Integrand size = 24, antiderivative size = 230 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^5} \, dx=-\frac {7 b e m n}{144 d x^3}+\frac {3 b e^2 m n}{32 d^2 x^2}-\frac {5 b e^3 m n}{16 d^3 x}-\frac {b e^4 m n \log (x)}{16 d^4}-\frac {b e n \log \left (f x^m\right )}{12 d x^3}+\frac {b e^2 n \log \left (f x^m\right )}{8 d^2 x^2}-\frac {b e^3 n \log \left (f x^m\right )}{4 d^3 x}+\frac {b e^4 n \log \left (1+\frac {d}{e x}\right ) \log \left (f x^m\right )}{4 d^4}+\frac {b e^4 m n \log (d+e x)}{16 d^4}-\frac {1}{16} \left (\frac {m}{x^4}+\frac {4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b e^4 m n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{4 d^4} \]
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Time = 0.17 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2473, 2380, 2341, 2379, 2438, 46} \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^5} \, dx=-\frac {1}{16} \left (\frac {4 \log \left (f x^m\right )}{x^4}+\frac {m}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e^4 n \log \left (\frac {d}{e x}+1\right ) \log \left (f x^m\right )}{4 d^4}-\frac {b e^4 m n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{4 d^4}-\frac {b e^4 m n \log (x)}{16 d^4}+\frac {b e^4 m n \log (d+e x)}{16 d^4}-\frac {b e^3 n \log \left (f x^m\right )}{4 d^3 x}-\frac {5 b e^3 m n}{16 d^3 x}+\frac {b e^2 n \log \left (f x^m\right )}{8 d^2 x^2}+\frac {3 b e^2 m n}{32 d^2 x^2}-\frac {b e n \log \left (f x^m\right )}{12 d x^3}-\frac {7 b e m n}{144 d x^3} \]
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Rule 46
Rule 2341
Rule 2379
Rule 2380
Rule 2438
Rule 2473
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{16} \left (\frac {m}{x^4}+\frac {4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {1}{4} (b e n) \int \frac {\log \left (f x^m\right )}{x^4 (d+e x)} \, dx+\frac {1}{16} (b e m n) \int \frac {1}{x^4 (d+e x)} \, dx \\ & = -\frac {1}{16} \left (\frac {m}{x^4}+\frac {4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {(b e n) \int \frac {\log \left (f x^m\right )}{x^4} \, dx}{4 d}-\frac {\left (b e^2 n\right ) \int \frac {\log \left (f x^m\right )}{x^3 (d+e x)} \, dx}{4 d}+\frac {1}{16} (b e m n) \int \left (\frac {1}{d x^4}-\frac {e}{d^2 x^3}+\frac {e^2}{d^3 x^2}-\frac {e^3}{d^4 x}+\frac {e^4}{d^4 (d+e x)}\right ) \, dx \\ & = -\frac {7 b e m n}{144 d x^3}+\frac {b e^2 m n}{32 d^2 x^2}-\frac {b e^3 m n}{16 d^3 x}-\frac {b e^4 m n \log (x)}{16 d^4}-\frac {b e n \log \left (f x^m\right )}{12 d x^3}+\frac {b e^4 m n \log (d+e x)}{16 d^4}-\frac {1}{16} \left (\frac {m}{x^4}+\frac {4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {\left (b e^2 n\right ) \int \frac {\log \left (f x^m\right )}{x^3} \, dx}{4 d^2}+\frac {\left (b e^3 n\right ) \int \frac {\log \left (f x^m\right )}{x^2 (d+e x)} \, dx}{4 d^2} \\ & = -\frac {7 b e m n}{144 d x^3}+\frac {3 b e^2 m n}{32 d^2 x^2}-\frac {b e^3 m n}{16 d^3 x}-\frac {b e^4 m n \log (x)}{16 d^4}-\frac {b e n \log \left (f x^m\right )}{12 d x^3}+\frac {b e^2 n \log \left (f x^m\right )}{8 d^2 x^2}+\frac {b e^4 m n \log (d+e x)}{16 d^4}-\frac {1}{16} \left (\frac {m}{x^4}+\frac {4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {\left (b e^3 n\right ) \int \frac {\log \left (f x^m\right )}{x^2} \, dx}{4 d^3}-\frac {\left (b e^4 n\right ) \int \frac {\log \left (f x^m\right )}{x (d+e x)} \, dx}{4 d^3} \\ & = -\frac {7 b e m n}{144 d x^3}+\frac {3 b e^2 m n}{32 d^2 x^2}-\frac {5 b e^3 m n}{16 d^3 x}-\frac {b e^4 m n \log (x)}{16 d^4}-\frac {b e n \log \left (f x^m\right )}{12 d x^3}+\frac {b e^2 n \log \left (f x^m\right )}{8 d^2 x^2}-\frac {b e^3 n \log \left (f x^m\right )}{4 d^3 x}+\frac {b e^4 n \log \left (1+\frac {d}{e x}\right ) \log \left (f x^m\right )}{4 d^4}+\frac {b e^4 m n \log (d+e x)}{16 d^4}-\frac {1}{16} \left (\frac {m}{x^4}+\frac {4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {\left (b e^4 m n\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{4 d^4} \\ & = -\frac {7 b e m n}{144 d x^3}+\frac {3 b e^2 m n}{32 d^2 x^2}-\frac {5 b e^3 m n}{16 d^3 x}-\frac {b e^4 m n \log (x)}{16 d^4}-\frac {b e n \log \left (f x^m\right )}{12 d x^3}+\frac {b e^2 n \log \left (f x^m\right )}{8 d^2 x^2}-\frac {b e^3 n \log \left (f x^m\right )}{4 d^3 x}+\frac {b e^4 n \log \left (1+\frac {d}{e x}\right ) \log \left (f x^m\right )}{4 d^4}+\frac {b e^4 m n \log (d+e x)}{16 d^4}-\frac {1}{16} \left (\frac {m}{x^4}+\frac {4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b e^4 m n \text {Li}_2\left (-\frac {d}{e x}\right )}{4 d^4} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.19 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^5} \, dx=-\frac {18 a d^4 m+14 b d^3 e m n x-27 b d^2 e^2 m n x^2+90 b d e^3 m n x^3-36 b e^4 m n x^4 \log ^2(x)+72 a d^4 \log \left (f x^m\right )+24 b d^3 e n x \log \left (f x^m\right )-36 b d^2 e^2 n x^2 \log \left (f x^m\right )+72 b d e^3 n x^3 \log \left (f x^m\right )-18 b e^4 m n x^4 \log (d+e x)-72 b e^4 n x^4 \log \left (f x^m\right ) \log (d+e x)+18 b d^4 m \log \left (c (d+e x)^n\right )+72 b d^4 \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )+18 b e^4 n x^4 \log (x) \left (m+4 \log \left (f x^m\right )+4 m \log (d+e x)-4 m \log \left (1+\frac {e x}{d}\right )\right )-72 b e^4 m n x^4 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{288 d^4 x^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 32.96 (sec) , antiderivative size = 1237, normalized size of antiderivative = 5.38
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\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^5} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^5} \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^5} \, dx=\frac {1}{288} \, {\left (\frac {72 \, {\left (\log \left (\frac {e x}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x}{d}\right )\right )} b e^{4} n}{d^{4}} + \frac {18 \, b e^{4} n \log \left (e x + d\right )}{d^{4}} - \frac {72 \, b e^{4} n x^{4} \log \left (e x + d\right ) \log \left (x\right ) - 36 \, b e^{4} n x^{4} \log \left (x\right )^{2} + 18 \, b e^{4} n x^{4} \log \left (x\right ) + 90 \, b d e^{3} n x^{3} - 27 \, b d^{2} e^{2} n x^{2} + 14 \, b d^{3} e n x + 18 \, b d^{4} \log \left ({\left (e x + d\right )}^{n}\right ) + 18 \, b d^{4} \log \left (c\right ) + 18 \, a d^{4}}{d^{4} x^{4}}\right )} m + \frac {1}{24} \, {\left (b e n {\left (\frac {6 \, e^{3} \log \left (e x + d\right )}{d^{4}} - \frac {6 \, e^{3} \log \left (x\right )}{d^{4}} - \frac {6 \, e^{2} x^{2} - 3 \, d e x + 2 \, d^{2}}{d^{3} x^{3}}\right )} - \frac {6 \, b \log \left ({\left (e x + d\right )}^{n} c\right )}{x^{4}} - \frac {6 \, a}{x^{4}}\right )} \log \left (f x^{m}\right ) \]
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\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^5} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^5} \, dx=\int \frac {\ln \left (f\,x^m\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x^5} \,d x \]
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