Integrand size = 18, antiderivative size = 85 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^3}\right )^p\right ) \, dx=-\frac {3 e f^2 p (f x)^{-2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2-m}{3},\frac {5-m}{3},-\frac {e}{d x^3}\right )}{d \left (2+m-m^2\right )}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (1+m)} \]
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Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2505, 16, 346, 371} \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^3}\right )^p\right ) \, dx=\frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (m+1)}-\frac {3 e f^2 p (f x)^{m-2} \operatorname {Hypergeometric2F1}\left (1,\frac {2-m}{3},\frac {5-m}{3},-\frac {e}{d x^3}\right )}{d \left (-m^2+m+2\right )} \]
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Rule 16
Rule 346
Rule 371
Rule 2505
Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (1+m)}+\frac {(3 e p) \int \frac {(f x)^{1+m}}{\left (d+\frac {e}{x^3}\right ) x^4} \, dx}{f (1+m)} \\ & = \frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (1+m)}+\frac {\left (3 e f^3 p\right ) \int \frac {(f x)^{-3+m}}{d+\frac {e}{x^3}} \, dx}{1+m} \\ & = \frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (1+m)}-\frac {\left (3 e f^2 p \left (\frac {1}{x}\right )^{-2+m} (f x)^{-2+m}\right ) \text {Subst}\left (\int \frac {x^{1-m}}{d+e x^3} \, dx,x,\frac {1}{x}\right )}{1+m} \\ & = -\frac {3 e f^2 p (f x)^{-2+m} \, _2F_1\left (1,\frac {2-m}{3};\frac {5-m}{3};-\frac {e}{d x^3}\right )}{d \left (2+m-m^2\right )}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (1+m)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^3}\right )^p\right ) \, dx=\frac {(f x)^m \left (3 e p \operatorname {Hypergeometric2F1}\left (1,\frac {2}{3}-\frac {m}{3},\frac {5}{3}-\frac {m}{3},-\frac {e}{d x^3}\right )+d (-2+m) x^3 \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )\right )}{d (-2+m) (1+m) x^2} \]
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\[\int \left (f x \right )^{m} \ln \left (c \left (d +\frac {e}{x^{3}}\right )^{p}\right )d x\]
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\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^3}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x^{3}}\right )}^{p}\right ) \,d x } \]
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Timed out. \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^3}\right )^p\right ) \, dx=\text {Timed out} \]
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\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^3}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x^{3}}\right )}^{p}\right ) \,d x } \]
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\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^3}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x^{3}}\right )}^{p}\right ) \,d x } \]
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Timed out. \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^3}\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+\frac {e}{x^3}\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]
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