Integrand size = 38, antiderivative size = 139 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx=-\frac {a (a e m-b d n q) x^{2 m}}{2 b m n q}+\left (\frac {b d}{m}-\frac {a e}{n q}\right ) x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (1+q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2625, 14, 2347, 2212} \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx=x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \left (\frac {b d}{m}-\frac {a e}{n q}\right ) \Gamma \left (q+1,-\frac {m \log \left (c x^n\right )}{n}\right )+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\frac {a x^{2 m} (a e m-b d n q)}{2 b m n q} \]
[In]
[Out]
Rule 14
Rule 2212
Rule 2347
Rule 2625
Rubi steps \begin{align*} \text {integral}& = \frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\left (-d+\frac {a e m}{b n q}\right ) \int x^{-1+m} \left (a x^m+b \log ^q\left (c x^n\right )\right ) \, dx \\ & = \frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\left (-d+\frac {a e m}{b n q}\right ) \int \left (a x^{-1+2 m}+b x^{-1+m} \log ^q\left (c x^n\right )\right ) \, dx \\ & = \frac {a \left (d-\frac {a e m}{b n q}\right ) x^{2 m}}{2 m}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\left (b \left (-d+\frac {a e m}{b n q}\right )\right ) \int x^{-1+m} \log ^q\left (c x^n\right ) \, dx \\ & = \frac {a \left (d-\frac {a e m}{b n q}\right ) x^{2 m}}{2 m}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\frac {\left (b \left (-d+\frac {a e m}{b n q}\right ) x^m \left (c x^n\right )^{-\frac {m}{n}}\right ) \text {Subst}\left (\int e^{\frac {m x}{n}} x^q \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {a \left (d-\frac {a e m}{b n q}\right ) x^{2 m}}{2 m}+\left (\frac {b d}{m}-\frac {a e}{n q}\right ) x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (1+q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx=\frac {\left (c x^n\right )^{-\frac {m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \left (-2 a e m q x^m \Gamma \left (q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right )+2 b d n q x^m \Gamma \left (1+q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right )+\left (c x^n\right )^{m/n} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (a d n q x^{2 m}+b e m \log ^{2 q}\left (c x^n\right )\right )\right )}{2 m n q} \]
[In]
[Out]
\[\int \frac {\left (d \,x^{m}+e \ln \left (c \,x^{n}\right )^{-1+q}\right ) \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )}{x}d x\]
[In]
[Out]
\[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} {\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx=\int \frac {\left (a x^{m} + b \log {\left (c x^{n} \right )}^{q}\right ) \left (d x^{m} + e \log {\left (c x^{n} \right )}^{q - 1}\right )}{x}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} {\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx=\int \frac {\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )\,\left (d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}\right )}{x} \,d x \]
[In]
[Out]