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

Optimal. Leaf size=70 \[ -\frac{1}{9} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{F\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{9 \sqrt{3}}-\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{3 \sqrt{3}} \]

[Out]

-(x*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/9 - EllipticE[ArcCos[Sqrt[3/2]*x], 2]/(3*S
qrt[3]) - EllipticF[ArcCos[Sqrt[3/2]*x], 2]/(9*Sqrt[3])

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Rubi [A]  time = 0.161637, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{1}{9} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{F\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{9 \sqrt{3}}-\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-(x*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/9 - EllipticE[ArcCos[Sqrt[3/2]*x], 2]/(3*S
qrt[3]) - EllipticF[ArcCos[Sqrt[3/2]*x], 2]/(9*Sqrt[3])

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Rubi in Sympy [A]  time = 25.6158, size = 61, normalized size = 0.87 \[ - \frac{x \sqrt{- 3 x^{2} + 2} \sqrt{3 x^{2} - 1}}{9} - \frac{\sqrt{3} E\left (\operatorname{acos}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | 2\right )}{9} - \frac{\sqrt{3} F\left (\operatorname{acos}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | 2\right )}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*sqrt(-3*x**2 + 2)*sqrt(3*x**2 - 1)/9 - sqrt(3)*elliptic_e(acos(sqrt(6)*x/2),
2)/9 - sqrt(3)*elliptic_f(acos(sqrt(6)*x/2), 2)/27

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Mathematica [A]  time = 0.145141, size = 86, normalized size = 1.23 \[ \frac{3 x \sqrt{2-3 x^2} \left (1-3 x^2\right )+\sqrt{3-9 x^2} F\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )-3 \sqrt{3-9 x^2} E\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{27 \sqrt{3 x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

(3*x*(1 - 3*x^2)*Sqrt[2 - 3*x^2] - 3*Sqrt[3 - 9*x^2]*EllipticE[ArcSin[Sqrt[3/2]*
x], 2] + Sqrt[3 - 9*x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 2])/(27*Sqrt[-1 + 3*x^2]
)

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Maple [A]  time = 0.017, size = 129, normalized size = 1.8 \[ -{\frac{\sqrt{2}}{972\,{x}^{4}-972\,{x}^{2}+216}\sqrt{3\,{x}^{2}-1}\sqrt{-6\,{x}^{2}+4} \left ( 54\,{x}^{5}+\sqrt{3}\sqrt{2}\sqrt{-6\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{3}\sqrt{2}}{2}},\sqrt{2} \right ) -3\,\sqrt{3}\sqrt{2}\sqrt{-6\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\it EllipticE} \left ( 1/2\,x\sqrt{3}\sqrt{2},\sqrt{2} \right ) -54\,{x}^{3}+12\,x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/108*(3*x^2-1)^(1/2)*2^(1/2)*(-6*x^2+4)^(1/2)*(54*x^5+3^(1/2)*2^(1/2)*(-6*x^2+
4)^(1/2)*(-3*x^2+1)^(1/2)*EllipticF(1/2*x*3^(1/2)*2^(1/2),2^(1/2))-3*3^(1/2)*2^(
1/2)*(-6*x^2+4)^(1/2)*(-3*x^2+1)^(1/2)*EllipticE(1/2*x*3^(1/2)*2^(1/2),2^(1/2))-
54*x^3+12*x)/(9*x^4-9*x^2+2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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