Integrand size = 20, antiderivative size = 69 \[ \int (f x)^{-1+n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {p (f x)^n}{f n}+\frac {d p x^{-n} (f x)^n \log \left (d+e x^n\right )}{e f n}+\frac {(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2505, 20, 272, 45} \[ \int (f x)^{-1+n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {d p x^{-n} (f x)^n \log \left (d+e x^n\right )}{e f n}-\frac {p (f x)^n}{f n} \]
[In]
[Out]
Rule 20
Rule 45
Rule 272
Rule 2505
Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {(e p) \int \frac {x^{-1+n} (f x)^n}{d+e x^n} \, dx}{f} \\ & = \frac {(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {\left (e p x^{-n} (f x)^n\right ) \int \frac {x^{-1+2 n}}{d+e x^n} \, dx}{f} \\ & = \frac {(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {\left (e p x^{-n} (f x)^n\right ) \text {Subst}\left (\int \frac {x}{d+e x} \, dx,x,x^n\right )}{f n} \\ & = \frac {(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {\left (e p x^{-n} (f x)^n\right ) \text {Subst}\left (\int \left (\frac {1}{e}-\frac {d}{e (d+e x)}\right ) \, dx,x,x^n\right )}{f n} \\ & = -\frac {p (f x)^n}{f n}+\frac {d p x^{-n} (f x)^n \log \left (d+e x^n\right )}{e f n}+\frac {(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.70 \[ \int (f x)^{-1+n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {x^{1-n} (f x)^{-1+n} \left (-p x^n+\frac {\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e}\right )}{n} \]
[In]
[Out]
\[\int \left (f x \right )^{n -1} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )d x\]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int (f x)^{-1+n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {{\left (e p - e \log \left (c\right )\right )} f^{n - 1} x^{n} - {\left (e f^{n - 1} p x^{n} + d f^{n - 1} p\right )} \log \left (e x^{n} + d\right )}{e n} \]
[In]
[Out]
\[ \int (f x)^{-1+n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int \left (f x\right )^{n - 1} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01 \[ \int (f x)^{-1+n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {e p {\left (\frac {f^{n} x^{n}}{e n} - \frac {d f^{n} \log \left (\frac {e x^{n} + d}{e}\right )}{e^{2} n}\right )}}{f} + \frac {\left (f x\right )^{n} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f n} \]
[In]
[Out]
\[ \int (f x)^{-1+n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int (f x)^{-1+n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f\,x\right )}^{n-1} \,d x \]
[In]
[Out]