Integrand size = 26, antiderivative size = 939 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\frac {b^2 e^2 m n^2 \log (x)}{d^2}-\frac {b^2 e^2 m n^2 \log ^2(x)}{2 d^2}+\frac {b^2 e^2 m n^2 \log \left (-\frac {e x}{d}\right )}{2 d^2}+\frac {b^2 e^2 n^2 \log (x) \log \left (f x^m\right )}{d^2}-\frac {3 b^2 e^2 m n^2 \log (d+e x)}{2 d^2}-\frac {3 b^2 e m n^2 \log (d+e x)}{2 d x}+\frac {b^2 e^2 m n^2 \log (x) \log (d+e x)}{d^2}+\frac {b^2 e^2 m n^2 \log ^2(x) \log (d+e x)}{2 d^2}-\frac {b^2 e^2 m n^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{2 d^2}-\frac {b^2 e^2 n^2 \log \left (f x^m\right ) \log (d+e x)}{d^2}-\frac {b^2 e n^2 \log \left (f x^m\right ) \log (d+e x)}{d x}-\frac {b^2 e^2 n^2 \log (x) \log \left (f x^m\right ) \log (d+e x)}{d^2}+\frac {b^2 e^2 m n^2 \log ^2(d+e x)}{4 d^2}-\frac {b^2 m n^2 \log ^2(d+e x)}{4 x^2}-\frac {b^2 e^2 m n^2 \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)}{2 d^2}+\frac {b^2 e^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{2 d^2}-\frac {b^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{2 x^2}+\frac {b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (e^2 x^2 \log \left (-\frac {e x}{d}\right )+(d+e x) (e x+(d-e x) \log (d+e x))\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{d^2 x^2}-\frac {m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{2 x^2}-\frac {\left (m-2 m \log (x)+2 \log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{4 x^2}-\frac {b^2 e^2 m n^2 \log (x) \log \left (1+\frac {e x}{d}\right )}{d^2}-\frac {b^2 e^2 m n^2 \log ^2(x) \log \left (1+\frac {e x}{d}\right )}{2 d^2}+\frac {b^2 e^2 n^2 \log (x) \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{d^2}-\frac {b^2 e^2 n^2 \left (m-\log \left (f x^m\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}-\frac {b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (e x (d+e x)+e^2 x^2 \log \left (-\frac {e x}{d}\right )+\left (d^2-e^2 x^2\right ) \log (d+e x)+2 d^2 \log (x) \log (d+e x)+e x \left (e x \log ^2(x)+2 d (1+\log (x))-2 e x \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )\right )}{2 d^2 x^2}-\frac {b^2 e^2 m n^2 (1+2 \log (d+e x)) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 d^2}-\frac {b^2 e^2 m n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^2}+\frac {b^2 e^2 m n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{d^2} \]
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\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 781, normalized size of antiderivative = 0.83 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\frac {4 b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (e^2 x^2 \log \left (-\frac {e x}{d}\right )+(d+e x) (e x+(d-e x) \log (d+e x))\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )-2 d^2 m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+d^2 \left (-m+2 m \log (x)-2 \log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-2 b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (e x (d+e x)+e^2 x^2 \log \left (-\frac {e x}{d}\right )+\left (d^2-e^2 x^2\right ) \log (d+e x)+2 d^2 \log (x) \log (d+e x)+e x \left (e x \log ^2(x)+2 d (1+\log (x))-2 e x \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )\right )+b^2 n^2 \left (4 e^2 m x^2 \log (x)-2 e^2 m x^2 \log ^2(x)+2 e^2 m x^2 \log \left (-\frac {e x}{d}\right )+4 e^2 x^2 \log (x) \log \left (f x^m\right )-6 d e m x \log (d+e x)-6 e^2 m x^2 \log (d+e x)+4 e^2 m x^2 \log (x) \log (d+e x)-2 e^2 m x^2 \log ^2(x) \log (d+e x)-2 e^2 m x^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)+4 e^2 m x^2 \log (x) \log \left (-\frac {e x}{d}\right ) \log (d+e x)-4 d e x \log \left (f x^m\right ) \log (d+e x)-4 e^2 x^2 \log \left (f x^m\right ) \log (d+e x)-4 e^2 x^2 \log \left (-\frac {e x}{d}\right ) \log \left (f x^m\right ) \log (d+e x)-d^2 m \log ^2(d+e x)+e^2 m x^2 \log ^2(d+e x)-2 e^2 m x^2 \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)-2 d^2 \log \left (f x^m\right ) \log ^2(d+e x)+2 e^2 x^2 \log \left (f x^m\right ) \log ^2(d+e x)-4 e^2 m x^2 \log (x) \log \left (1+\frac {e x}{d}\right )+2 e^2 m x^2 \log ^2(x) \log \left (1+\frac {e x}{d}\right )+4 e^2 m x^2 (-1+\log (x)) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-2 e^2 x^2 \left (m-2 m \log (x)+2 \log \left (f x^m\right )+2 m \log (d+e x)\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-4 e^2 m x^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+4 e^2 m x^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )}{4 d^2 x^2} \]
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\[\int \frac {\ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{x^{3}}d x\]
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\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int \frac {\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^3} \,d x \]
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