Optimal. Leaf size=150 \[ \frac{2 b n r \text{PolyLog}\left (3,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac{r \text{PolyLog}\left (2,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac{2 b^2 n^2 r \text{PolyLog}\left (4,-\frac{f x^m}{e}\right )}{m^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \log \left (\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
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Rubi [A] time = 0.248844, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2375, 2337, 2374, 2383, 6589} \[ \frac{2 b n r \text{PolyLog}\left (3,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac{r \text{PolyLog}\left (2,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac{2 b^2 n^2 r \text{PolyLog}\left (4,-\frac{f x^m}{e}\right )}{m^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \log \left (\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
Antiderivative was successfully verified.
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Rule 2375
Rule 2337
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 \left (e+f x^m\right )^r\right )}{x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{(f m r) \int \frac{x^{-1+m} \left (a+b \log \left (c x^n\right )\right )^3}{e+f x^m} \, dx}{3 b n}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x^m}{e}\right )}{3 b n}+r \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^m}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x^m}{e}\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{m}+\frac{(2 b n r) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x^m}{e}\right )}{x} \, dx}{m}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x^m}{e}\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{m}+\frac{2 b n r \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x^m}{e}\right )}{m^2}-\frac{\left (2 b^2 n^2 r\right ) \int \frac{\text{Li}_3\left (-\frac{f x^m}{e}\right )}{x} \, dx}{m^2}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x^m}{e}\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{m}+\frac{2 b n r \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x^m}{e}\right )}{m^2}-\frac{2 b^2 n^2 r \text{Li}_4\left (-\frac{f x^m}{e}\right )}{m^3}\\ \end{align*}
Mathematica [B] time = 0.360166, size = 741, normalized size = 4.94 \[ \frac{b n r \log (x) \text{PolyLog}\left (2,-\frac{e x^{-m}}{f}\right ) \left (2 \left (a+b \log \left (c x^n\right )\right )-b n \log (x)\right )}{m}+\frac{r \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2}{m}+\frac{2 a b n r \text{PolyLog}\left (3,-\frac{e x^{-m}}{f}\right )}{m^2}+\frac{2 b^2 n r \log \left (c x^n\right ) \text{PolyLog}\left (3,-\frac{e x^{-m}}{f}\right )}{m^2}+\frac{2 b^2 n^2 r \text{PolyLog}\left (4,-\frac{e x^{-m}}{f}\right )}{m^3}+a^2 \log (x) \log \left (d \left (e+f x^m\right )^r\right )-a^2 r \log (x) \log \left (e+f x^m\right )+\frac{a^2 r \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}+2 a b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )-2 a b r \log (x) \log \left (c x^n\right ) \log \left (e+f x^m\right )+\frac{2 a b r \log \left (c x^n\right ) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-a b n \log ^2(x) \log \left (d \left (e+f x^m\right )^r\right )-a b n r \log ^2(x) \log \left (\frac{e x^{-m}}{f}+1\right )+2 a b n r \log ^2(x) \log \left (e+f x^m\right )-\frac{2 a b n r \log (x) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac{1}{3} a b m n r \log ^3(x)-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )+b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )-b^2 n r \log ^2(x) \log \left (c x^n\right ) \log \left (\frac{e x^{-m}}{f}+1\right )+2 b^2 n r \log ^2(x) \log \left (c x^n\right ) \log \left (e+f x^m\right )-b^2 r \log (x) \log ^2\left (c x^n\right ) \log \left (e+f x^m\right )+\frac{b^2 r \log ^2\left (c x^n\right ) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac{2 b^2 n r \log (x) \log \left (c x^n\right ) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac{1}{3} b^2 m n r \log ^3(x) \log \left (c x^n\right )+\frac{1}{3} b^2 n^2 \log ^3(x) \log \left (d \left (e+f x^m\right )^r\right )+\frac{2}{3} b^2 n^2 r \log ^3(x) \log \left (\frac{e x^{-m}}{f}+1\right )-b^2 n^2 r \log ^3(x) \log \left (e+f x^m\right )+\frac{b^2 n^2 r \log ^2(x) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}+\frac{1}{4} b^2 m n^2 r \log ^4(x) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( e+f{x}^{m} \right ) ^{r} \right ) }{x}}\, dx \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^{m} + e\right )}^{r}\right ) - \int -\frac{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} e \log \left (d\right ) -{\left (b^{2} f m r \log \left (x\right ) - b^{2} f \log \left (d\right )\right )} x^{m}\right )} \log \left (x^{n}\right )^{2} -{\left (b^{2} f m n^{2} r \log \left (x\right )^{3} - 3 \, b^{2} f \log \left (c\right )^{2} \log \left (d\right ) - 6 \, a b f \log \left (c\right ) \log \left (d\right ) - 3 \, a^{2} f \log \left (d\right ) - 3 \,{\left (b^{2} f m n r \log \left (c\right ) + a b f m n r\right )} \log \left (x\right )^{2} + 3 \,{\left (b^{2} f m r \log \left (c\right )^{2} + 2 \, a b f m r \log \left (c\right ) + a^{2} f m r\right )} \log \left (x\right )\right )} x^{m} + 3 \,{\left (2 \, b^{2} e \log \left (c\right ) \log \left (d\right ) + 2 \, a b e \log \left (d\right ) +{\left (b^{2} f m n r \log \left (x\right )^{2} + 2 \, b^{2} f \log \left (c\right ) \log \left (d\right ) + 2 \, a b f \log \left (d\right ) - 2 \,{\left (b^{2} f m r \log \left (c\right ) + a b f m r\right )} \log \left (x\right )\right )} x^{m}\right )} \log \left (x^{n}\right )}{3 \,{\left (f x x^{m} + e x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.10778, size = 995, normalized size = 6.63 \begin{align*} \frac{b^{2} m^{3} n^{2} \log \left (d\right ) \log \left (x\right )^{3} - 6 \, b^{2} n^{2} r{\rm polylog}\left (4, -\frac{f x^{m}}{e}\right ) + 3 \,{\left (b^{2} m^{3} n \log \left (c\right ) + a b m^{3} n\right )} \log \left (d\right ) \log \left (x\right )^{2} + 3 \,{\left (b^{2} m^{3} \log \left (c\right )^{2} + 2 \, a b m^{3} \log \left (c\right ) + a^{2} m^{3}\right )} \log \left (d\right ) \log \left (x\right ) - 3 \,{\left (b^{2} m^{2} n^{2} r \log \left (x\right )^{2} + b^{2} m^{2} r \log \left (c\right )^{2} + 2 \, a b m^{2} r \log \left (c\right ) + a^{2} m^{2} r + 2 \,{\left (b^{2} m^{2} n r \log \left (c\right ) + a b m^{2} n r\right )} \log \left (x\right )\right )}{\rm Li}_2\left (-\frac{f x^{m} + e}{e} + 1\right ) +{\left (b^{2} m^{3} n^{2} r \log \left (x\right )^{3} + 3 \,{\left (b^{2} m^{3} n r \log \left (c\right ) + a b m^{3} n r\right )} \log \left (x\right )^{2} + 3 \,{\left (b^{2} m^{3} r \log \left (c\right )^{2} + 2 \, a b m^{3} r \log \left (c\right ) + a^{2} m^{3} r\right )} \log \left (x\right )\right )} \log \left (f x^{m} + e\right ) -{\left (b^{2} m^{3} n^{2} r \log \left (x\right )^{3} + 3 \,{\left (b^{2} m^{3} n r \log \left (c\right ) + a b m^{3} n r\right )} \log \left (x\right )^{2} + 3 \,{\left (b^{2} m^{3} r \log \left (c\right )^{2} + 2 \, a b m^{3} r \log \left (c\right ) + a^{2} m^{3} r\right )} \log \left (x\right )\right )} \log \left (\frac{f x^{m} + e}{e}\right ) + 6 \,{\left (b^{2} m n^{2} r \log \left (x\right ) + b^{2} m n r \log \left (c\right ) + a b m n r\right )}{\rm polylog}\left (3, -\frac{f x^{m}}{e}\right )}{3 \, m^{3}} \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^{m} + e\right )}^{r} d\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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