3.1083 \(\int \frac{x^3}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=48 \[ \frac{2}{9} \sqrt [4]{3 x^2-1}-\frac{2}{9} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{2}{9} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.115188, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2}{9} \sqrt [4]{3 x^2-1}-\frac{2}{9} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{2}{9} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.2907, size = 42, normalized size = 0.88 \[ \frac{2 \sqrt [4]{3 x^{2} - 1}}{9} - \frac{2 \operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{9} - \frac{2 \operatorname{atanh}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(3*x**2 - 1)**(1/4)/9 - 2*atan((3*x**2 - 1)**(1/4))/9 - 2*atanh((3*x**2 - 1)**
(1/4))/9

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Mathematica [C]  time = 0.0280881, size = 34, normalized size = 0.71 \[ \frac{2}{9} \sqrt [4]{3 x^2-1} \left (1-2 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};3 x^2-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(-1 + 3*x^2)^(1/4)*(1 - 2*Hypergeometric2F1[1/4, 1, 5/4, -1 + 3*x^2]))/9

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{3\,{x}^{2}-2} \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50785, size = 70, normalized size = 1.46 \[ \frac{2}{9} \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - \frac{2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.226596, size = 70, normalized size = 1.46 \[ \frac{2}{9} \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - \frac{2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23715, size = 72, normalized size = 1.5 \[ \frac{2}{9} \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - \frac{2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{9} \,{\rm ln}\left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{9} \,{\rm ln}\left ({\left |{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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