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

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 = 0.0371693, 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, Rules used = {266, 58, 618, 204, 31} \[ \frac{1}{42} \log \left (\sqrt [3]{2 x^7-27}+3\right )-\frac{\tan ^{-1}\left (\frac{3-2 \sqrt [3]{2 x^7-27}}{3 \sqrt{3}}\right )}{21 \sqrt{3}}-\frac{\log (x)}{18} \]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d
*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x
] && NegQ[(b*c - a*d)/b]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{x \left (-27+2 x^7\right )^{2/3}} \, dx &=\frac{1}{7} \operatorname{Subst}\left (\int \frac{1}{x (-27+2 x)^{2/3}} \, dx,x,x^7\right )\\ &=-\frac{\log (x)}{18}+\frac{1}{42} \operatorname{Subst}\left (\int \frac{1}{3+x} \, dx,x,\sqrt [3]{-27+2 x^7}\right )+\frac{1}{14} \operatorname{Subst}\left (\int \frac{1}{9-3 x+x^2} \, dx,x,\sqrt [3]{-27+2 x^7}\right )\\ &=-\frac{\log (x)}{18}+\frac{1}{42} \log \left (3+\sqrt [3]{-27+2 x^7}\right )-\frac{1}{7} \operatorname{Subst}\left (\int \frac{1}{-27-x^2} \, dx,x,-3+2 \sqrt [3]{-27+2 x^7}\right )\\ &=-\frac{\tan ^{-1}\left (\frac{3-2 \sqrt [3]{-27+2 x^7}}{3 \sqrt{3}}\right )}{21 \sqrt{3}}-\frac{\log (x)}{18}+\frac{1}{42} \log \left (3+\sqrt [3]{-27+2 x^7}\right )\\ \end{align*}

Mathematica [A]  time = 0.0221199, size = 84, normalized size = 1.42 \[ \frac{1}{63} \log \left (\sqrt [3]{2 x^7-27}+3\right )-\frac{1}{126} \log \left (\left (2 x^7-27\right )^{2/3}-3 \sqrt [3]{2 x^7-27}+9\right )+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{2 x^7-27}-3}{3 \sqrt{3}}\right )}{21 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[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[3 + (-27 + 2*x^7)^(1/3)]/63 - Log[9 - 3*(-
27 + 2*x^7)^(1/3) + (-27 + 2*x^7)^(2/3)]/126

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Maple [C]  time = 0.058, size = 74, normalized size = 1.3 \begin{align*}{\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}}}} \end{align*}

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)*GAM
MA(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 = 1.4436, size = 86, normalized size = 1.46 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2*x^7-27)^(2/3),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 = 1.94792, size = 217, normalized size = 3.68 \begin{align*} \frac{1}{63} \, \sqrt{3} \arctan \left (\frac{2}{9} \, \sqrt{3}{\left (2 \, x^{7} - 27\right )}^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/63*sqrt(3)*arctan(2/9*sqrt(3)*(2*x^7 - 27)^(1/3) - 1/3*sqrt(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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Sympy [C]  time = 1.03546, size = 42, normalized size = 0.71 \begin{align*} - \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 )} \end{align*}

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))/(14*x**(14/3)*gamma(5/3))

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Giac [A]  time = 1.09344, size = 86, normalized size = 1.46 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)