3.301 \(\int \frac{1}{x \left (-27+2 x^7\right )^{2/3}} \, dx\)

Optimal. Leaf size=59 \[ \frac{1}{42} \log \left (\sqrt [3]{2 x^7-27}+3\right )-\frac{\tan ^{-1}\left (\frac{3-2 \sqrt [3]{2 x^7-27}}{3 \sqrt{3}}\right )}{21 \sqrt{3}}-\frac{\log (x)}{18} \]

[Out]

-ArcTan[(3 - 2*(-27 + 2*x^7)^(1/3))/(3*Sqrt[3])]/(21*Sqrt[3]) - Log[x]/18 + Log[
3 + (-27 + 2*x^7)^(1/3)]/42

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Rubi [A]  time = 0.0804597, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{42} \log \left (\sqrt [3]{2 x^7-27}+3\right )-\frac{\tan ^{-1}\left (\frac{3-2 \sqrt [3]{2 x^7-27}}{3 \sqrt{3}}\right )}{21 \sqrt{3}}-\frac{\log (x)}{18} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(-27 + 2*x^7)^(2/3)),x]

[Out]

-ArcTan[(3 - 2*(-27 + 2*x^7)^(1/3))/(3*Sqrt[3])]/(21*Sqrt[3]) - Log[x]/18 + Log[
3 + (-27 + 2*x^7)^(1/3)]/42

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Rubi in Sympy [A]  time = 2.34342, size = 51, normalized size = 0.86 \[ - \frac{\log{\left (x^{7} \right )}}{126} + \frac{\log{\left (\sqrt [3]{2 x^{7} - 27} + 3 \right )}}{42} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2 x^{7} - 27}}{9} - \frac{1}{3}\right ) \right )}}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(2*x**7-27)**(2/3),x)

[Out]

-log(x**7)/126 + log((2*x**7 - 27)**(1/3) + 3)/42 + sqrt(3)*atan(sqrt(3)*(2*(2*x
**7 - 27)**(1/3)/9 - 1/3))/63

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Mathematica [C]  time = 0.0272546, size = 43, normalized size = 0.73 \[ -\frac{3 \left (2-\frac{27}{x^7}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{27}{2 x^7}\right )}{14 \left (4 x^7-54\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(-27 + 2*x^7)^(2/3)),x]

[Out]

(-3*(2 - 27/x^7)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, 27/(2*x^7)])/(14*(-54 +
4*x^7)^(2/3))

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Maple [C]  time = 0.113, size = 74, normalized size = 1.3 \[{\frac{1}{63\,\Gamma \left ( 2/3 \right ) } \left ( -{\it signum} \left ( -1+{\frac{2\,{x}^{7}}{27}} \right ) \right ) ^{{\frac{2}{3}}} \left ( \left ({\frac{\pi \,\sqrt{3}}{6}}-{\frac{9\,\ln \left ( 3 \right ) }{2}}+7\,\ln \left ( x \right ) +\ln \left ( 2 \right ) +i\pi \right ) \Gamma \left ({\frac{2}{3}} \right ) +{\frac{4\,\Gamma \left ( 2/3 \right ){x}^{7}}{81}{\mbox{$_3$F$_2$}(1,1,{\frac{5}{3}};\,2,2;\,{\frac{2\,{x}^{7}}{27}})}} \right ) \left ({\it signum} \left ( -1+{\frac{2\,{x}^{7}}{27}} \right ) \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(2*x^7-27)^(2/3),x)

[Out]

1/63/signum(-1+2/27*x^7)^(2/3)*(-signum(-1+2/27*x^7))^(2/3)*((1/6*Pi*3^(1/2)-9/2
*ln(3)+7*ln(x)+ln(2)+I*Pi)*GAMMA(2/3)+4/81*GAMMA(2/3)*x^7*hypergeom([1,1,5/3],[2
,2],2/27*x^7))/GAMMA(2/3)

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Maxima [A]  time = 1.72589, size = 86, normalized size = 1.46 \[ \frac{1}{63} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (2 \,{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} - 3\right )}\right ) - \frac{1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac{2}{3}} - 3 \,{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 9\right ) + \frac{1}{63} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*x^7 - 27)^(2/3)*x),x, algorithm="maxima")

[Out]

1/63*sqrt(3)*arctan(1/9*sqrt(3)*(2*(2*x^7 - 27)^(1/3) - 3)) - 1/126*log((2*x^7 -
27)^(2/3) - 3*(2*x^7 - 27)^(1/3) + 9) + 1/63*log((2*x^7 - 27)^(1/3) + 3)

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Fricas [A]  time = 0.225627, size = 99, normalized size = 1.68 \[ -\frac{1}{378} \, \sqrt{3}{\left (\sqrt{3} \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac{2}{3}} - 3 \,{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 9\right ) - 2 \, \sqrt{3} \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 3\right ) - 6 \, \arctan \left (\frac{2}{9} \, \sqrt{3}{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*x^7 - 27)^(2/3)*x),x, algorithm="fricas")

[Out]

-1/378*sqrt(3)*(sqrt(3)*log((2*x^7 - 27)^(2/3) - 3*(2*x^7 - 27)^(1/3) + 9) - 2*s
qrt(3)*log((2*x^7 - 27)^(1/3) + 3) - 6*arctan(2/9*sqrt(3)*(2*x^7 - 27)^(1/3) - 1
/3*sqrt(3)))

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Sympy [A]  time = 1.72816, size = 42, normalized size = 0.71 \[ - \frac{\sqrt [3]{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{27 e^{2 i \pi }}{2 x^{7}}} \right )}}{14 x^{\frac{14}{3}} \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(2*x**7-27)**(2/3),x)

[Out]

-2**(1/3)*gamma(2/3)*hyper((2/3, 2/3), (5/3,), 27*exp_polar(2*I*pi)/(2*x**7))/(1
4*x**(14/3)*gamma(5/3))

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GIAC/XCAS [A]  time = 0.203055, size = 86, normalized size = 1.46 \[ \frac{1}{63} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (2 \,{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} - 3\right )}\right ) - \frac{1}{126} \,{\rm ln}\left ({\left (2 \, x^{7} - 27\right )}^{\frac{2}{3}} - 3 \,{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 9\right ) + \frac{1}{63} \,{\rm ln}\left ({\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*x^7 - 27)^(2/3)*x),x, algorithm="giac")

[Out]

1/63*sqrt(3)*arctan(1/9*sqrt(3)*(2*(2*x^7 - 27)^(1/3) - 3)) - 1/126*ln((2*x^7 -
27)^(2/3) - 3*(2*x^7 - 27)^(1/3) + 9) + 1/63*ln((2*x^7 - 27)^(1/3) + 3)