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

Optimal. Leaf size=31 \[ \frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|-\frac{3}{2}\right )-\frac{1}{3} \sqrt{2} F\left (\sin ^{-1}(x)|-\frac{3}{2}\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.107223, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|-\frac{3}{2}\right )-\frac{1}{3} \sqrt{2} F\left (\sin ^{-1}(x)|-\frac{3}{2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.4008, size = 29, normalized size = 0.94 \[ \frac{\sqrt{2} E\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{3}{2}\right )}{3} - \frac{\sqrt{2} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{3}{2}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0441813, size = 24, normalized size = 0.77 \[ \frac{1}{3} \sqrt{2} \left (E\left (\sin ^{-1}(x)|-\frac{3}{2}\right )-F\left (\sin ^{-1}(x)|-\frac{3}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 31, normalized size = 1. \[ -{\frac{ \left ({\it EllipticF} \left ( x,{\frac{i}{2}}\sqrt{3}\sqrt{2} \right ) -{\it EllipticE} \left ( x,{\frac{i}{2}}\sqrt{3}\sqrt{2} \right ) \right ) \sqrt{2}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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