Integrand size = 26, antiderivative size = 607 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=-\frac {b^2 e m n^2 \log ^2(x) \log (d+e x)}{d}+\frac {2 b^2 e m n^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {2 b^2 e n^2 \log (x) \log \left (f x^m\right ) \log (d+e x)}{d}-\frac {b^2 e m n^2 \log ^2(d+e x)}{d}-\frac {b^2 m n^2 \log ^2(d+e x)}{x}+\frac {b^2 e m n^2 \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)}{d}-\frac {b^2 e n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{d}-\frac {b^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{x}-\frac {2 b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (e x \log \left (-\frac {e x}{d}\right )-(d+e x) \log (d+e x)\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{d x}-\frac {m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{x}-\frac {\left (m-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{x}+\frac {b^2 e m n^2 \log ^2(x) \log \left (1+\frac {e x}{d}\right )}{d}-\frac {2 b^2 e n^2 \log (x) \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{d}-\frac {2 b^2 e n^2 \log \left (f x^m\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d}+\frac {b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (2 e x \log \left (-\frac {e x}{d}\right )-2 (d+e x) \log (d+e x)-2 d \log (x) \log (d+e x)+e x \left (\log ^2(x)-2 \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )\right )}{d x}+\frac {2 b^2 e m n^2 (1+\log (d+e x)) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d}+\frac {2 b^2 e m n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d}-\frac {2 b^2 e m n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{d} \]
[Out]
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx \]
[In]
[Out]
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^2} \, dx \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\frac {2 b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (-e x \log \left (-\frac {e x}{d}\right )+(d+e x) \log (d+e x)\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )-d m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+d \left (-m+m \log (x)-\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (-2 e x \log \left (-\frac {e x}{d}\right )+2 (d+e x) \log (d+e x)+2 d \log (x) \log (d+e x)-e x \left (\log ^2(x)-2 \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )\right )+b^2 n^2 \left (e m x \log ^2(x) \log (d+e x)+2 e m x \log \left (-\frac {e x}{d}\right ) \log (d+e x)-2 e m x \log (x) \log \left (-\frac {e x}{d}\right ) \log (d+e x)+2 e x \log \left (-\frac {e x}{d}\right ) \log \left (f x^m\right ) \log (d+e x)-d m \log ^2(d+e x)-e m x \log ^2(d+e x)+e m x \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)-d \log \left (f x^m\right ) \log ^2(d+e x)-e x \log \left (f x^m\right ) \log ^2(d+e x)-e m x \log ^2(x) \log \left (1+\frac {e x}{d}\right )-2 e m x \log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+2 e x \left (m-m \log (x)+\log \left (f x^m\right )+m \log (d+e x)\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+2 e m x \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )-2 e m x \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )}{d x} \]
[In]
[Out]
\[\int \frac {\ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{x^{2}}d x\]
[In]
[Out]
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx=\int \frac {\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^2} \,d x \]
[In]
[Out]