Integrand size = 25, antiderivative size = 267 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^3} \, dx=\frac {b e n x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2 \left (d+e x^r\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}-\frac {b^2 n^2 \log \left (d+e x^r\right )}{d^3 r^3}-\frac {3 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^3 r^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^3 r^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d x^{-r}}{e}\right )}{d^3 r^3} \]
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Time = 0.59 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2391, 2379, 2421, 6724, 2376, 2438, 2373, 266} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^3} \, dx=\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2}+\frac {3 b n \log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2}+\frac {b e n x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2 \left (d+e x^r\right )}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3 r}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}-\frac {3 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^3 r^3}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d x^{-r}}{e}\right )}{d^3 r^3}-\frac {b^2 n^2 \log \left (d+e x^r\right )}{d^3 r^3} \]
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Rule 266
Rule 2373
Rule 2376
Rule 2379
Rule 2391
Rule 2421
Rule 2438
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^2} \, dx}{d}-\frac {e \int \frac {x^{-1+r} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^r\right )^3} \, dx}{d} \\ & = \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}+\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx}{d^2}-\frac {e \int \frac {x^{-1+r} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^r\right )^2} \, dx}{d^2}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^2} \, dx}{d r} \\ & = \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{x} \, dx}{d^3 r}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx}{d^2 r}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx}{d^2 r}+\frac {(b e n) \int \frac {x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx}{d^2 r} \\ & = \frac {b e n x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2 \left (d+e x^r\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2}-\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+\frac {d x^{-r}}{e}\right )}{x} \, dx}{d^3 r^2}-\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {d x^{-r}}{e}\right )}{x} \, dx}{d^3 r^2}-\frac {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{x} \, dx}{d^3 r^2}-\frac {\left (b^2 e n^2\right ) \int \frac {x^{-1+r}}{d+e x^r} \, dx}{d^3 r^2} \\ & = \frac {b e n x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2 \left (d+e x^r\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}-\frac {b^2 n^2 \log \left (d+e x^r\right )}{d^3 r^3}-\frac {3 b^2 n^2 \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2}+\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^3} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^3} \, dx=\frac {\frac {d^2 r^2 \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^r\right )^2}+\frac {2 d r \left (a+b \log \left (c x^n\right )\right ) \left (-b n+a r+b r \log \left (c x^n\right )\right )}{d+e x^r}-2 b^2 n^2 \log \left (d-d x^r\right )+6 a b n r \log \left (d-d x^r\right )-2 a^2 r^2 \log \left (d-d x^r\right )+4 a b r^2 \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+6 b^2 n r \left (-n \log (x)+\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )-2 b^2 r^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2 \log \left (d-d x^r\right )-6 b^2 n^2 \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )\right )+4 a b n r \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )\right )+4 b^2 n r \left (-n \log (x)+\log \left (c x^n\right )\right ) \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )\right )-2 b^2 n^2 \left (r^2 \log ^2(x) \log \left (1+\frac {d x^{-r}}{e}\right )-2 r \log (x) \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )-2 \operatorname {PolyLog}\left (3,-\frac {d x^{-r}}{e}\right )\right )}{2 d^3 r^3} \]
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\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{x \left (d +e \,x^{r}\right )^{3}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 1165 vs. \(2 (263) = 526\).
Time = 0.28 (sec) , antiderivative size = 1165, normalized size of antiderivative = 4.36 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x^{r} + d\right )}^{3} x} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x^{r} + d\right )}^{3} x} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x\,{\left (d+e\,x^r\right )}^3} \,d x \]
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