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

Optimal. Leaf size=99 \[ -\frac{7}{135} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{1}{15} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x^3-\frac{2 F\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{27 \sqrt{3}}-\frac{8 E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{45 \sqrt{3}} \]

[Out]

(-7*x*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/135 - (x^3*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x
^2])/15 - (8*EllipticE[ArcCos[Sqrt[3/2]*x], 2])/(45*Sqrt[3]) - (2*EllipticF[ArcC
os[Sqrt[3/2]*x], 2])/(27*Sqrt[3])

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Rubi [A]  time = 0.298323, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{7}{135} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{1}{15} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x^3-\frac{2 F\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{27 \sqrt{3}}-\frac{8 E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{45 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-7*x*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/135 - (x^3*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x
^2])/15 - (8*EllipticE[ArcCos[Sqrt[3/2]*x], 2])/(45*Sqrt[3]) - (2*EllipticF[ArcC
os[Sqrt[3/2]*x], 2])/(27*Sqrt[3])

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Rubi in Sympy [A]  time = 34.8414, size = 92, normalized size = 0.93 \[ - \frac{x^{3} \sqrt{- 3 x^{2} + 2} \sqrt{3 x^{2} - 1}}{15} - \frac{7 x \sqrt{- 3 x^{2} + 2} \sqrt{3 x^{2} - 1}}{135} - \frac{8 \sqrt{3} E\left (\operatorname{acos}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | 2\right )}{135} - \frac{2 \sqrt{3} F\left (\operatorname{acos}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | 2\right )}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**3*sqrt(-3*x**2 + 2)*sqrt(3*x**2 - 1)/15 - 7*x*sqrt(-3*x**2 + 2)*sqrt(3*x**2
- 1)/135 - 8*sqrt(3)*elliptic_e(acos(sqrt(6)*x/2), 2)/135 - 2*sqrt(3)*elliptic_f
(acos(sqrt(6)*x/2), 2)/81

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Mathematica [A]  time = 0.143801, size = 92, normalized size = 0.93 \[ \frac{10 \sqrt{3-9 x^2} F\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )-24 \sqrt{3-9 x^2} E\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )-3 x \sqrt{2-3 x^2} \left (27 x^4+12 x^2-7\right )}{405 \sqrt{3 x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*x*Sqrt[2 - 3*x^2]*(-7 + 12*x^2 + 27*x^4) - 24*Sqrt[3 - 9*x^2]*EllipticE[ArcS
in[Sqrt[3/2]*x], 2] + 10*Sqrt[3 - 9*x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 2])/(405
*Sqrt[-1 + 3*x^2])

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Maple [A]  time = 0.031, size = 135, normalized size = 1.4 \[ -{\frac{\sqrt{2}}{7290\,{x}^{4}-7290\,{x}^{2}+1620}\sqrt{3\,{x}^{2}-1}\sqrt{-6\,{x}^{2}+4} \left ( 243\,{x}^{7}-54\,{x}^{5}+5\,\sqrt{3}\sqrt{2}\sqrt{-6\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\it EllipticF} \left ( 1/2\,x\sqrt{3}\sqrt{2},\sqrt{2} \right ) -12\,\sqrt{3}\sqrt{2}\sqrt{-6\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\it EllipticE} \left ( 1/2\,x\sqrt{3}\sqrt{2},\sqrt{2} \right ) -135\,{x}^{3}+42\,x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/810*(3*x^2-1)^(1/2)*2^(1/2)*(-6*x^2+4)^(1/2)*(243*x^7-54*x^5+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))-12*
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))-135*x^3+42*x)/(9*x^4-9*x^2+2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{3 \, x^{2} - 1} x^{4}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x^2 - 1)*x^4/sqrt(-3*x^2 + 2),x, algorithm="maxima")

[Out]

integrate(sqrt(3*x^2 - 1)*x^4/sqrt(-3*x^2 + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{3 \, x^{2} - 1} x^{4}}{\sqrt{-3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x^2 - 1)*x^4/sqrt(-3*x^2 + 2),x, algorithm="fricas")

[Out]

integral(sqrt(3*x^2 - 1)*x^4/sqrt(-3*x^2 + 2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{3 \, x^{2} - 1} x^{4}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x^2 - 1)*x^4/sqrt(-3*x^2 + 2),x, algorithm="giac")

[Out]

integrate(sqrt(3*x^2 - 1)*x^4/sqrt(-3*x^2 + 2), x)