3.1081 \(\int \frac{x^7}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=78 \[ \frac{2}{729} \left (3 x^2-1\right )^{9/4}+\frac{8}{405} \left (3 x^2-1\right )^{5/4}+\frac{14}{81} \sqrt [4]{3 x^2-1}-\frac{8}{81} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{8}{81} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

[Out]

(14*(-1 + 3*x^2)^(1/4))/81 + (8*(-1 + 3*x^2)^(5/4))/405 + (2*(-1 + 3*x^2)^(9/4))
/729 - (8*ArcTan[(-1 + 3*x^2)^(1/4)])/81 - (8*ArcTanh[(-1 + 3*x^2)^(1/4)])/81

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Rubi [A]  time = 0.159706, antiderivative size = 78, 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 \[ \frac{2}{729} \left (3 x^2-1\right )^{9/4}+\frac{8}{405} \left (3 x^2-1\right )^{5/4}+\frac{14}{81} \sqrt [4]{3 x^2-1}-\frac{8}{81} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{8}{81} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^7/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]

[Out]

(14*(-1 + 3*x^2)^(1/4))/81 + (8*(-1 + 3*x^2)^(5/4))/405 + (2*(-1 + 3*x^2)^(9/4))
/729 - (8*ArcTan[(-1 + 3*x^2)^(1/4)])/81 - (8*ArcTanh[(-1 + 3*x^2)^(1/4)])/81

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Rubi in Sympy [A]  time = 15.8558, size = 70, normalized size = 0.9 \[ \frac{2 \left (3 x^{2} - 1\right )^{\frac{9}{4}}}{729} + \frac{8 \left (3 x^{2} - 1\right )^{\frac{5}{4}}}{405} + \frac{14 \sqrt [4]{3 x^{2} - 1}}{81} - \frac{8 \operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{81} - \frac{8 \operatorname{atanh}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(3*x**2-2)/(3*x**2-1)**(3/4),x)

[Out]

2*(3*x**2 - 1)**(9/4)/729 + 8*(3*x**2 - 1)**(5/4)/405 + 14*(3*x**2 - 1)**(1/4)/8
1 - 8*atan((3*x**2 - 1)**(1/4))/81 - 8*atanh((3*x**2 - 1)**(1/4))/81

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Mathematica [C]  time = 0.0881131, size = 74, normalized size = 0.95 \[ \frac{2 \left (-120 \left (\frac{1-3 x^2}{2-3 x^2}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2-3 x^2}\right )+135 x^6+189 x^4+774 x^2-284\right )}{3645 \left (3 x^2-1\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]

[Out]

(2*(-284 + 774*x^2 + 189*x^4 + 135*x^6 - 120*((1 - 3*x^2)/(2 - 3*x^2))^(3/4)*Hyp
ergeometric2F1[3/4, 3/4, 7/4, (2 - 3*x^2)^(-1)]))/(3645*(-1 + 3*x^2)^(3/4))

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Maple [F]  time = 0.106, size = 0, normalized size = 0. \[ \int{\frac{{x}^{7}}{3\,{x}^{2}-2} \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(3*x^2-2)/(3*x^2-1)^(3/4),x)

[Out]

int(x^7/(3*x^2-2)/(3*x^2-1)^(3/4),x)

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Maxima [A]  time = 1.5056, size = 100, normalized size = 1.28 \[ \frac{2}{729} \,{\left (3 \, x^{2} - 1\right )}^{\frac{9}{4}} + \frac{8}{405} \,{\left (3 \, x^{2} - 1\right )}^{\frac{5}{4}} + \frac{14}{81} \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - \frac{8}{81} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x, algorithm="maxima")

[Out]

2/729*(3*x^2 - 1)^(9/4) + 8/405*(3*x^2 - 1)^(5/4) + 14/81*(3*x^2 - 1)^(1/4) - 8/
81*arctan((3*x^2 - 1)^(1/4)) - 4/81*log((3*x^2 - 1)^(1/4) + 1) + 4/81*log((3*x^2
- 1)^(1/4) - 1)

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Fricas [A]  time = 0.22634, size = 86, normalized size = 1.1 \[ \frac{2}{3645} \,{\left (45 \, x^{4} + 78 \, x^{2} + 284\right )}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - \frac{8}{81} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x, algorithm="fricas")

[Out]

2/3645*(45*x^4 + 78*x^2 + 284)*(3*x^2 - 1)^(1/4) - 8/81*arctan((3*x^2 - 1)^(1/4)
) - 4/81*log((3*x^2 - 1)^(1/4) + 1) + 4/81*log((3*x^2 - 1)^(1/4) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(3*x**2-2)/(3*x**2-1)**(3/4),x)

[Out]

Integral(x**7/((3*x**2 - 2)*(3*x**2 - 1)**(3/4)), x)

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GIAC/XCAS [A]  time = 0.236653, size = 101, normalized size = 1.29 \[ \frac{2}{729} \,{\left (3 \, x^{2} - 1\right )}^{\frac{9}{4}} + \frac{8}{405} \,{\left (3 \, x^{2} - 1\right )}^{\frac{5}{4}} + \frac{14}{81} \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - \frac{8}{81} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{4}{81} \,{\rm ln}\left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{4}{81} \,{\rm ln}\left ({\left |{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x, algorithm="giac")

[Out]

2/729*(3*x^2 - 1)^(9/4) + 8/405*(3*x^2 - 1)^(5/4) + 14/81*(3*x^2 - 1)^(1/4) - 8/
81*arctan((3*x^2 - 1)^(1/4)) - 4/81*ln((3*x^2 - 1)^(1/4) + 1) + 4/81*ln(abs((3*x
^2 - 1)^(1/4) - 1))