Optimal. Leaf size=398 \[ -\frac{8 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^5}-\frac{26 b^2 d n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{3 e^5}+\frac{8 b^2 d n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e^5}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac{b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac{5 d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5}+\frac{10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}-\frac{4 d \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^5}-\frac{26 b d n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^5}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac{2 a b n x}{e^4}-\frac{2 b^2 n x \log \left (c x^n\right )}{e^4}-\frac{b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac{b^2 d n^2 \log (x)}{3 e^5}-\frac{3 b^2 d n^2 \log (d+e x)}{e^5}+\frac{2 b^2 n^2 x}{e^4} \]
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Rubi [A] time = 0.825625, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 15, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.652, Rules used = {2353, 2296, 2295, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 44, 2318, 2374, 6589} \[ -\frac{8 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^5}-\frac{26 b^2 d n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{3 e^5}+\frac{8 b^2 d n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e^5}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac{b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac{5 d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5}+\frac{10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}-\frac{4 d \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^5}-\frac{26 b d n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^5}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac{2 a b n x}{e^4}-\frac{2 b^2 n x \log \left (c x^n\right )}{e^4}-\frac{b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac{b^2 d n^2 \log (x)}{3 e^5}-\frac{3 b^2 d n^2 \log (d+e x)}{e^5}+\frac{2 b^2 n^2 x}{e^4} \]
Antiderivative was successfully verified.
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Rule 2353
Rule 2296
Rule 2295
Rule 2319
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 44
Rule 2318
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx &=\int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{e^4}+\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)^4}-\frac{4 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)^3}+\frac{6 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)^2}-\frac{4 d \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}\right ) \, dx\\ &=\frac{\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^4}-\frac{(4 d) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^4}+\frac{\left (6 d^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e^4}-\frac{\left (4 d^3\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e^4}+\frac{d^4 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx}{e^4}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac{4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^5}+\frac{(8 b d n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^5}-\frac{\left (4 b d^3 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^5}+\frac{\left (2 b d^4 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 e^5}-\frac{(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^4}-\frac{(12 b d n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^4}\\ &=-\frac{2 a b n x}{e^4}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac{12 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^5}-\frac{4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^5}-\frac{8 b d n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^5}-\frac{\left (4 b d^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{e^5}+\frac{\left (2 b d^3 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 e^5}-\frac{\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e^4}+\frac{\left (4 b d^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^4}-\frac{\left (2 b d^3 n\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 e^4}+\frac{\left (8 b^2 d n^2\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{e^5}+\frac{\left (12 b^2 d n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^5}\\ &=-\frac{2 a b n x}{e^4}+\frac{2 b^2 n^2 x}{e^4}-\frac{2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac{b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac{4 b d n x \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac{12 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^5}-\frac{4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^5}-\frac{12 b^2 d n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{e^5}-\frac{8 b d n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^5}+\frac{8 b^2 d n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e^5}-\frac{(4 b d n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{e^5}+\frac{\left (2 b d^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 e^5}+\frac{(4 b d n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^4}-\frac{\left (2 b d^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 e^4}-\frac{\left (b^2 d^3 n^2\right ) \int \frac{1}{x (d+e x)^2} \, dx}{3 e^5}-\frac{\left (4 b^2 d n^2\right ) \int \frac{1}{d+e x} \, dx}{e^4}\\ &=-\frac{2 a b n x}{e^4}+\frac{2 b^2 n^2 x}{e^4}-\frac{2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac{b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac{10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}-\frac{2 d \left (a+b \log \left (c x^n\right )\right )^2}{e^5}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac{4 b^2 d n^2 \log (d+e x)}{e^5}-\frac{8 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^5}-\frac{4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^5}-\frac{12 b^2 d n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{e^5}-\frac{8 b d n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^5}+\frac{8 b^2 d n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e^5}+\frac{(2 b d n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{3 e^5}-\frac{(2 b d n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{3 e^4}-\frac{\left (4 b^2 d n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^5}-\frac{\left (b^2 d^3 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 e^5}+\frac{\left (2 b^2 d n^2\right ) \int \frac{1}{d+e x} \, dx}{3 e^4}\\ &=-\frac{2 a b n x}{e^4}+\frac{2 b^2 n^2 x}{e^4}-\frac{b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac{b^2 d n^2 \log (x)}{3 e^5}-\frac{2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac{b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac{10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}-\frac{5 d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac{3 b^2 d n^2 \log (d+e x)}{e^5}-\frac{26 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{3 e^5}-\frac{4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^5}-\frac{8 b^2 d n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{e^5}-\frac{8 b d n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^5}+\frac{8 b^2 d n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e^5}+\frac{\left (2 b^2 d n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{3 e^5}\\ &=-\frac{2 a b n x}{e^4}+\frac{2 b^2 n^2 x}{e^4}-\frac{b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac{b^2 d n^2 \log (x)}{3 e^5}-\frac{2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac{b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac{10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}-\frac{5 d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac{3 b^2 d n^2 \log (d+e x)}{e^5}-\frac{26 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{3 e^5}-\frac{4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^5}-\frac{26 b^2 d n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{3 e^5}-\frac{8 b d n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^5}+\frac{8 b^2 d n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e^5}\\ \end{align*}
Mathematica [A] time = 0.640398, size = 344, normalized size = 0.86 \[ -\frac{24 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+26 b^2 d n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )-24 b^2 d n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )+\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}-\frac{6 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-\frac{b d^3 n \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac{18 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac{10 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{d+e x}+12 d \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+26 b d n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-13 d \left (a+b \log \left (c x^n\right )\right )^2-3 e x \left (a+b \log \left (c x^n\right )\right )^2+6 b e n x \left (a+b \log \left (c x^n\right )-b n\right )-10 b^2 d n^2 (\log (x)-\log (d+e x))+\frac{b^2 d n^2 (\log (x) (d+e x)-(d+e x) \log (d+e x)+d)}{d+e x}}{3 e^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.868, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{ \left ( ex+d \right ) ^{4}}}\, 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}{3} \, a^{2}{\left (\frac{18 \, d^{2} e^{2} x^{2} + 30 \, d^{3} e x + 13 \, d^{4}}{e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}} - \frac{3 \, x}{e^{4}} + \frac{12 \, d \log \left (e x + d\right )}{e^{5}}\right )} + \int \frac{b^{2} x^{4} \log \left (x^{n}\right )^{2} + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} x^{4} \log \left (x^{n}\right ) +{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right )\right )} x^{4}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}\,{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^{4} \log \left (c x^{n}\right )^{2} + 2 \, a b x^{4} \log \left (c x^{n}\right ) + a^{2} x^{4}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, 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^{4} \left (a + b \log{\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{4}}\, 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^{4}}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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