3.1063 \(\int \frac{x^3}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=158 \[ \frac{2}{9} \sqrt [4]{2-3 x^2}+\frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{9 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{9 \sqrt [4]{2}}-\frac{1}{9} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{1}{9} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]

[Out]

(2*(2 - 3*x^2)^(1/4))/9 - (2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)])/9 + (2^(3/4)*A
rcTan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4)])/9 + Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4
) + Sqrt[2 - 3*x^2]]/(9*2^(1/4)) - Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqr
t[2 - 3*x^2]]/(9*2^(1/4))

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Rubi [A]  time = 0.373465, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{2}{9} \sqrt [4]{2-3 x^2}+\frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{9 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{9 \sqrt [4]{2}}-\frac{1}{9} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{1}{9} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(2 - 3*x^2)^(1/4))/9 - (2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)])/9 + (2^(3/4)*A
rcTan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4)])/9 + Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4
) + Sqrt[2 - 3*x^2]]/(9*2^(1/4)) - Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqr
t[2 - 3*x^2]]/(9*2^(1/4))

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Rubi in Sympy [A]  time = 31.4117, size = 141, normalized size = 0.89 \[ \frac{2 \sqrt [4]{- 3 x^{2} + 2}}{9} + \frac{2^{\frac{3}{4}} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{18} - \frac{2^{\frac{3}{4}} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{18} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{9} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-3*x**2 + 2)**(1/4)/9 + 2**(3/4)*log(-2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3
*x**2 + 2) + sqrt(2))/18 - 2**(3/4)*log(2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*
x**2 + 2) + sqrt(2))/18 - 2**(3/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4) - 1)/9 - 2
**(3/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4) + 1)/9

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Mathematica [C]  time = 0.0528609, size = 66, normalized size = 0.42 \[ -\frac{2 \left (-4 \left (\frac{2-3 x^2}{4-3 x^2}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{2}{4-3 x^2}\right )+9 x^2-6\right )}{27 \left (2-3 x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-6 + 9*x^2 - 4*((2 - 3*x^2)/(4 - 3*x^2))^(3/4)*Hypergeometric2F1[3/4, 3/4,
7/4, 2/(4 - 3*x^2)]))/(27*(2 - 3*x^2)^(3/4))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{-3\,{x}^{2}+4} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^3/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)

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Maxima [A]  time = 1.53668, size = 174, normalized size = 1.1 \[ -\frac{1}{9} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{9} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{18} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{1}{18} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{9} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 1/9*2^(3/4)*
arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 1/18*2^(3/4)*log(2^(3/4)
*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 1/18*2^(3/4)*log(-2^(3/4)*(-
3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/9*(-3*x^2 + 2)^(1/4)

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Fricas [A]  time = 0.241383, size = 261, normalized size = 1.65 \[ \frac{2}{9} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{2^{\frac{3}{4}}}{2^{\frac{3}{4}} + 2 \, \sqrt{2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) + \frac{2}{9} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{2^{\frac{3}{4}}}{2^{\frac{3}{4}} - 2 \, \sqrt{-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) - \frac{1}{18} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{1}{18} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{9} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*2^(3/4)*arctan(2^(3/4)/(2^(3/4) + 2*sqrt(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2
) + sqrt(-3*x^2 + 2)) + 2*(-3*x^2 + 2)^(1/4))) + 2/9*2^(3/4)*arctan(-2^(3/4)/(2^
(3/4) - 2*sqrt(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 2*(-3
*x^2 + 2)^(1/4))) - 1/18*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt
(-3*x^2 + 2)) + 1/18*2^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3
*x^2 + 2)) + 2/9*(-3*x^2 + 2)^(1/4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{3}}{3 x^{2} \left (- 3 x^{2} + 2\right )^{\frac{3}{4}} - 4 \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x**3/(3*x**2*(-3*x**2 + 2)**(3/4) - 4*(-3*x**2 + 2)**(3/4)), x)

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GIAC/XCAS [A]  time = 0.241099, size = 174, normalized size = 1.1 \[ -\frac{1}{9} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{9} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{18} \cdot 2^{\frac{3}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{1}{18} \cdot 2^{\frac{3}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{9} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 1/9*2^(3/4)*
arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 1/18*2^(3/4)*ln(2^(3/4)*
(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 1/18*2^(3/4)*ln(-2^(3/4)*(-3*
x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/9*(-3*x^2 + 2)^(1/4)