Integrand size = 20, antiderivative size = 107 \[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a (1+m)}{b}} \left (c \sqrt {x}\right )^{-2 (1+m)} (d x)^{1+m} \Gamma \left (1+p,-\frac {2 (1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p}}{d (1+m)} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2347, 2212} \[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a (m+1)}{b}} \left (c \sqrt {x}\right )^{-2 (m+1)} (d x)^{m+1} \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 (m+1) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )}{d (m+1)} \]
[In]
[Out]
Rule 2212
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 \left (c \sqrt {x}\right )^{-2 (1+m)} (d x)^{1+m}\right ) \text {Subst}\left (\int e^{2 (1+m) x} (a+b x)^p \, dx,x,\log \left (c \sqrt {x}\right )\right )}{d} \\ & = \frac {2^{-p} e^{-\frac {2 a (1+m)}{b}} \left (c \sqrt {x}\right )^{-2 (1+m)} (d x)^{1+m} \Gamma \left (1+p,-\frac {2 (1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p}}{d (1+m)} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96 \[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a (1+m)}{b}} \left (c \sqrt {x}\right )^{-2 m} (d x)^m \Gamma \left (1+p,-\frac {2 (1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 (1+m)} \]
[In]
[Out]
\[\int \left (d x \right )^{m} \left (a +b \ln \left (c \sqrt {x}\right )\right )^{p}d x\]
[In]
[Out]
\[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\int { \left (d x\right )^{m} {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p} \,d x } \]
[In]
[Out]
\[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\int \left (d x\right )^{m} \left (a + b \log {\left (c \sqrt {x} \right )}\right )^{p}\, dx \]
[In]
[Out]
\[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\int { \left (d x\right )^{m} {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p} \,d x } \]
[In]
[Out]
\[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\int { \left (d x\right )^{m} {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\int {\left (d\,x\right )}^m\,{\left (a+b\,\ln \left (c\,\sqrt {x}\right )\right )}^p \,d x \]
[In]
[Out]