Optimal. Leaf size=156 \[ \frac{b e^2 g n^2 \text{PolyLog}\left (2,\frac{d}{d+e x}\right )}{d^2}-\frac{e^2 n \log \left (1-\frac{d}{d+e x}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{2 d^2}-\frac{e n (d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{2 d^2 x}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{2 x^2}+\frac{b e^2 g n^2 \log (x)}{d^2} \]
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Rubi [A] time = 0.556868, antiderivative size = 265, normalized size of antiderivative = 1.7, number of steps used = 17, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used = {2439, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31} \[ -\frac{b e^2 g n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d^2}+\frac{e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 b d^2}-\frac{e^2 g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2}-\frac{e g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{2 x^2}+\frac{b e^2 \left (g \log \left (c (d+e x)^n\right )+f\right )^2}{4 d^2 g}-\frac{b e^2 n \log \left (-\frac{e x}{d}\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{2 d^2}-\frac{b e n (d+e x) \left (g \log \left (c (d+e x)^n\right )+f\right )}{2 d^2 x}+\frac{b e^2 g n^2 \log (x)}{d^2} \]
Antiderivative was successfully verified.
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Rule 2439
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^3} \, dx &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}+\frac{1}{2} (b e n) \int \frac{f+g \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx+\frac{1}{2} (e g n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}+\frac{1}{2} (b n) \operatorname{Subst}\left (\int \frac{f+g \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x\right )+\frac{1}{2} (g n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x\right )\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}+\frac{(b n) \operatorname{Subst}\left (\int \frac{f+g \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x\right )}{2 d}-\frac{(b e n) \operatorname{Subst}\left (\int \frac{f+g \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e x\right )}{2 d}+\frac{(g n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x\right )}{2 d}-\frac{(e g n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e x\right )}{2 d}\\ &=-\frac{e g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac{b e n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}-\frac{(b e n) \operatorname{Subst}\left (\int \frac{f+g \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{2 d^2}+\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{f+g \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{2 d^2}-\frac{(e g n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{2 d^2}+\frac{\left (e^2 g n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{2 d^2}+2 \frac{\left (b e g n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{2 d^2}\\ &=\frac{b e^2 g n^2 \log (x)}{d^2}-\frac{e g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac{e^2 g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2}+\frac{e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 b d^2}-\frac{b e n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac{b e^2 n \log \left (-\frac{e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 d^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}+\frac{b e^2 \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{4 d^2 g}+2 \frac{\left (b e^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{2 d^2}\\ &=\frac{b e^2 g n^2 \log (x)}{d^2}-\frac{e g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac{e^2 g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2}+\frac{e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 b d^2}-\frac{b e n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac{b e^2 n \log \left (-\frac{e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 d^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}+\frac{b e^2 \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{4 d^2 g}-\frac{b e^2 g n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.128616, size = 254, normalized size = 1.63 \[ b e g n \left (-\frac{e n \text{PolyLog}\left (2,\frac{d+e x}{d}\right )}{d^2}+\frac{e \log ^2\left (c (d+e x)^n\right )}{2 d^2 n}-\frac{e \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{d^2}-\frac{\log \left (c (d+e x)^n\right )}{d x}+\frac{e n \left (\frac{\log (x)}{d}-\frac{\log (d+e x)}{d}\right )}{d}\right )-\frac{a g \log \left (c (d+e x)^n\right )}{2 x^2}+\frac{1}{2} a e g n \left (-\frac{e \log (x)}{d^2}+\frac{e \log (d+e x)}{d^2}-\frac{1}{d x}\right )-\frac{a f}{2 x^2}-\frac{b f \log \left (c (d+e x)^n\right )}{2 x^2}-\frac{b g \log ^2\left (c (d+e x)^n\right )}{2 x^2}+\frac{1}{2} b e f n \left (-\frac{e \log (x)}{d^2}+\frac{e \log (d+e x)}{d^2}-\frac{1}{d x}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.593, size = 1201, normalized size = 7.7 \begin{align*} \text{result 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{1}{2} \, b e f n{\left (\frac{e \log \left (e x + d\right )}{d^{2}} - \frac{e \log \left (x\right )}{d^{2}} - \frac{1}{d x}\right )} + \frac{1}{2} \, a e g n{\left (\frac{e \log \left (e x + d\right )}{d^{2}} - \frac{e \log \left (x\right )}{d^{2}} - \frac{1}{d x}\right )} - \frac{1}{2} \, b g{\left (\frac{\log \left ({\left (e x + d\right )}^{n}\right )^{2}}{x^{2}} - 2 \, \int \frac{e x \log \left (c\right )^{2} + d \log \left (c\right )^{2} +{\left ({\left (e n + 2 \, e \log \left (c\right )\right )} x + 2 \, d \log \left (c\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{e x^{4} + d x^{3}}\,{d x}\right )} - \frac{b f \log \left ({\left (e x + d\right )}^{n} c\right )}{2 \, x^{2}} - \frac{a g \log \left ({\left (e x + d\right )}^{n} c\right )}{2 \, x^{2}} - \frac{a f}{2 \, x^{2}} \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 g \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a f +{\left (b f + a g\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}{x^{3}}, 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 ) \left (f + g \log{\left (c \left (d + e x\right )^{n} \right )}\right )}{x^{3}}\, 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 )}{\left (g \log \left ({\left (e x + d\right )}^{n} c\right ) + f\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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