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

Optimal. Leaf size=20 \[ \frac{E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{8}{3}\right )}{\sqrt{3}} \]

[Out]

EllipticE[ArcSin[Sqrt[3/2]*x], 8/3]/Sqrt[3]

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Rubi [A]  time = 0.0264847, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{8}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

EllipticE[ArcSin[Sqrt[3/2]*x], 8/3]/Sqrt[3]

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Rubi in Sympy [A]  time = 5.26896, size = 19, normalized size = 0.95 \[ \frac{\sqrt{3} E\left (\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | \frac{8}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0280647, size = 20, normalized size = 1. \[ \frac{E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{8}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

EllipticE[ArcSin[Sqrt[3/2]*x], 8/3]/Sqrt[3]

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Maple [A]  time = 0.111, size = 35, normalized size = 1.8 \[ -{\frac{\sqrt{2}}{12} \left ( 5\,{\it EllipticF} \left ( 2\,x,1/4\,\sqrt{3}\sqrt{2} \right ) -8\,{\it EllipticE} \left ( 2\,x,1/4\,\sqrt{3}\sqrt{2} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*2^(1/2)*(5*EllipticF(2*x,1/4*3^(1/2)*2^(1/2))-8*EllipticE(2*x,1/4*3^(1/2)*
2^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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