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

Optimal. Leaf size=19 \[ -\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]

[Out]

-(EllipticE[ArcCos[Sqrt[3/2]*x], 2]/Sqrt[3])

_______________________________________________________________________________________

Rubi [A]  time = 0.0261532, antiderivative size = 19, 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 \[ -\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-(EllipticE[ArcCos[Sqrt[3/2]*x], 2]/Sqrt[3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.26547, size = 19, normalized size = 1. \[ - \frac{\sqrt{3} E\left (\operatorname{acos}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | 2\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0353428, size = 35, normalized size = 1.84 \[ \frac{\sqrt{3 x^2-1} E\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3-9 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-1 + 3*x^2]*EllipticE[ArcSin[Sqrt[3/2]*x], 2])/Sqrt[3 - 9*x^2]

_______________________________________________________________________________________

Maple [A]  time = 0.027, size = 37, normalized size = 2. \[ -{\frac{\sqrt{3}}{3}{\it EllipticE} \left ({\frac{x\sqrt{3}\sqrt{2}}{2}},\sqrt{2} \right ) \sqrt{-3\,{x}^{2}+1}{\frac{1}{\sqrt{3\,{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*EllipticE(1/2*x*3^(1/2)*2^(1/2),2^(1/2))*(-3*x^2+1)^(1/2)*3^(1/2)/(3*x^2-1)
^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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