Integrand size = 23, antiderivative size = 322 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^3} \, dx=-\frac {2 b^2 n^2}{d^3 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}+\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{b d^4 n}+\frac {b^2 e n^2 \log (d+e x)}{d^4}-\frac {5 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}+\frac {3 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^4}-\frac {5 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}+\frac {6 b e n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}-\frac {6 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^4} \]
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Time = 0.34 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2395, 2342, 2341, 2356, 2389, 2379, 2438, 2351, 31, 2355, 2354, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^3} \, dx=\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac {b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}-\frac {6 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {3 e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac {b e n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {4 b e n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}+\frac {b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^4}-\frac {4 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}-\frac {6 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^4}+\frac {b^2 e n^2 \log (d+e x)}{d^4}-\frac {2 b^2 n^2}{d^3 x} \]
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Rule 31
Rule 2341
Rule 2342
Rule 2351
Rule 2354
Rule 2355
Rule 2356
Rule 2379
Rule 2389
Rule 2395
Rule 2421
Rule 2438
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^3 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^2 (d+e x)^3}+\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{d^3}-\frac {(3 e) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx}{d^3}+\frac {\left (2 e^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{d^3}+\frac {e^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{d^2} \\ & = -\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}+\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {3 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^3}-\frac {(6 b e n) \int \frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{d^4}+\frac {(b e n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{d^2}-\frac {\left (4 b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^4} \\ & = -\frac {2 b^2 n^2}{d^3 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}+\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {3 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac {4 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}-\frac {6 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4}+\frac {(b e n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^3}-\frac {\left (b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^3}+\frac {\left (4 b^2 e n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^4}+\frac {\left (6 b^2 e n^2\right ) \int \frac {\text {Li}_2\left (-\frac {d}{e x}\right )}{x} \, dx}{d^4} \\ & = -\frac {2 b^2 n^2}{d^3 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}-\frac {b e n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}+\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {3 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac {4 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}-\frac {6 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4}-\frac {4 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^4}-\frac {6 b^2 e n^2 \text {Li}_3\left (-\frac {d}{e x}\right )}{d^4}+\frac {\left (b^2 e n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^4}+\frac {\left (b^2 e^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{d^4} \\ & = -\frac {2 b^2 n^2}{d^3 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}-\frac {b e n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}+\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {3 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac {b^2 e n^2 \log (d+e x)}{d^4}-\frac {4 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}+\frac {b^2 e n^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4}-\frac {6 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4}-\frac {4 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^4}-\frac {6 b^2 e n^2 \text {Li}_3\left (-\frac {d}{e x}\right )}{d^4} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^3} \, dx=-\frac {\frac {4 b^2 d n^2}{x}+\frac {4 b d n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 b d e n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-5 e \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {4 d e \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^3}{b n}+2 b^2 e n^2 (\log (x)-\log (d+e x))+10 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-6 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+10 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 b e n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+12 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{2 d^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.50 (sec) , antiderivative size = 908, normalized size of antiderivative = 2.82
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3} x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{2} \left (d + e x\right )^{3}}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3} x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2\,{\left (d+e\,x\right )}^3} \,d x \]
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