Optimal. Leaf size=234 \[ -\frac{b f^2 m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{2 e^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac{f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{f^2 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac{b n \log \left (d (e+f x)^m\right )}{4 x^2}+\frac{b f^2 m n \log ^2(x)}{4 e^2}-\frac{b f^2 m n \log (x)}{4 e^2}+\frac{b f^2 m n \log (e+f x)}{4 e^2}-\frac{b f^2 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{2 e^2}-\frac{3 b f m n}{4 e x} \]
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Rubi [A] time = 0.162672, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2395, 44, 2376, 2301, 2394, 2315} \[ -\frac{b f^2 m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{2 e^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac{f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{f^2 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac{b n \log \left (d (e+f x)^m\right )}{4 x^2}+\frac{b f^2 m n \log ^2(x)}{4 e^2}-\frac{b f^2 m n \log (x)}{4 e^2}+\frac{b f^2 m n \log (e+f x)}{4 e^2}-\frac{b f^2 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{2 e^2}-\frac{3 b f m n}{4 e x} \]
Antiderivative was successfully verified.
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Rule 2395
Rule 44
Rule 2376
Rule 2301
Rule 2394
Rule 2315
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^3} \, dx &=-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac{f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-(b n) \int \left (-\frac{f m}{2 e x^2}-\frac{f^2 m \log (x)}{2 e^2 x}+\frac{f^2 m \log (e+f x)}{2 e^2 x}-\frac{\log \left (d (e+f x)^m\right )}{2 x^3}\right ) \, dx\\ &=-\frac{b f m n}{2 e x}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac{f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{1}{2} (b n) \int \frac{\log \left (d (e+f x)^m\right )}{x^3} \, dx+\frac{\left (b f^2 m n\right ) \int \frac{\log (x)}{x} \, dx}{2 e^2}-\frac{\left (b f^2 m n\right ) \int \frac{\log (e+f x)}{x} \, dx}{2 e^2}\\ &=-\frac{b f m n}{2 e x}+\frac{b f^2 m n \log ^2(x)}{4 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac{f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{b f^2 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{2 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac{b n \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{1}{4} (b f m n) \int \frac{1}{x^2 (e+f x)} \, dx+\frac{\left (b f^3 m n\right ) \int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx}{2 e^2}\\ &=-\frac{b f m n}{2 e x}+\frac{b f^2 m n \log ^2(x)}{4 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac{f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{b f^2 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{2 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac{b n \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac{b f^2 m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{2 e^2}+\frac{1}{4} (b f m n) \int \left (\frac{1}{e x^2}-\frac{f}{e^2 x}+\frac{f^2}{e^2 (e+f x)}\right ) \, dx\\ &=-\frac{3 b f m n}{4 e x}-\frac{b f^2 m n \log (x)}{4 e^2}+\frac{b f^2 m n \log ^2(x)}{4 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac{f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{b f^2 m n \log (e+f x)}{4 e^2}-\frac{b f^2 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{2 e^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac{b n \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac{b f^2 m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{2 e^2}\\ \end{align*}
Mathematica [A] time = 0.148011, size = 232, normalized size = 0.99 \[ -\frac{-2 b f^2 m n x^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right )+f^2 m x^2 \log (x) \left (2 a+2 b \log \left (c x^n\right )+2 b n \log (e+f x)-2 b n \log \left (\frac{f x}{e}+1\right )+b n\right )+2 a e^2 \log \left (d (e+f x)^m\right )-2 a f^2 m x^2 \log (e+f x)+2 a e f m x+2 b e^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-2 b f^2 m x^2 \log \left (c x^n\right ) \log (e+f x)+2 b e f m x \log \left (c x^n\right )+b e^2 n \log \left (d (e+f x)^m\right )-b f^2 m n x^2 \log (e+f x)+3 b e f m n x-b f^2 m n x^2 \log ^2(x)}{4 e^2 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.348, size = 2100, normalized size = 9. \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 [A] time = 1.69233, size = 385, normalized size = 1.65 \begin{align*} \frac{{\left (\log \left (\frac{f x}{e} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{f x}{e}\right )\right )} b f^{2} m n}{2 \, e^{2}} + \frac{{\left (2 \, a f^{2} m +{\left (f^{2} m n + 2 \, f^{2} m \log \left (c\right )\right )} b\right )} \log \left (f x + e\right )}{4 \, e^{2}} - \frac{2 \, b f^{2} m n x^{2} \log \left (f x + e\right ) \log \left (x\right ) - b f^{2} m n x^{2} \log \left (x\right )^{2} + 2 \, a e^{2} \log \left (d\right ) +{\left (2 \, a f^{2} m +{\left (f^{2} m n + 2 \, f^{2} m \log \left (c\right )\right )} b\right )} x^{2} \log \left (x\right ) +{\left (e^{2} n \log \left (d\right ) + 2 \, e^{2} \log \left (c\right ) \log \left (d\right )\right )} b +{\left (2 \, a e f m +{\left (3 \, e f m n + 2 \, e f m \log \left (c\right )\right )} b\right )} x +{\left (2 \, b e^{2} \log \left (x^{n}\right ) + 2 \, a e^{2} +{\left (e^{2} n + 2 \, e^{2} \log \left (c\right )\right )} b\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - 2 \,{\left (b f^{2} m x^{2} \log \left (f x + e\right ) - b f^{2} m x^{2} \log \left (x\right ) - b e f m x - b e^{2} \log \left (d\right )\right )} \log \left (x^{n}\right )}{4 \, e^{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{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{3}}, 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 )} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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