Optimal. Leaf size=232 \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{3 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^3}-\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e^3}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac{3 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac{2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}+\frac{b^2 n^2 \log (d+e x)}{e^3} \]
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Rubi [A] time = 0.445967, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {2353, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2318, 2374, 6589} \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{3 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^3}-\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e^3}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac{3 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac{2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}+\frac{b^2 n^2 \log (d+e x)}{e^3} \]
Antiderivative was successfully verified.
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Rule 2353
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{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx &=\int \left (\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)^3}-\frac{2 d \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^2}-\frac{(2 d) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e^2}+\frac{d^2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e^2}\\ &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac{2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^3}-\frac{(2 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^3}+\frac{\left (b d^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^3}+\frac{(4 b n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^2}\\ &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac{2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^3}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^3}+\frac{(b d n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{e^3}-\frac{(b d n) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^2}-\frac{\left (2 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{e^3}-\frac{\left (4 b^2 n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^3}\\ &=-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac{2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^3}+\frac{4 b^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{e^3}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^3}-\frac{2 b^2 n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e^3}+\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{e^3}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^2}+\frac{\left (b^2 n^2\right ) \int \frac{1}{d+e x} \, dx}{e^2}\\ &=-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac{2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac{b^2 n^2 \log (d+e x)}{e^3}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^3}+\frac{4 b^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{e^3}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^3}-\frac{2 b^2 n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e^3}+\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^3}\\ &=-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac{2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac{b^2 n^2 \log (d+e x)}{e^3}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^3}+\frac{3 b^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{e^3}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^3}-\frac{2 b^2 n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.227929, size = 212, normalized size = 0.91 \[ \frac{4 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )-4 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac{2 b d n \left (a+b \log \left (c x^n\right )\right )}{d+e x}+6 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{4 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-3 \left (a+b \log \left (c x^n\right )\right )^2-2 b^2 n^2 (\log (x)-\log (d+e x))}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.727, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{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{4 \, d e x + 3 \, d^{2}}{e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}} + \frac{2 \, \log \left (e x + d\right )}{e^{3}}\right )} + \int \frac{b^{2} x^{2} \log \left (x^{n}\right )^{2} + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} x^{2} \log \left (x^{n}\right ) +{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right )\right )} x^{2}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}\,{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} x^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b x^{2} \log \left (c x^{n}\right ) + a^{2} x^{2}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, 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{x^{2} \left (a + b \log{\left (c x^{n} \right )}\right )^{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} x^{2}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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