Integrand size = 24, antiderivative size = 139 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x} \, dx=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )+\frac {9}{2} b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )-9 b^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {e x^{2/3}}{d}\right )+9 b^3 n^3 \operatorname {PolyLog}\left (4,1+\frac {e x^{2/3}}{d}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2504, 2443, 2481, 2421, 2430, 6724} \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x} \, dx=-9 b^2 n^2 \operatorname {PolyLog}\left (3,\frac {x^{2/3} e}{d}+1\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {9}{2} b n \operatorname {PolyLog}\left (2,\frac {x^{2/3} e}{d}+1\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {3}{2} \log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+9 b^3 n^3 \operatorname {PolyLog}\left (4,\frac {x^{2/3} e}{d}+1\right ) \]
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Rule 2421
Rule 2430
Rule 2443
Rule 2481
Rule 2504
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {3}{2} \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x} \, dx,x,x^{2/3}\right ) \\ & = \frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )-\frac {1}{2} (9 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx,x,x^{2/3}\right ) \\ & = \frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )-\frac {1}{2} (9 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right ) \\ & = \frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )+\frac {9}{2} b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-\left (9 b^2 n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right ) \\ & = \frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )+\frac {9}{2} b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-9 b^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e x^{2/3}}{d}\right )+\left (9 b^3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right ) \\ & = \frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log \left (-\frac {e x^{2/3}}{d}\right )+\frac {9}{2} b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-9 b^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e x^{2/3}}{d}\right )+9 b^3 n^3 \text {Li}_4\left (1+\frac {e x^{2/3}}{d}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(339\) vs. \(2(139)=278\).
Time = 0.14 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.44 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x} \, dx=\left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \log (x)+3 b n \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (\left (\log \left (d+e x^{2/3}\right )-\log \left (1+\frac {e x^{2/3}}{d}\right )\right ) \log (x)-\frac {3}{2} \operatorname {PolyLog}\left (2,-\frac {e x^{2/3}}{d}\right )\right )+\frac {9}{2} b^2 n^2 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (\log ^2\left (d+e x^{2/3}\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+2 \log \left (d+e x^{2/3}\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e x^{2/3}}{d}\right )\right )+\frac {3}{2} b^3 n^3 \left (\log ^3\left (d+e x^{2/3}\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+3 \log ^2\left (d+e x^{2/3}\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )-6 \log \left (d+e x^{2/3}\right ) \operatorname {PolyLog}\left (3,1+\frac {e x^{2/3}}{d}\right )+6 \operatorname {PolyLog}\left (4,1+\frac {e x^{2/3}}{d}\right )\right ) \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{3}}{x}d x\]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x^{\frac {2}{3}}\right )^{n} \right )}\right )^{3}}{x}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3}{x} \,d x \]
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