Optimal. Leaf size=131 \[ -m \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 b m n \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-2 b^2 m n^2 \text{PolyLog}\left (4,-\frac{f x}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac{m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
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Rubi [A] time = 0.141113, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2375, 2317, 2374, 2383, 6589} \[ -m \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 b m n \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-2 b^2 m n^2 \text{PolyLog}\left (4,-\frac{f x}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac{m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
Antiderivative was successfully verified.
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Rule 2375
Rule 2317
Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac{(f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx}{3 b n}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{3 b n}+m \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{3 b n}-m \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )+(2 b m n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{3 b n}-m \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )+2 b m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )-\left (2 b^2 m n^2\right ) \int \frac{\text{Li}_3\left (-\frac{f x}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{3 b n}-m \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )+2 b m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )-2 b^2 m n^2 \text{Li}_4\left (-\frac{f x}{e}\right )\\ \end{align*}
Mathematica [B] time = 0.161831, size = 329, normalized size = 2.51 \[ -m \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 b m n \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-2 b^2 m n^2 \text{PolyLog}\left (4,-\frac{f x}{e}\right )+a^2 \log (x) \log \left (d (e+f x)^m\right )-a^2 m \log (x) \log \left (\frac{f x}{e}+1\right )+2 a b \log (x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-2 a b m \log (x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )-a b n \log ^2(x) \log \left (d (e+f x)^m\right )+a b m n \log ^2(x) \log \left (\frac{f x}{e}+1\right )-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )-b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )+\frac{1}{3} b^2 n^2 \log ^3(x) \log \left (d (e+f x)^m\right )-\frac{1}{3} b^2 m n^2 \log ^3(x) \log \left (\frac{f x}{e}+1\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.796, size = 21792, normalized size = 166.4 \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{1}{3} \,{\left (b^{2} n^{2} \log \left (x\right )^{3} + 3 \, b^{2} \log \left (x\right ) \log \left (x^{n}\right )^{2} - 3 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )^{2} - 3 \,{\left (b^{2} n \log \left (x\right )^{2} - 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x\right )\right )} \log \left (x^{n}\right ) + 3 \,{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2}\right )} \log \left (x\right )\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - \int \frac{b^{2} f m n^{2} x \log \left (x\right )^{3} - 3 \, b^{2} e \log \left (c\right )^{2} \log \left (d\right ) - 6 \, a b e \log \left (c\right ) \log \left (d\right ) - 3 \, a^{2} e \log \left (d\right ) - 3 \,{\left (b^{2} f m n \log \left (c\right ) + a b f m n\right )} x \log \left (x\right )^{2} + 3 \,{\left (b^{2} f m \log \left (c\right )^{2} + 2 \, a b f m \log \left (c\right ) + a^{2} f m\right )} x \log \left (x\right ) + 3 \,{\left (b^{2} f m x \log \left (x\right ) - b^{2} f x \log \left (d\right ) - b^{2} e \log \left (d\right )\right )} \log \left (x^{n}\right )^{2} - 3 \,{\left (b^{2} f \log \left (c\right )^{2} \log \left (d\right ) + 2 \, a b f \log \left (c\right ) \log \left (d\right ) + a^{2} f \log \left (d\right )\right )} x - 3 \,{\left (b^{2} f m n x \log \left (x\right )^{2} + 2 \, b^{2} e \log \left (c\right ) \log \left (d\right ) + 2 \, a b e \log \left (d\right ) - 2 \,{\left (b^{2} f m \log \left (c\right ) + a b f m\right )} x \log \left (x\right ) + 2 \,{\left (b^{2} f \log \left (c\right ) \log \left (d\right ) + a b f \log \left (d\right )\right )} x\right )} \log \left (x^{n}\right )}{3 \,{\left (f x^{2} + e x\right )}}\,{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{{\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left ({\left (f x + e\right )}^{m} d\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x + e\right )}^{m} d\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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