Integrand size = 24, antiderivative size = 263 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=-\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {3 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt {x}}\right )}{d^2} \]
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Time = 0.32 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\frac {6 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}+\frac {6 b^2 e^2 n^2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}-\frac {3 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2}-\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (2,\frac {\sqrt {x} e}{d}+1\right )}{d^2}+\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt {x}}\right )}{d^2} \]
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Rule 2354
Rule 2355
Rule 2379
Rule 2389
Rule 2421
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rule 6724
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^3} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{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 )^3}{x}+(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{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 )^3}{x}+\frac {(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right )}{d}-\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e \sqrt {x}\right )}{d} \\ & = -\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {3 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 )^2}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\frac {\left (6 b^2 e n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt {x}\right )}{d^2}+\frac {\left (6 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^2} \\ & = -\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {3 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 )^2}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\frac {6 b^2 e^2 n^2 \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 {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{d^2}-\frac {\left (6 b^3 e^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^2}-\frac {\left (6 b^3 e^2 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d}{x}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^2} \\ & = -\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {3 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 )^2}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\frac {6 b^2 e^2 n^2 \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 {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {Li}_3\left (\frac {d}{d+e \sqrt {x}}\right )}{d^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(536\) vs. \(2(263)=526\).
Time = 0.53 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\frac {-3 b d e n \sqrt {x} \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-3 b d^2 n \log \left (d+e \sqrt {x}\right ) \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+3 b e^2 n x \log \left (d+e \sqrt {x}\right ) \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-d^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3-\frac {3}{2} b e^2 n x \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log (x)+3 b^2 n^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (\left (d+e \sqrt {x}\right ) \log \left (d+e \sqrt {x}\right ) \left (-2 e \sqrt {x}+\left (-d+e \sqrt {x}\right ) \log \left (d+e \sqrt {x}\right )\right )-2 e^2 x \left (-1+\log \left (d+e \sqrt {x}\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )-2 e^2 x \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )\right )+b^3 n^3 \left (\left (d+e \sqrt {x}\right ) \log ^2\left (d+e \sqrt {x}\right ) \left (-3 e \sqrt {x}+\left (-d+e \sqrt {x}\right ) \log \left (d+e \sqrt {x}\right )\right )-3 e^2 x \left (-2+\log \left (d+e \sqrt {x}\right )\right ) \log \left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )-6 e^2 x \left (-1+\log \left (d+e \sqrt {x}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )+6 e^2 x \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right )\right )}{d^2 x} \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{3}}{x^{2}}d x\]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}}{x^{2}}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^3}{x^2} \,d x \]
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