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

Optimal. Leaf size=65 \[ -\frac{1}{36} \sqrt{2-3 x^2} \left (3 x^2-1\right )^{3/2}-\frac{7}{72} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{7}{144} \sin ^{-1}\left (3-6 x^2\right ) \]

[Out]

(-7*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/72 - (Sqrt[2 - 3*x^2]*(-1 + 3*x^2)^(3/2))/
36 - (7*ArcSin[3 - 6*x^2])/144

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Rubi [A]  time = 0.146526, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{36} \sqrt{2-3 x^2} \left (3 x^2-1\right )^{3/2}-\frac{7}{72} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{7}{144} \sin ^{-1}\left (3-6 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-7*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/72 - (Sqrt[2 - 3*x^2]*(-1 + 3*x^2)^(3/2))/
36 - (7*ArcSin[3 - 6*x^2])/144

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Rubi in Sympy [A]  time = 14.324, size = 75, normalized size = 1.15 \[ - \frac{\sqrt{- 3 x^{2} + 2} \left (3 x^{2} - 1\right )^{\frac{3}{2}}}{36} - \frac{7 \sqrt{- 3 x^{2} + 2} \sqrt{3 x^{2} - 1}}{72} - \frac{7 \operatorname{atan}{\left (\frac{- 18 x^{2} + 9}{6 \sqrt{- 9 x^{4} + 9 x^{2} - 2}} \right )}}{144} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-3*x**2 + 2)*(3*x**2 - 1)**(3/2)/36 - 7*sqrt(-3*x**2 + 2)*sqrt(3*x**2 - 1)
/72 - 7*atan((-18*x**2 + 9)/(6*sqrt(-9*x**4 + 9*x**2 - 2)))/144

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Mathematica [A]  time = 0.0678985, size = 60, normalized size = 0.92 \[ \frac{1}{144} \left (-2 \sqrt{-9 x^4+9 x^2-2} \left (6 x^2+5\right )-7 \tan ^{-1}\left (\frac{3-6 x^2}{2 \sqrt{-9 x^4+9 x^2-2}}\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*(5 + 6*x^2)*Sqrt[-2 + 9*x^2 - 9*x^4] - 7*ArcTan[(3 - 6*x^2)/(2*Sqrt[-2 + 9*x
^2 - 9*x^4])])/144

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Maple [A]  time = 0.035, size = 81, normalized size = 1.3 \[{\frac{1}{144}\sqrt{-3\,{x}^{2}+2}\sqrt{3\,{x}^{2}-1} \left ( -12\,{x}^{2}\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2}+7\,\arcsin \left ( 6\,{x}^{2}-3 \right ) -10\,\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2} \right ){\frac{1}{\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/144*(3*x^2-1)^(1/2)*(-3*x^2+2)^(1/2)*(-12*x^2*(-9*x^4+9*x^2-2)^(1/2)+7*arcsin(
6*x^2-3)-10*(-9*x^4+9*x^2-2)^(1/2))/(-9*x^4+9*x^2-2)^(1/2)

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Maxima [A]  time = 1.51679, size = 62, normalized size = 0.95 \[ -\frac{1}{12} \, \sqrt{-9 \, x^{4} + 9 \, x^{2} - 2} x^{2} - \frac{5}{72} \, \sqrt{-9 \, x^{4} + 9 \, x^{2} - 2} + \frac{7}{144} \, \arcsin \left (6 \, x^{2} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(-9*x^4 + 9*x^2 - 2)*x^2 - 5/72*sqrt(-9*x^4 + 9*x^2 - 2) + 7/144*arcsi
n(6*x^2 - 3)

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Fricas [A]  time = 0.240719, size = 78, normalized size = 1.2 \[ -\frac{1}{72} \,{\left (6 \, x^{2} + 5\right )} \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} + \frac{7}{144} \, \arctan \left (\frac{3 \,{\left (2 \, x^{2} - 1\right )}}{2 \, \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/72*(6*x^2 + 5)*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2) + 7/144*arctan(3/2*(2*x^2 - 1
)/(sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229491, size = 54, normalized size = 0.83 \[ -\frac{1}{72} \,{\left (6 \, x^{2} + 5\right )} \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} + \frac{7}{72} \, \arcsin \left (\sqrt{3 \, x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/72*(6*x^2 + 5)*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2) + 7/72*arcsin(sqrt(3*x^2 - 1)
)