Integrand size = 24, antiderivative size = 93 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x} \, dx=-2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d \sqrt {x}}\right )-4 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt {x}}\right )+4 b^2 n^2 \operatorname {PolyLog}\left (3,1+\frac {e}{d \sqrt {x}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2504, 2443, 2481, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x} \, dx=-4 b n \operatorname {PolyLog}\left (2,\frac {e}{d \sqrt {x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-2 \log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+4 b^2 n^2 \operatorname {PolyLog}\left (3,\frac {e}{d \sqrt {x}}+1\right ) \]
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Rule 2421
Rule 2443
Rule 2481
Rule 2504
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d \sqrt {x}}\right )+(4 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d \sqrt {x}}\right )+(4 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right ) \\ & = -2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d \sqrt {x}}\right )-4 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )+\left (4 b^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right ) \\ & = -2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d \sqrt {x}}\right )-4 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )+4 b^2 n^2 \text {Li}_3\left (1+\frac {e}{d \sqrt {x}}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(386\) vs. \(2(93)=186\).
Time = 0.23 (sec) , antiderivative size = 386, normalized size of antiderivative = 4.15 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x} \, dx=\left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \log (x)+2 b n \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (\left (\log \left (d+\frac {e}{\sqrt {x}}\right )-\log \left (1+\frac {e}{d \sqrt {x}}\right )\right ) \log (x)+2 \operatorname {PolyLog}\left (2,-\frac {e}{d \sqrt {x}}\right )\right )+\frac {1}{12} b^2 n^2 \left (24 \log ^2\left (\frac {e}{d}+\sqrt {x}\right ) \log \left (-\frac {d \sqrt {x}}{e}\right )+12 \log ^2\left (d+\frac {e}{\sqrt {x}}\right ) \log (x)-12 \log ^2\left (\frac {e}{d}+\sqrt {x}\right ) \log (x)-24 \log \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (1+\frac {d \sqrt {x}}{e}\right ) \log (x)+24 \log \left (\frac {e}{d}+\sqrt {x}\right ) \log \left (1+\frac {d \sqrt {x}}{e}\right ) \log (x)+6 \log \left (d+\frac {e}{\sqrt {x}}\right ) \log ^2(x)-6 \log \left (1+\frac {d \sqrt {x}}{e}\right ) \log ^2(x)+\log ^3(x)+48 \log \left (\frac {e}{d}+\sqrt {x}\right ) \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {x}}{e}\right )-48 \left (\log \left (d+\frac {e}{\sqrt {x}}\right )-\log \left (\frac {e}{d}+\sqrt {x}\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d \sqrt {x}}{e}\right )-48 \operatorname {PolyLog}\left (3,1+\frac {d \sqrt {x}}{e}\right )-48 \operatorname {PolyLog}\left (3,-\frac {d \sqrt {x}}{e}\right )\right ) \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{2}}{x}d x\]
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\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2}}{x} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{2}}{x}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2}}{x} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^2}{x} \,d x \]
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