Optimal. Leaf size=243 \[ -\frac{b e^3 m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{3 f^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{e^3 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac{4 b e^2 m n x}{9 f^2}-\frac{b e^3 m n \log (e+f x)}{9 f^3}-\frac{b e^3 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{3 f^3}-\frac{5 b e m n x^2}{36 f}+\frac{2}{27} b m n x^3 \]
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Rubi [A] time = 0.166235, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2395, 43, 2376, 2394, 2315} \[ -\frac{b e^3 m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{3 f^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{e^3 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac{4 b e^2 m n x}{9 f^2}-\frac{b e^3 m n \log (e+f x)}{9 f^3}-\frac{b e^3 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{3 f^3}-\frac{5 b e m n x^2}{36 f}+\frac{2}{27} b m n x^3 \]
Antiderivative was successfully verified.
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Rule 2395
Rule 43
Rule 2376
Rule 2394
Rule 2315
Rubi steps
\begin{align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx &=-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-(b n) \int \left (-\frac{e^2 m}{3 f^2}+\frac{e m x}{6 f}-\frac{m x^2}{9}+\frac{e^3 m \log (e+f x)}{3 f^3 x}+\frac{1}{3} x^2 \log \left (d (e+f x)^m\right )\right ) \, dx\\ &=\frac{b e^2 m n x}{3 f^2}-\frac{b e m n x^2}{12 f}+\frac{1}{27} b m n x^3-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{1}{3} (b n) \int x^2 \log \left (d (e+f x)^m\right ) \, dx-\frac{\left (b e^3 m n\right ) \int \frac{\log (e+f x)}{x} \, dx}{3 f^3}\\ &=\frac{b e^2 m n x}{3 f^2}-\frac{b e m n x^2}{12 f}+\frac{1}{27} b m n x^3-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{b e^3 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac{e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}-\frac{1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{\left (b e^3 m n\right ) \int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx}{3 f^2}+\frac{1}{9} (b f m n) \int \frac{x^3}{e+f x} \, dx\\ &=\frac{b e^2 m n x}{3 f^2}-\frac{b e m n x^2}{12 f}+\frac{1}{27} b m n x^3-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{b e^3 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac{e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}-\frac{1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{b e^3 m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{3 f^3}+\frac{1}{9} (b f m n) \int \left (\frac{e^2}{f^3}-\frac{e x}{f^2}+\frac{x^2}{f}-\frac{e^3}{f^3 (e+f x)}\right ) \, dx\\ &=\frac{4 b e^2 m n x}{9 f^2}-\frac{5 b e m n x^2}{36 f}+\frac{2}{27} b m n x^3-\frac{e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac{e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{b e^3 m n \log (e+f x)}{9 f^3}-\frac{b e^3 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac{e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}-\frac{1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{b e^3 m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{3 f^3}\\ \end{align*}
Mathematica [A] time = 0.152889, size = 252, normalized size = 1.04 \[ \frac{36 b e^3 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right )+36 a f^3 x^3 \log \left (d (e+f x)^m\right )-36 a e^2 f m x+36 a e^3 m \log (e+f x)+18 a e f^2 m x^2-12 a f^3 m x^3-6 b \log \left (c x^n\right ) \left (-6 f^3 x^3 \log \left (d (e+f x)^m\right )+f m x \left (6 e^2-3 e f x+2 f^2 x^2\right )-6 e^3 m \log (e+f x)\right )-12 b f^3 n x^3 \log \left (d (e+f x)^m\right )+48 b e^2 f m n x-12 b e^3 m n \log (e+f x)-36 b e^3 m n \log (x) \log (e+f x)+36 b e^3 m n \log (x) \log \left (\frac{f x}{e}+1\right )-15 b e f^2 m n x^2+8 b f^3 m n x^3}{108 f^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.318, size = 2222, normalized size = 9.1 \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 [A] time = 1.6929, size = 443, normalized size = 1.82 \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 e^{3} m n}{3 \, f^{3}} + \frac{{\left (3 \, a e^{3} m -{\left (e^{3} m n - 3 \, e^{3} m \log \left (c\right )\right )} b\right )} \log \left (f x + e\right )}{9 \, f^{3}} - \frac{36 \, b e^{3} m n \log \left (f x + e\right ) \log \left (x\right ) + 4 \,{\left (3 \,{\left (f^{3} m - 3 \, f^{3} \log \left (d\right )\right )} a -{\left (2 \, f^{3} m n - 3 \, f^{3} n \log \left (d\right ) - 3 \,{\left (f^{3} m - 3 \, f^{3} \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x^{3} - 3 \,{\left (6 \, a e f^{2} m -{\left (5 \, e f^{2} m n - 6 \, e f^{2} m \log \left (c\right )\right )} b\right )} x^{2} + 12 \,{\left (3 \, a e^{2} f m -{\left (4 \, e^{2} f m n - 3 \, e^{2} f m \log \left (c\right )\right )} b\right )} x - 12 \,{\left (3 \, b f^{3} x^{3} \log \left (x^{n}\right ) +{\left (3 \, a f^{3} -{\left (f^{3} n - 3 \, f^{3} \log \left (c\right )\right )} b\right )} x^{3}\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - 6 \,{\left (3 \, b e f^{2} m x^{2} - 6 \, b e^{2} f m x + 6 \, b e^{3} m \log \left (f x + e\right ) - 2 \,{\left (f^{3} m - 3 \, f^{3} \log \left (d\right )\right )} b x^{3}\right )} \log \left (x^{n}\right )}{108 \, f^{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 ({\left (b x^{2} \log \left (c x^{n}\right ) + a x^{2}\right )} \log \left ({\left (f x + e\right )}^{m} d\right ), 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{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f x + e\right )}^{m} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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