Optimal. Leaf size=158 \[ -\frac{6 b^2 n^2 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{3 b n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}+\frac{6 b^3 n^3 \text{PolyLog}\left (4,-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g} \]
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Rubi [A] time = 0.179892, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2396, 2433, 2374, 2383, 6589} \[ -\frac{6 b^2 n^2 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{3 b n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}+\frac{6 b^3 n^3 \text{PolyLog}\left (4,-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g} \]
Antiderivative was successfully verified.
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Rule 2396
Rule 2433
Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}-\frac{(3 b e n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}-\frac{(3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac{e \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}+\frac{3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}-\frac{\left (6 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}+\frac{3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}-\frac{6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{\left (6 b^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}+\frac{3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}-\frac{6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{6 b^3 n^3 \text{Li}_4\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}\\ \end{align*}
Mathematica [B] time = 0.174223, size = 335, normalized size = 2.12 \[ \frac{6 b^2 n^2 \left (-\text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right )+\log (d+e x) \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )+\frac{1}{2} \log ^2(d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )+3 b n \left (\text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )+\log (d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+b^3 n^3 \left (6 \text{PolyLog}\left (4,\frac{g (d+e x)}{d g-e f}\right )+3 \log ^2(d+e x) \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )-6 \log (d+e x) \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right )+\log ^3(d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )\right )+\log (f+g x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^3}{g} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.194, size = 9538, normalized size = 60.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{3} \log \left (g x + f\right )}{g} + \int \frac{b^{3} \log \left ({\left (e x + d\right )}^{n}\right )^{3} + b^{3} \log \left (c\right )^{3} + 3 \, a b^{2} \log \left (c\right )^{2} + 3 \, a^{2} b \log \left (c\right ) + 3 \,{\left (b^{3} \log \left (c\right ) + a b^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 3 \,{\left (b^{3} \log \left (c\right )^{2} + 2 \, a b^{2} \log \left (c\right ) + a^{2} b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{3}}{g x + f}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}{f + g x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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