3.1082 \(\int \frac{x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{135} \left (3 x^2-1\right )^{5/4}+\frac{2}{9} \sqrt [4]{3 x^2-1}-\frac{4}{27} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{4}{27} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

[Out]

(2*(-1 + 3*x^2)^(1/4))/9 + (2*(-1 + 3*x^2)^(5/4))/135 - (4*ArcTan[(-1 + 3*x^2)^(
1/4)])/27 - (4*ArcTanh[(-1 + 3*x^2)^(1/4)])/27

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Rubi [A]  time = 0.146888, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2}{135} \left (3 x^2-1\right )^{5/4}+\frac{2}{9} \sqrt [4]{3 x^2-1}-\frac{4}{27} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{4}{27} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(-1 + 3*x^2)^(1/4))/9 + (2*(-1 + 3*x^2)^(5/4))/135 - (4*ArcTan[(-1 + 3*x^2)^(
1/4)])/27 - (4*ArcTanh[(-1 + 3*x^2)^(1/4)])/27

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Rubi in Sympy [A]  time = 14.5365, size = 56, normalized size = 0.89 \[ \frac{2 \left (3 x^{2} - 1\right )^{\frac{5}{4}}}{135} + \frac{2 \sqrt [4]{3 x^{2} - 1}}{9} - \frac{4 \operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{27} - \frac{4 \operatorname{atanh}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(3*x**2-2)/(3*x**2-1)**(3/4),x)

[Out]

2*(3*x**2 - 1)**(5/4)/135 + 2*(3*x**2 - 1)**(1/4)/9 - 4*atan((3*x**2 - 1)**(1/4)
)/27 - 4*atanh((3*x**2 - 1)**(1/4))/27

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Mathematica [C]  time = 0.0706535, size = 69, normalized size = 1.1 \[ \frac{2 \left (-20 \left (\frac{1-3 x^2}{2-3 x^2}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2-3 x^2}\right )+27 x^4+117 x^2-42\right )}{405 \left (3 x^2-1\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-42 + 117*x^2 + 27*x^4 - 20*((1 - 3*x^2)/(2 - 3*x^2))^(3/4)*Hypergeometric2F
1[3/4, 3/4, 7/4, (2 - 3*x^2)^(-1)]))/(405*(-1 + 3*x^2)^(3/4))

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \[ \int{\frac{{x}^{5}}{3\,{x}^{2}-2} \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(3*x^2-2)/(3*x^2-1)^(3/4),x)

[Out]

int(x^5/(3*x^2-2)/(3*x^2-1)^(3/4),x)

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Maxima [A]  time = 1.50096, size = 85, normalized size = 1.35 \[ \frac{2}{135} \,{\left (3 \, x^{2} - 1\right )}^{\frac{5}{4}} + \frac{2}{9} \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - \frac{4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x, algorithm="maxima")

[Out]

2/135*(3*x^2 - 1)^(5/4) + 2/9*(3*x^2 - 1)^(1/4) - 4/27*arctan((3*x^2 - 1)^(1/4))
- 2/27*log((3*x^2 - 1)^(1/4) + 1) + 2/27*log((3*x^2 - 1)^(1/4) - 1)

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Fricas [A]  time = 0.22489, size = 80, normalized size = 1.27 \[ \frac{2}{135} \,{\left (3 \, x^{2} + 14\right )}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - \frac{4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x, algorithm="fricas")

[Out]

2/135*(3*x^2 + 14)*(3*x^2 - 1)^(1/4) - 4/27*arctan((3*x^2 - 1)^(1/4)) - 2/27*log
((3*x^2 - 1)^(1/4) + 1) + 2/27*log((3*x^2 - 1)^(1/4) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(3*x**2-2)/(3*x**2-1)**(3/4),x)

[Out]

Integral(x**5/((3*x**2 - 2)*(3*x**2 - 1)**(3/4)), x)

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GIAC/XCAS [A]  time = 0.249449, size = 86, normalized size = 1.37 \[ \frac{2}{135} \,{\left (3 \, x^{2} - 1\right )}^{\frac{5}{4}} + \frac{2}{9} \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - \frac{4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{2}{27} \,{\rm ln}\left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{2}{27} \,{\rm ln}\left ({\left |{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x, algorithm="giac")

[Out]

2/135*(3*x^2 - 1)^(5/4) + 2/9*(3*x^2 - 1)^(1/4) - 4/27*arctan((3*x^2 - 1)^(1/4))
- 2/27*ln((3*x^2 - 1)^(1/4) + 1) + 2/27*ln(abs((3*x^2 - 1)^(1/4) - 1))