Optimal. Leaf size=161 \[ -6 b^2 m n^2 \text{PolyLog}\left (4,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-m \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b m n \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+6 b^3 m n^3 \text{PolyLog}\left (5,-\frac{f x}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac{m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]
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Rubi [A] time = 0.189815, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -6 b^2 m n^2 \text{PolyLog}\left (4,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-m \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b m n \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+6 b^3 m n^3 \text{PolyLog}\left (5,-\frac{f x}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac{m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]
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 )^3 \log \left (d (e+f x)^m\right )}{x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac{(f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^4}{e+f x} \, dx}{4 b n}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x}{e}\right )}{4 b n}+m \int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x}{e}\right )+(3 b m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-\frac{f x}{e}\right )-\left (6 b^2 m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-\frac{f x}{e}\right )-6 b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_4\left (-\frac{f x}{e}\right )+\left (6 b^3 m n^3\right ) \int \frac{\text{Li}_4\left (-\frac{f x}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac{m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac{f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_2\left (-\frac{f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_3\left (-\frac{f x}{e}\right )-6 b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_4\left (-\frac{f x}{e}\right )+6 b^3 m n^3 \text{Li}_5\left (-\frac{f x}{e}\right )\\ \end{align*}
Mathematica [B] time = 0.253731, size = 602, normalized size = 3.74 \[ -6 a b^2 m n^2 \text{PolyLog}\left (4,-\frac{f x}{e}\right )-m \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b m n \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2-6 b^3 m n^2 \log \left (c x^n\right ) \text{PolyLog}\left (4,-\frac{f x}{e}\right )+6 b^3 m n^3 \text{PolyLog}\left (5,-\frac{f x}{e}\right )+3 a^2 b \log (x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 a^2 b m \log (x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )-\frac{3}{2} a^2 b n \log ^2(x) \log \left (d (e+f x)^m\right )+\frac{3}{2} a^2 b m n \log ^2(x) \log \left (\frac{f x}{e}+1\right )+a^3 \log (x) \log \left (d (e+f x)^m\right )-a^3 m \log (x) \log \left (\frac{f x}{e}+1\right )-3 a b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+3 a b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+3 a b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )-3 a b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )+a b^2 n^2 \log ^3(x) \log \left (d (e+f x)^m\right )-a b^2 m n^2 \log ^3(x) \log \left (\frac{f x}{e}+1\right )+b^3 n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-\frac{3}{2} b^3 n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 \log (x) \log ^3\left (c x^n\right ) \log \left (d (e+f x)^m\right )-b^3 m n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )+\frac{3}{2} b^3 m n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )-b^3 m \log (x) \log ^3\left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )-\frac{1}{4} b^3 n^3 \log ^4(x) \log \left (d (e+f x)^m\right )+\frac{1}{4} b^3 m n^3 \log ^4(x) \log \left (\frac{f x}{e}+1\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 2.217, size = 60520, normalized size = 375.9 \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*} \text{result too large to display} \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^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\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 )}^{3} \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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