3.1033 \(\int \frac{x^3}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=106 \[ \frac{2}{27} \left (2-3 x^2\right )^{3/4}+\frac{2}{9} \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac{2}{9} \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \]

[Out]

(2*(2 - 3*x^2)^(3/4))/27 + (2*2^(1/4)*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4
)*(2 - 3*x^2)^(1/4))])/9 + (2*2^(1/4)*ArcTanh[(Sqrt[2] + Sqrt[2 - 3*x^2])/(2^(3/
4)*(2 - 3*x^2)^(1/4))])/9

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Rubi [A]  time = 0.145825, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2}{27} \left (2-3 x^2\right )^{3/4}+\frac{2}{9} \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac{2}{9} \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(2 - 3*x^2)^(3/4))/27 + (2*2^(1/4)*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4
)*(2 - 3*x^2)^(1/4))])/9 + (2*2^(1/4)*ArcTanh[(Sqrt[2] + Sqrt[2 - 3*x^2])/(2^(3/
4)*(2 - 3*x^2)^(1/4))])/9

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Rubi in Sympy [A]  time = 30.9311, size = 144, normalized size = 1.36 \[ \frac{2 \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{27} - \frac{\sqrt [4]{2} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{9} + \frac{\sqrt [4]{2} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{9} - \frac{2 \sqrt [4]{2} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{9} - \frac{2 \sqrt [4]{2} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-3*x**2 + 2)**(3/4)/27 - 2**(1/4)*log(-2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-
3*x**2 + 2) + sqrt(2))/9 + 2**(1/4)*log(2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*
x**2 + 2) + sqrt(2))/9 - 2*2**(1/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4) - 1)/9 -
2*2**(1/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4) + 1)/9

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Mathematica [C]  time = 0.0465022, size = 66, normalized size = 0.62 \[ \frac{24 \sqrt [4]{\frac{2-3 x^2}{4-3 x^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{2}{4-3 x^2}\right )-6 x^2+4}{27 \sqrt [4]{2-3 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(4 - 6*x^2 + 24*((2 - 3*x^2)/(4 - 3*x^2))^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4,
2/(4 - 3*x^2)])/(27*(2 - 3*x^2)^(1/4))

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{-3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50078, size = 174, normalized size = 1.64 \[ -\frac{2}{9} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{2}{9} \cdot 2^{\frac{1}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{1}{9} \cdot 2^{\frac{1}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{1}{9} \cdot 2^{\frac{1}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{27} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*2^(1/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 2/9*2^(1/4)*
arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) + 1/9*2^(1/4)*log(2^(3/4)*
(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 1/9*2^(1/4)*log(-2^(3/4)*(-3*
x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/27*(-3*x^2 + 2)^(3/4)

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Fricas [A]  time = 0.268293, size = 335, normalized size = 3.16 \[ \frac{2}{9} \cdot 8^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{8^{\frac{3}{4}} \sqrt{2}}{8^{\frac{3}{4}} \sqrt{2} + 4 \, \sqrt{8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} + 4 \, \sqrt{-3 \, x^{2} + 2}} + 8 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) + \frac{2}{9} \cdot 8^{\frac{1}{4}} \sqrt{2} \arctan \left (-\frac{8^{\frac{3}{4}} \sqrt{2}}{8^{\frac{3}{4}} \sqrt{2} - 2 \, \sqrt{-4 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 16 \, \sqrt{2} + 16 \, \sqrt{-3 \, x^{2} + 2}} - 8 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) + \frac{1}{18} \cdot 8^{\frac{1}{4}} \sqrt{2} \log \left (4 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 16 \, \sqrt{2} + 16 \, \sqrt{-3 \, x^{2} + 2}\right ) - \frac{1}{18} \cdot 8^{\frac{1}{4}} \sqrt{2} \log \left (-4 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 16 \, \sqrt{2} + 16 \, \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{27} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*8^(1/4)*sqrt(2)*arctan(8^(3/4)*sqrt(2)/(8^(3/4)*sqrt(2) + 4*sqrt(8^(3/4)*sqr
t(2)*(-3*x^2 + 2)^(1/4) + 4*sqrt(2) + 4*sqrt(-3*x^2 + 2)) + 8*(-3*x^2 + 2)^(1/4)
)) + 2/9*8^(1/4)*sqrt(2)*arctan(-8^(3/4)*sqrt(2)/(8^(3/4)*sqrt(2) - 2*sqrt(-4*8^
(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 16*sqrt(2) + 16*sqrt(-3*x^2 + 2)) - 8*(-3*x^2
+ 2)^(1/4))) + 1/18*8^(1/4)*sqrt(2)*log(4*8^(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) +
16*sqrt(2) + 16*sqrt(-3*x^2 + 2)) - 1/18*8^(1/4)*sqrt(2)*log(-4*8^(3/4)*sqrt(2)*
(-3*x^2 + 2)^(1/4) + 16*sqrt(2) + 16*sqrt(-3*x^2 + 2)) + 2/27*(-3*x^2 + 2)^(3/4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x**3/(3*x**2*(-3*x**2 + 2)**(1/4) - 4*(-3*x**2 + 2)**(1/4)), x)

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GIAC/XCAS [A]  time = 0.240753, size = 174, normalized size = 1.64 \[ -\frac{2}{9} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{2}{9} \cdot 2^{\frac{1}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{1}{9} \cdot 2^{\frac{1}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{1}{9} \cdot 2^{\frac{1}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{27} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*2^(1/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 2/9*2^(1/4)*
arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) + 1/9*2^(1/4)*ln(2^(3/4)*(
-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 1/9*2^(1/4)*ln(-2^(3/4)*(-3*x^
2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/27*(-3*x^2 + 2)^(3/4)