Integrand size = 18, antiderivative size = 89 \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\frac {b p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {b (d+e x)}{b d-a e}\right )}{e (b d-a e) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c (a+b x)^p\right )}{e (1+m)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 89, 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 = {2442, 70} \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\frac {(d+e x)^{m+1} \log \left (c (a+b x)^p\right )}{e (m+1)}+\frac {b p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {b (d+e x)}{b d-a e}\right )}{e (m+1) (m+2) (b d-a e)} \]
[In]
[Out]
Rule 70
Rule 2442
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{1+m} \log \left (c (a+b x)^p\right )}{e (1+m)}-\frac {(b p) \int \frac {(d+e x)^{1+m}}{a+b x} \, dx}{e (1+m)} \\ & = \frac {b p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {b (d+e x)}{b d-a e}\right )}{e (b d-a e) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c (a+b x)^p\right )}{e (1+m)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.87 \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\frac {(d+e x)^{1+m} \left (\frac {b p (d+e x) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e) (2+m)}+\log \left (c (a+b x)^p\right )\right )}{e (1+m)} \]
[In]
[Out]
\[\int \left (e x +d \right )^{m} \ln \left (c \left (b x +a \right )^{p}\right )d x\]
[In]
[Out]
\[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x + a\right )}^{p} c\right ) \,d x } \]
[In]
[Out]
Exception generated. \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \]
[In]
[Out]
\[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x + a\right )}^{p} c\right ) \,d x } \]
[In]
[Out]
\[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x + a\right )}^{p} c\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\int \ln \left (c\,{\left (a+b\,x\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \]
[In]
[Out]