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

Optimal. Leaf size=35 \[ \frac{7}{3} \sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right )-\frac{1}{3} \sqrt{2} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right ) \]

[Out]

-(Sqrt[2]*EllipticE[ArcSin[x/2], -6])/3 + (7*Sqrt[2]*EllipticF[ArcSin[x/2], -6])
/3

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Rubi [A]  time = 0.0670297, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{7}{3} \sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right )-\frac{1}{3} \sqrt{2} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[2]*EllipticE[ArcSin[x/2], -6])/3 + (7*Sqrt[2]*EllipticF[ArcSin[x/2], -6])
/3

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Rubi in Sympy [A]  time = 12.424, size = 31, normalized size = 0.89 \[ - \frac{\sqrt{2} E\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -6\right )}{3} + \frac{7 \sqrt{2} F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -6\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*elliptic_e(asin(x/2), -6)/3 + 7*sqrt(2)*elliptic_f(asin(x/2), -6)/3

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Mathematica [C]  time = 0.0267695, size = 27, normalized size = 0.77 \[ -\frac{2 i E\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.035, size = 37, normalized size = 1.1 \[{\frac{\sqrt{2}}{3} \left ( 7\,{\it EllipticF} \left ( x/2,i\sqrt{3}\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{x}{2}},i\sqrt{3}\sqrt{2} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(7*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. \[ \int \frac{\sqrt{-x^{2} + 4}}{\sqrt{3 \, x^{2} + 2}}\,{d x} \]

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. \[ \int \frac{\sqrt{- \left (x - 2\right ) \left (x + 2\right )}}{\sqrt{3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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)