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

Optimal. Leaf size=35 \[ \frac{11 F\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right )}{6 \sqrt{2}}-\frac{2}{3} \sqrt{2} E\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right ) \]

[Out]

(-2*Sqrt[2]*EllipticE[ArcSin[2*x], -3/8])/3 + (11*EllipticF[ArcSin[2*x], -3/8])/
(6*Sqrt[2])

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Rubi [A]  time = 0.0671248, 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 \[ \frac{11 F\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right )}{6 \sqrt{2}}-\frac{2}{3} \sqrt{2} E\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[2]*EllipticE[ArcSin[2*x], -3/8])/3 + (11*EllipticF[ArcSin[2*x], -3/8])/
(6*Sqrt[2])

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Rubi in Sympy [A]  time = 12.4344, size = 36, normalized size = 1.03 \[ - \frac{2 \sqrt{2} E\left (\operatorname{asin}{\left (2 x \right )}\middle | - \frac{3}{8}\right )}{3} + \frac{11 \sqrt{2} F\left (\operatorname{asin}{\left (2 x \right )}\middle | - \frac{3}{8}\right )}{12} \]

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]

-2*sqrt(2)*elliptic_e(asin(2*x), -3/8)/3 + 11*sqrt(2)*elliptic_f(asin(2*x), -3/8
)/12

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Mathematica [C]  time = 0.0263359, size = 27, normalized size = 0.77 \[ -\frac{i E\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{8}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], -8/3])/Sqrt[3]

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Maple [A]  time = 0.029, size = 37, normalized size = 1.1 \[{\frac{ \left ( 11\,{\it EllipticF} \left ( 2\,x,i/4\sqrt{3}\sqrt{2} \right ) -8\,{\it EllipticE} \left ( 2\,x,i/4\sqrt{3}\sqrt{2} \right ) \right ) \sqrt{2}}{12}} \]

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*(11*EllipticF(2*x,1/4*I*3^(1/2)*2^(1/2))-8*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. \[ \int \frac{\sqrt{-4 \, x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}\,{d x} \]

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)