Integrand size = 18, antiderivative size = 67 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\frac {e p (f x)^m \operatorname {Hypergeometric2F1}\left (1,-m,1-m,-\frac {e}{d x}\right )}{d m (1+m)}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (1+m)} \]
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Time = 0.03 (sec) , antiderivative size = 67, 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, 66} \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (m+1)}+\frac {e p (f x)^m \operatorname {Hypergeometric2F1}\left (1,-m,1-m,-\frac {e}{d x}\right )}{d m (m+1)} \]
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Rule 16
Rule 66
Rule 346
Rule 2505
Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (1+m)}+\frac {(e p) \int \frac {(f x)^{1+m}}{\left (d+\frac {e}{x}\right ) x^2} \, dx}{f (1+m)} \\ & = \frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (1+m)}+\frac {(e f p) \int \frac {(f x)^{-1+m}}{d+\frac {e}{x}} \, dx}{1+m} \\ & = \frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (1+m)}-\frac {\left (e p \left (\frac {1}{x}\right )^m (f x)^m\right ) \text {Subst}\left (\int \frac {x^{-1-m}}{d+e x} \, dx,x,\frac {1}{x}\right )}{1+m} \\ & = \frac {e p (f x)^m \, _2F_1\left (1,-m;1-m;-\frac {e}{d x}\right )}{d m (1+m)}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (1+m)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\frac {(f x)^m \left (e p \operatorname {Hypergeometric2F1}\left (1,-m,1-m,-\frac {e}{d x}\right )+d m x \log \left (c \left (d+\frac {e}{x}\right )^p\right )\right )}{d m (1+m)} \]
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\[\int \left (f x \right )^{m} \ln \left (c \left (d +\frac {e}{x}\right )^{p}\right )d x\]
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\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x}\right )}^{p}\right ) \,d x } \]
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Time = 9.21 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.45 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=e p \left (\begin {cases} \frac {0^{m} \log {\left (d x + e \right )}}{d} & \text {for}\: \left (f = 0 \wedge m \neq -1\right ) \vee f = 0 \\\frac {d^{m - 1} f^{m + 1} m x^{m} \Phi \left (\frac {e e^{i \pi }}{d x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{d^{m} f m \Gamma \left (1 - m\right ) + d^{m} f \Gamma \left (1 - m\right )} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq -1 \\\frac {\begin {cases} - \frac {1}{d x} & \text {for}\: e = 0 \\\frac {\begin {cases} \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}}{f} - \frac {\left (\begin {cases} \frac {1}{d x} & \text {for}\: e = 0 \\\frac {\log {\left (d + \frac {e}{x} \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (f x \right )}}{f} & \text {otherwise} \end {cases}\right ) + \left (\begin {cases} 0^{m} x & \text {for}\: f = 0 \\\frac {\begin {cases} \frac {\left (f x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (f x \right )} & \text {otherwise} \end {cases}}{f} & \text {otherwise} \end {cases}\right ) \log {\left (c \left (d + \frac {e}{x}\right )^{p} \right )} \]
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\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x}\right )}^{p}\right ) \,d x } \]
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\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x}\right )}^{p}\right ) \,d x } \]
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Timed out. \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+\frac {e}{x}\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]
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