Integrand size = 10, antiderivative size = 56 \[ \int (a+b \log (c x))^p \, dx=\frac {e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log (c x)}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2336, 2212} \[ \int (a+b \log (c x))^p \, dx=\frac {e^{-\frac {a}{b}} (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log (c x)}{b}\right )}{c} \]
[In]
[Out]
Rule 2212
Rule 2336
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log (c x)\right )}{c} \\ & = \frac {e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log (c x)}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int (a+b \log (c x))^p \, dx=\frac {e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log (c x)}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c} \]
[In]
[Out]
\[\int \left (a +b \ln \left (x c \right )\right )^{p}d x\]
[In]
[Out]
none
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68 \[ \int (a+b \log (c x))^p \, dx=\frac {e^{\left (-\frac {b p \log \left (-\frac {1}{b}\right ) + a}{b}\right )} \Gamma \left (p + 1, -\frac {b \log \left (c x\right ) + a}{b}\right )}{c} \]
[In]
[Out]
\[ \int (a+b \log (c x))^p \, dx=\int \left (a + b \log {\left (c x \right )}\right )^{p}\, dx \]
[In]
[Out]
none
Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int (a+b \log (c x))^p \, dx=-\frac {{\left (b \log \left (c x\right ) + a\right )}^{p + 1} e^{\left (-\frac {a}{b}\right )} E_{-p}\left (-\frac {b \log \left (c x\right ) + a}{b}\right )}{b c} \]
[In]
[Out]
\[ \int (a+b \log (c x))^p \, dx=\int { {\left (b \log \left (c x\right ) + a\right )}^{p} \,d x } \]
[In]
[Out]
Timed out. \[ \int (a+b \log (c x))^p \, dx=\int {\left (a+b\,\ln \left (c\,x\right )\right )}^p \,d x \]
[In]
[Out]