Integrand size = 18, antiderivative size = 87 \[ \int (f x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {e n p x^{1+n} (f x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {1+m+n}{n},\frac {1+m+2 n}{n},-\frac {e x^n}{d}\right )}{d (1+m) (1+m+n)}+\frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)} \]
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Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2505, 20, 371} \[ \int (f x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {(f x)^{m+1} \log \left (c \left (d+e x^n\right )^p\right )}{f (m+1)}-\frac {e n p x^{n+1} (f x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m+n+1}{n},\frac {m+2 n+1}{n},-\frac {e x^n}{d}\right )}{d (m+1) (m+n+1)} \]
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Rule 20
Rule 371
Rule 2505
Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)}-\frac {(e n p) \int \frac {x^{-1+n} (f x)^{1+m}}{d+e x^n} \, dx}{f (1+m)} \\ & = \frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)}-\frac {\left (e n p x^{-m} (f x)^m\right ) \int \frac {x^{m+n}}{d+e x^n} \, dx}{1+m} \\ & = -\frac {e n p x^{1+n} (f x)^m \, _2F_1\left (1,\frac {1+m+n}{n};\frac {1+m+2 n}{n};-\frac {e x^n}{d}\right )}{d (1+m) (1+m+n)}+\frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int (f x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {x (f x)^m \left (-e n p x^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+m+n}{n},\frac {1+m+2 n}{n},-\frac {e x^n}{d}\right )+d (1+m+n) \log \left (c \left (d+e x^n\right )^p\right )\right )}{d (1+m) (1+m+n)} \]
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\[\int \left (f x \right )^{m} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )d x\]
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\[ \int (f x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
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\[ \int (f x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int \left (f x\right )^{m} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}\, dx \]
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\[ \int (f x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
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\[ \int (f x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
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Timed out. \[ \int (f x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]
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