Integrand size = 22, antiderivative size = 51 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+2 b n \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2441, 2352} \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+2 b n \operatorname {PolyLog}\left (2,\frac {\sqrt {x} e}{d}+1\right ) \]
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Rule 2352
Rule 2441
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )-(2 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+2 b n \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=2 b \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+a \log (x)+2 b n \operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right ) \]
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\[\int \frac {a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )}{x}d x\]
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\[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a}{x} \,d x } \]
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\[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=\int \frac {a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (44) = 88\).
Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.12 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=-2 \, {\left (\log \left (\frac {e \sqrt {x}}{d} + 1\right ) \log \left (\sqrt {x}\right ) + {\rm Li}_2\left (-\frac {e \sqrt {x}}{d}\right )\right )} b n + \frac {2 \, {\left (b e n \sqrt {x} \log \left (\sqrt {x}\right ) - b e n \sqrt {x}\right )}}{d} + \frac {b d \log \left ({\left (e \sqrt {x} + d\right )}^{n}\right ) \log \left (x\right ) + {\left (b d \log \left (c\right ) + a d\right )} \log \left (x\right ) - \frac {b e n x \log \left (x\right ) - 2 \, b e n x}{\sqrt {x}}}{d} \]
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\[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a}{x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{x} \,d x \]
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