 + 1)), x)