Integrand size = 24, antiderivative size = 155 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^2} \, dx=-\frac {2 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2 \sqrt {x}}-\frac {2 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+\frac {b^2 e^2 n^2 \log (x)}{d^2}+\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^2} \]
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Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2445, 2458, 2389, 2379, 2438, 2351, 31} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^2} \, dx=-\frac {2 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}-\frac {2 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2 \sqrt {x}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^2}+\frac {b^2 e^2 n^2 \log (x)}{d^2} \]
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Rule 31
Rule 2351
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+\frac {(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right )}{d}-\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e \sqrt {x}\right )}{d} \\ & = -\frac {2 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2 \sqrt {x}}-\frac {2 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+\frac {\left (2 b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt {x}\right )}{d^2}+\frac {\left (2 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^2} \\ & = -\frac {2 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2 \sqrt {x}}-\frac {2 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x}+\frac {b^2 e^2 n^2 \log (x)}{d^2}+\frac {2 b^2 e^2 n^2 \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{d^2} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^2} \, dx=2 \left (-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x}+b e n \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{d \sqrt {x}}+\frac {e \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 b d^2 n}-\frac {e \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {b n \left (-\frac {e \log \left (d+e \sqrt {x}\right )}{d}+\frac {e \log (x)}{2 d}\right )}{d}-\frac {b e n \operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right )}{d^2}\right )\right ) \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{2}}{x^{2}}d x\]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{2}}{x^{2}}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^2}{x^2} \,d x \]
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