Integrand size = 18, antiderivative size = 106 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {e^{-\frac {a (1+m)}{b n}} (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{d (1+m)} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2347, 2212} \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {(d x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{d (m+1)} \]
[In]
[Out]
Rule 2212
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int e^{\frac {(1+m) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{d n} \\ & = \frac {e^{-\frac {a (1+m)}{b n}} (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{d (1+m)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.01 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {e^{-\frac {(1+m) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} x^{-m} (d x)^m \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m} \]
[In]
[Out]
\[\int \left (d x \right )^{m} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]
[In]
[Out]
\[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { \left (d x\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} \,d x } \]
[In]
[Out]
\[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int \left (d x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )^{p}\, dx \]
[In]
[Out]
Exception generated. \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { \left (d x\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int {\left (d\,x\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]
[In]
[Out]