Integrand size = 20, antiderivative size = 83 \[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=-\frac {e p x^{3/2} (f x)^m \operatorname {Hypergeometric2F1}\left (1,3+2 m,2 (2+m),-\frac {e \sqrt {x}}{d}\right )}{d \left (3+5 m+2 m^2\right )}+\frac {(f x)^{1+m} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (1+m)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2505, 20, 348, 66} \[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\frac {(f x)^{m+1} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (m+1)}-\frac {e p x^{3/2} (f x)^m \operatorname {Hypergeometric2F1}\left (1,2 m+3,2 (m+2),-\frac {e \sqrt {x}}{d}\right )}{d \left (2 m^2+5 m+3\right )} \]
[In]
[Out]
Rule 20
Rule 66
Rule 348
Rule 2505
Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (1+m)}-\frac {(e p) \int \frac {(f x)^{1+m}}{\left (d+e \sqrt {x}\right ) \sqrt {x}} \, dx}{2 f (1+m)} \\ & = \frac {(f x)^{1+m} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (1+m)}-\frac {\left (e p x^{-m} (f x)^m\right ) \int \frac {x^{\frac {1}{2}+m}}{d+e \sqrt {x}} \, dx}{2 (1+m)} \\ & = \frac {(f x)^{1+m} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (1+m)}-\frac {\left (e p x^{-m} (f x)^m\right ) \text {Subst}\left (\int \frac {x^{-1+2 \left (\frac {3}{2}+m\right )}}{d+e x} \, dx,x,\sqrt {x}\right )}{1+m} \\ & = -\frac {e p x^{3/2} (f x)^m \, _2F_1\left (1,3+2 m;2 (2+m);-\frac {e \sqrt {x}}{d}\right )}{d \left (3+5 m+2 m^2\right )}+\frac {(f x)^{1+m} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (1+m)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.92 \[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\frac {x (f x)^m \left (-e p \sqrt {x} \operatorname {Hypergeometric2F1}\left (1,3+2 m,4+2 m,-\frac {e \sqrt {x}}{d}\right )+d (3+2 m) \log \left (c \left (d+e \sqrt {x}\right )^p\right )\right )}{d (1+m) (3+2 m)} \]
[In]
[Out]
\[\int \left (f x \right )^{m} \ln \left (c \left (d +e \sqrt {x}\right )^{p}\right )d x\]
[In]
[Out]
\[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right ) \,d x } \]
[In]
[Out]
\[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\int \left (f x\right )^{m} \log {\left (c \left (d + e \sqrt {x}\right )^{p} \right )}\, dx \]
[In]
[Out]
\[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right ) \,d x } \]
[In]
[Out]
\[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]
[In]
[Out]