Integrand size = 22, antiderivative size = 158 \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-\frac {3 b d m n x}{4 e}+\frac {1}{4} b m n x^2+\frac {b d n x \log \left (f x^m\right )}{2 e}-\frac {1}{4} b n x^2 \log \left (f x^m\right )+\frac {b d^2 m n \log (d+e x)}{4 e^2}-\frac {1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b d^2 n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{2 e^2}-\frac {b d^2 m n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 e^2} \]
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Time = 0.09 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2473, 45, 2393, 2332, 2341, 2354, 2438} \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-\frac {1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b d^2 n \log \left (\frac {e x}{d}+1\right ) \log \left (f x^m\right )}{2 e^2}-\frac {b d^2 m n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 e^2}+\frac {b d^2 m n \log (d+e x)}{4 e^2}+\frac {b d n x \log \left (f x^m\right )}{2 e}-\frac {3 b d m n x}{4 e}-\frac {1}{4} b n x^2 \log \left (f x^m\right )+\frac {1}{4} b m n x^2 \]
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Rule 45
Rule 2332
Rule 2341
Rule 2354
Rule 2393
Rule 2438
Rule 2473
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{2} (b e n) \int \frac {x^2 \log \left (f x^m\right )}{d+e x} \, dx+\frac {1}{4} (b e m n) \int \frac {x^2}{d+e x} \, dx \\ & = -\frac {1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{2} (b e n) \int \left (-\frac {d \log \left (f x^m\right )}{e^2}+\frac {x \log \left (f x^m\right )}{e}+\frac {d^2 \log \left (f x^m\right )}{e^2 (d+e x)}\right ) \, dx+\frac {1}{4} (b e m n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx \\ & = -\frac {b d m n x}{4 e}+\frac {1}{8} b m n x^2+\frac {b d^2 m n \log (d+e x)}{4 e^2}-\frac {1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{2} (b n) \int x \log \left (f x^m\right ) \, dx+\frac {(b d n) \int \log \left (f x^m\right ) \, dx}{2 e}-\frac {\left (b d^2 n\right ) \int \frac {\log \left (f x^m\right )}{d+e x} \, dx}{2 e} \\ & = -\frac {3 b d m n x}{4 e}+\frac {1}{4} b m n x^2+\frac {b d n x \log \left (f x^m\right )}{2 e}-\frac {1}{4} b n x^2 \log \left (f x^m\right )+\frac {b d^2 m n \log (d+e x)}{4 e^2}-\frac {1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b d^2 n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{2 e^2}+\frac {\left (b d^2 m n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{2 e^2} \\ & = -\frac {3 b d m n x}{4 e}+\frac {1}{4} b m n x^2+\frac {b d n x \log \left (f x^m\right )}{2 e}-\frac {1}{4} b n x^2 \log \left (f x^m\right )+\frac {b d^2 m n \log (d+e x)}{4 e^2}-\frac {1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b d^2 n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{2 e^2}-\frac {b d^2 m n \text {Li}_2\left (-\frac {e x}{d}\right )}{2 e^2} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.04 \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {\log \left (f x^m\right ) \left (-2 b d^2 n \log (d+e x)+e x \left (2 b d n+2 a e x-b e n x+2 b e x \log \left (c (d+e x)^n\right )\right )\right )+m \left (-3 b d e n x-a e^2 x^2+b e^2 n x^2+b d^2 n (1+2 \log (x)) \log (d+e x)-b e^2 x^2 \log \left (c (d+e x)^n\right )-2 b d^2 n \log (x) \log \left (1+\frac {e x}{d}\right )\right )-2 b d^2 m n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{4 e^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 13.04 (sec) , antiderivative size = 843, normalized size of antiderivative = 5.34
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\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x \log \left (f x^{m}\right ) \,d x } \]
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Timed out. \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.13 \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {1}{4} \, {\left (\frac {2 \, {\left (\log \left (e x + d\right ) \log \left (-\frac {e x + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {e x + d}{d}\right )\right )} b d^{2} n}{e^{2}} - \frac {b e^{2} x^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + 3 \, b d e n x - b d^{2} n \log \left (e x + d\right ) + {\left (a e^{2} - {\left (e^{2} n - e^{2} \log \left (c\right )\right )} b\right )} x^{2}}{e^{2}}\right )} m - \frac {1}{4} \, {\left (b e n {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} - 2 \, b x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) - 2 \, a x^{2}\right )} \log \left (f x^{m}\right ) \]
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\[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x \log \left (f x^{m}\right ) \,d x } \]
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Timed out. \[ \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\int x\,\ln \left (f\,x^m\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right ) \,d x \]
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