Optimal. Leaf size=234 \[ -\frac{2 b e^3 g n^2 \text{PolyLog}\left (2,\frac{d}{d+e x}\right )}{3 d^3}+\frac{e^3 n \log \left (1-\frac{d}{d+e x}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{3 d^3}+\frac{e^2 n (d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{3 d^3 x}-\frac{e n \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{6 d x^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}-\frac{b e^2 g n^2}{3 d^2 x}-\frac{b e^3 g n^2 \log (x)}{d^3}+\frac{b e^3 g n^2 \log (d+e x)}{3 d^3} \]
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Rubi [A] time = 0.822259, antiderivative size = 365, normalized size of antiderivative = 1.56, number of steps used = 25, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2439, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ \frac{2 b e^3 g n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{3 d^3}-\frac{e^3 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{6 b d^3}+\frac{e^3 g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3}+\frac{e^2 g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}-\frac{e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 d x^2}-\frac{b e^3 \left (g \log \left (c (d+e x)^n\right )+f\right )^2}{6 d^3 g}+\frac{b e^3 n \log \left (-\frac{e x}{d}\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 d^3}+\frac{b e^2 n (d+e x) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 d^3 x}-\frac{b e n \left (g \log \left (c (d+e x)^n\right )+f\right )}{6 d x^2}-\frac{b e^2 g n^2}{3 d^2 x}-\frac{b e^3 g n^2 \log (x)}{d^3}+\frac{b e^3 g n^2 \log (d+e x)}{3 d^3} \]
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
Rule 2319
Rule 44
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^4} \, 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 )}{3 x^3}+\frac{1}{3} (b e n) \int \frac{f+g \log \left (c (d+e x)^n\right )}{x^3 (d+e x)} \, dx+\frac{1}{3} (e g n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x^3 (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 )}{3 x^3}+\frac{1}{3} (b n) \operatorname{Subst}\left (\int \frac{f+g \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e x\right )+\frac{1}{3} (g n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, 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 )}{3 x^3}+\frac{(b n) \operatorname{Subst}\left (\int \frac{f+g \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e x\right )}{3 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 )^2} \, dx,x,d+e x\right )}{3 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 )^3} \, dx,x,d+e x\right )}{3 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 )^2} \, dx,x,d+e x\right )}{3 d}\\ &=-\frac{e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 d x^2}-\frac{b e n \left (f+g \log \left (c (d+e x)^n\right )\right )}{6 d x^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 )}{3 x^3}-\frac{(b e 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 )}{3 d^2}+\frac{\left (b e^2 n\right ) \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 )}{3 d^2}-\frac{(e 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 )}{3 d^2}+\frac{\left (e^2 g n\right ) \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 )}{3 d^2}+2 \frac{\left (b e g n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x\right )}{6 d}\\ &=-\frac{e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac{e^2 g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}-\frac{b e n \left (f+g \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac{b e^2 n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 d^3 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 )}{3 x^3}+\frac{\left (b e^2 n\right ) \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 )}{3 d^3}-\frac{\left (b e^3 n\right ) \operatorname{Subst}\left (\int \frac{f+g \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{3 d^3}+\frac{\left (e^2 g n\right ) \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 )}{3 d^3}-\frac{\left (e^3 g n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{3 d^3}+2 \frac{\left (b e g n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d (d-x)^2}+\frac{e^2}{d^2 (d-x)}+\frac{e^2}{d^2 x}\right ) \, dx,x,d+e x\right )}{6 d}-2 \frac{\left (b e^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{3 d^3}\\ &=-\frac{2 b e^3 g n^2 \log (x)}{3 d^3}+2 \left (-\frac{b e^2 g n^2}{6 d^2 x}-\frac{b e^3 g n^2 \log (x)}{6 d^3}+\frac{b e^3 g n^2 \log (d+e x)}{6 d^3}\right )-\frac{e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac{e^2 g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}+\frac{e^3 g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3}-\frac{e^3 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{6 b d^3}-\frac{b e n \left (f+g \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac{b e^2 n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}+\frac{b e^3 n \log \left (-\frac{e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 d^3}-\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 )}{3 x^3}-\frac{b e^3 \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{6 d^3 g}-2 \frac{\left (b e^3 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{3 d^3}\\ &=-\frac{2 b e^3 g n^2 \log (x)}{3 d^3}+2 \left (-\frac{b e^2 g n^2}{6 d^2 x}-\frac{b e^3 g n^2 \log (x)}{6 d^3}+\frac{b e^3 g n^2 \log (d+e x)}{6 d^3}\right )-\frac{e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac{e^2 g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}+\frac{e^3 g n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3}-\frac{e^3 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{6 b d^3}-\frac{b e n \left (f+g \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac{b e^2 n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}+\frac{b e^3 n \log \left (-\frac{e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 d^3}-\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 )}{3 x^3}-\frac{b e^3 \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{6 d^3 g}+\frac{2 b e^3 g n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{3 d^3}\\ \end{align*}
Mathematica [A] time = 0.176225, size = 351, normalized size = 1.5 \[ \frac{2 b e^3 g n^2 \text{PolyLog}\left (2,\frac{d+e x}{d}\right )}{3 d^3}-\frac{a g \log \left (c (d+e x)^n\right )}{3 x^3}+\frac{1}{3} a e g n \left (\frac{e^2 \log (x)}{d^3}-\frac{e^2 \log (d+e x)}{d^3}+\frac{e}{d^2 x}-\frac{1}{2 d x^2}\right )-\frac{a f}{3 x^3}-\frac{b e^3 g \log ^2\left (c (d+e x)^n\right )}{3 d^3}+\frac{2 b e^3 g n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{3 d^3}+\frac{2 b e^2 g n \log \left (c (d+e x)^n\right )}{3 d^2 x}-\frac{b f \log \left (c (d+e x)^n\right )}{3 x^3}-\frac{b g \log ^2\left (c (d+e x)^n\right )}{3 x^3}-\frac{b e g n \log \left (c (d+e x)^n\right )}{3 d x^2}+\frac{1}{3} b e f n \left (\frac{e^2 \log (x)}{d^3}-\frac{e^2 \log (d+e x)}{d^3}+\frac{e}{d^2 x}-\frac{1}{2 d x^2}\right )-\frac{b e^2 g n^2}{3 d^2 x}-\frac{b e^3 g n^2 \log (x)}{d^3}+\frac{b e^3 g n^2 \log (d+e x)}{d^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.539, size = 1437, normalized size = 6.1 \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}{6} \, b e f n{\left (\frac{2 \, e^{2} \log \left (e x + d\right )}{d^{3}} - \frac{2 \, e^{2} \log \left (x\right )}{d^{3}} - \frac{2 \, e x - d}{d^{2} x^{2}}\right )} - \frac{1}{6} \, a e g n{\left (\frac{2 \, e^{2} \log \left (e x + d\right )}{d^{3}} - \frac{2 \, e^{2} \log \left (x\right )}{d^{3}} - \frac{2 \, e x - d}{d^{2} x^{2}}\right )} - \frac{1}{3} \, b g{\left (\frac{\log \left ({\left (e x + d\right )}^{n}\right )^{2}}{x^{3}} - 3 \, \int \frac{3 \, e x \log \left (c\right )^{2} + 3 \, d \log \left (c\right )^{2} + 2 \,{\left ({\left (e n + 3 \, e \log \left (c\right )\right )} x + 3 \, d \log \left (c\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{3 \,{\left (e x^{5} + d x^{4}\right )}}\,{d x}\right )} - \frac{b f \log \left ({\left (e x + d\right )}^{n} c\right )}{3 \, x^{3}} - \frac{a g \log \left ({\left (e x + d\right )}^{n} c\right )}{3 \, x^{3}} - \frac{a f}{3 \, x^{3}} \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^{4}}, 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^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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