Integrand size = 22, antiderivative size = 168 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx=-\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac {6 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e}{d (f+g x)}\right )}{d g}+\frac {6 b^3 e p^3 \operatorname {PolyLog}\left (3,1+\frac {e}{d (f+g x)}\right )}{d g} \]
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Time = 0.14 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2533, 2499, 2504, 2443, 2481, 2421, 6724} \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx=-\frac {6 b^2 e p^2 \operatorname {PolyLog}\left (2,\frac {e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )}{d g}-\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac {6 b^3 e p^3 \operatorname {PolyLog}\left (3,\frac {e}{d (f+g x)}+1\right )}{d g} \]
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Rule 2421
Rule 2443
Rule 2481
Rule 2499
Rule 2504
Rule 2533
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^p\right )\right )^3 \, dx,x,f+g x\right )}{g} \\ & = \frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac {(3 b e p) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x}\right )^p\right )\right )^2}{x} \, dx,x,f+g x\right )}{d g} \\ & = \frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac {(3 b e p) \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^p\right )\right )^2}{x} \, dx,x,\frac {1}{f+g x}\right )}{d g} \\ & = -\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac {\left (6 b^2 e^2 p^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^p\right )\right )}{d+e x} \, dx,x,\frac {1}{f+g x}\right )}{d g} \\ & = -\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac {\left (6 b^2 e p^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^p\right )\right ) \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+\frac {e}{f+g x}\right )}{d g} \\ & = -\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac {6 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \text {Li}_2\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}+\frac {\left (6 b^3 e p^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{f+g x}\right )}{d g} \\ & = -\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac {6 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \text {Li}_2\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}+\frac {6 b^3 e p^3 \text {Li}_3\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(441\) vs. \(2(168)=336\).
Time = 0.43 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.62 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx=\frac {3 b d p (f+g x) \log \left (d+\frac {e}{f+g x}\right ) \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2+d (f+g x) \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3+3 b e p \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2 \log (e+d (f+g x))+3 b^2 p^2 \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \left (d (f+g x) \log ^2\left (d+\frac {e}{f+g x}\right )+e \left (\log ^2\left (\frac {e}{d}+f+g x\right )+2 \left (\log (f+g x)-\log \left (\frac {e}{d}+f+g x\right )+\log \left (d+\frac {e}{f+g x}\right )\right ) \log (e+d (f+g x))-2 \left (\log (f+g x) \log \left (1+\frac {d (f+g x)}{e}\right )+\operatorname {PolyLog}\left (2,-\frac {d (f+g x)}{e}\right )\right )\right )\right )+b^3 p^3 \left (\log ^2\left (d+\frac {e}{f+g x}\right ) \left (-3 e \log \left (-\frac {e}{d f+d g x}\right )+(e+d f+d g x) \log \left (d+\frac {e}{f+g x}\right )\right )-6 e \log \left (d+\frac {e}{f+g x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e}{d f+d g x}\right )+6 e \operatorname {PolyLog}\left (3,1+\frac {e}{d f+d g x}\right )\right )}{d g} \]
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\[\int {\left (a +b \ln \left (c \left (d +\frac {e}{g x +f}\right )^{p}\right )\right )}^{3}d x\]
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\[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{3} \,d x } \]
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\[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx=\int \left (a + b \log {\left (c \left (d + \frac {e}{f + g x}\right )^{p} \right )}\right )^{3}\, dx \]
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\[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{3} \,d x } \]
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\[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{3} \,d x } \]
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Timed out. \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{f+g\,x}\right )}^p\right )\right )}^3 \,d x \]
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