Integrand size = 23, antiderivative size = 351 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\frac {b^2 n^2}{3 d^3 (d+e x)}+\frac {b^2 n^2 \log (x)}{3 d^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {5 b e n x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 (d+e x)}-\frac {5 \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b d^4 n}-\frac {2 b^2 n^2 \log (d+e x)}{d^4}+\frac {11 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^4}+\frac {11 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d^4}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^4} \]
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Time = 0.55 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.03, number of steps used = 25, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31, 46} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {5 b n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4}+\frac {5 b e n x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 (d+e x)}+\frac {2 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}-\frac {5 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{3 d^4}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^4}-\frac {2 b^2 n^2 \log (d+e x)}{d^4}+\frac {b^2 n^2 \log (x)}{3 d^4}+\frac {b^2 n^2}{3 d^3 (d+e x)} \]
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Rule 31
Rule 46
Rule 2351
Rule 2354
Rule 2355
Rule 2356
Rule 2379
Rule 2389
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 (d+e x)^3} \, dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx}{d} \\ & = \frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx}{d^2}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{d^2}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 d} \\ & = \frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}+\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx}{d^3}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{d^3}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 d^2}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{d^2}+\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 d^2} \\ & = -\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac {\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 {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{d^4}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 d^3}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^3}+\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^4}+\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 d^3}+\frac {(b e n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^3}+\frac {\left (b^2 n^2\right ) \int \frac {1}{x (d+e x)^2} \, dx}{3 d^2} \\ & = -\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {5 b e n x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 (d+e x)}+\frac {5 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac {\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 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4}-\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{3 d^4}-\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^4}-\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^4}-\frac {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {d}{e x}\right )}{x} \, dx}{d^4}+\frac {\left (b^2 n^2\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{3 d^2}-\frac {\left (2 b^2 e n^2\right ) \int \frac {1}{d+e x} \, dx}{3 d^4}-\frac {\left (b^2 e n^2\right ) \int \frac {1}{d+e x} \, dx}{d^4} \\ & = \frac {b^2 n^2}{3 d^3 (d+e x)}+\frac {b^2 n^2 \log (x)}{3 d^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {5 b e n x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 (d+e x)}+\frac {5 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac {2 b^2 n^2 \log (d+e x)}{d^4}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}-\frac {5 b^2 n^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{3 d^4}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4}+\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^4}+\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {d}{e x}\right )}{d^4} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\frac {-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {10 b d n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-11 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}+\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {6 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3}{b n}+10 b^2 n^2 (\log (x)-\log (d+e x))+\frac {2 b^2 n^2 (d+(d+e x) \log (x)-(d+e x) \log (d+e x))}{d+e x}+22 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-6 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+22 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+12 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{6 d^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.68 (sec) , antiderivative size = 894, normalized size of antiderivative = 2.55
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x \left (d + e x\right )^{4}}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x\,{\left (d+e\,x\right )}^4} \,d x \]
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