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

Optimal. Leaf size=21 \[ \frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0239248, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.86515, size = 22, normalized size = 1.05 \[ \frac{2 \sqrt{3} E\left (\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | - \frac{1}{6}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(3)*elliptic_e(asin(sqrt(6)*x/2), -1/6)/3

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Mathematica [A]  time = 0.0260063, size = 21, normalized size = 1. \[ \frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.025, size = 25, normalized size = 1.2 \[{\frac{2\,\sqrt{3}}{3}{\it EllipticE} \left ({\frac{x\sqrt{3}\sqrt{2}}{2}},{\frac{i}{6}}\sqrt{3}\sqrt{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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