Optimal. Leaf size=322 \[ \frac{6 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac{5 b^2 e n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^4}-\frac{6 b^2 e n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^4}+\frac{2 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac{b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{b d^4 n}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4}+\frac{3 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}-\frac{5 b e n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}+\frac{b^2 e n^2 \log (d+e x)}{d^4}-\frac{2 b^2 n^2}{d^3 x} \]
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Rubi [A] time = 0.537709, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 16, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.696, Rules used = {2353, 2305, 2304, 2302, 30, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2318, 2374, 6589} \[ \frac{6 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac{5 b^2 e n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^4}-\frac{6 b^2 e n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^4}+\frac{2 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac{b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{b d^4 n}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4}+\frac{3 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}-\frac{5 b e n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}+\frac{b^2 e n^2 \log (d+e x)}{d^4}-\frac{2 b^2 n^2}{d^3 x} \]
Antiderivative was successfully verified.
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Rule 2353
Rule 2305
Rule 2304
Rule 2302
Rule 30
Rule 2319
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 2318
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^3} \, dx &=\int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^3 x^2}-\frac{3 e \left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}+\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^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (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} \, dx}{d^4}+\frac{\left (3 e^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d^4}+\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 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^4}-\frac{(3 e) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d^4 n}+\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{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\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{e \left (a+b \log \left (c x^n\right )\right )^3}{b d^4 n}-\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{3 e \left (a+b \log \left (c x^n\right )\right )^2 \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{e x}{d}\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{e x}{d}\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{\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{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{3 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^4}-\frac{4 b^2 e n^2 \text{Li}_2\left (-\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{e x}{d}\right )}{d^4}-\frac{6 b^2 e n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^4}+\frac{(b e n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^4}-\frac{\left (b e^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e 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{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{4 b^2 e n^2 \text{Li}_2\left (-\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{e x}{d}\right )}{d^4}-\frac{6 b^2 e n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^4}+\frac{\left (b^2 e n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\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{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 \text{Li}_2\left (-\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{e x}{d}\right )}{d^4}-\frac{6 b^2 e n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 0.412947, size = 290, normalized size = 0.9 \[ -\frac{-12 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+10 b^2 e n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )+12 b^2 e n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )+\frac{d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-6 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{4 d e \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+10 b e n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b d e n \left (a+b \log \left (c x^n\right )\right )}{d+e x}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{4 b d n \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )^3}{b n}-5 e \left (a+b \log \left (c x^n\right )\right )^2+2 b^2 e n^2 (\log (x)-\log (d+e x))+\frac{4 b^2 d n^2}{x}}{2 d^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.768, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{{x}^{2} \left ( ex+d \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, a^{2}{\left (\frac{6 \, e^{2} x^{2} + 9 \, d e x + 2 \, d^{2}}{d^{3} e^{2} x^{3} + 2 \, d^{4} e x^{2} + d^{5} x} - \frac{6 \, e \log \left (e x + d\right )}{d^{4}} + \frac{6 \, e \log \left (x\right )}{d^{4}}\right )} + \int \frac{b^{2} \log \left (c\right )^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \left (c\right ) + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x^{n}\right )}{e^{3} x^{5} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{3} + d^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{e^{3} x^{5} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{3} + d^{3} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c x^{n} \right )}\right )^{2}}{x^{2} \left (d + e x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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