Optimal. Leaf size=210 \[ -\frac{b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{3 d^2 e^2}-\frac{b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 e^2}+\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 d e^2 (d+e x)}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}+\frac{b^2 n^2}{3 d e^2 (d+e x)} \]
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Rubi [A] time = 0.620166, antiderivative size = 229, normalized size of antiderivative = 1.09, number of steps used = 22, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {2353, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 44} \[ -\frac{b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{3 d^2 e^2}-\frac{b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 e^2}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}+\frac{b^2 n^2 \log (x)}{3 d^2 e^2}+\frac{b^2 n^2}{3 d e^2 (d+e x)} \]
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 44
Rubi steps
\begin{align*} \int \frac{x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx &=\int \left (-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)^4}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)^3}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e}-\frac{d \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx}{e}\\ &=\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^2}-\frac{(2 b d n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 e^2}\\ &=\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}-\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 e^2}+\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d e^2}+\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 e}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d e}\\ &=-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}+\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^2 e^2}-\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 d e^2}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2 e}+\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 d e}+\frac{\left (b^2 n^2\right ) \int \frac{1}{x (d+e x)^2} \, dx}{3 e^2}+\frac{\left (b^2 n^2\right ) \int \frac{1}{d+e x} \, dx}{d^2 e}\\ &=-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac{b^2 n^2 \log (d+e x)}{d^2 e^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^2 e^2}-\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{3 d^2 e^2}+\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{3 d^2 e}+\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 e^2}+\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^2 e^2}-\frac{\left (2 b^2 n^2\right ) \int \frac{1}{d+e x} \, dx}{3 d^2 e}\\ &=\frac{b^2 n^2}{3 d e^2 (d+e x)}+\frac{b^2 n^2 \log (x)}{3 d^2 e^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{3 d^2 e^2}-\frac{b^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^2 e^2}-\frac{\left (2 b^2 n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{3 d^2 e^2}\\ &=\frac{b^2 n^2}{3 d e^2 (d+e x)}+\frac{b^2 n^2 \log (x)}{3 d^2 e^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{3 d^2 e^2}-\frac{b^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{3 d^2 e^2}\\ \end{align*}
Mathematica [A] time = 0.23456, size = 281, normalized size = 1.34 \[ \frac{-2 b^2 n^2 (d+e x)^3 \text{PolyLog}\left (2,-\frac{e x}{d}\right )+3 a^2 d e^2 x^2+a^2 e^3 x^3-2 b \log \left (c x^n\right ) \left (b n (d+e x)^3 \log \left (\frac{e x}{d}+1\right )-e x (a e x (3 d+e x)+b d n (d+e x))\right )+2 a b d^2 e n x-2 a b d^3 n \log \left (\frac{e x}{d}+1\right )-6 a b d^2 e n x \log \left (\frac{e x}{d}+1\right )+2 a b d e^2 n x^2-6 a b d e^2 n x^2 \log \left (\frac{e x}{d}+1\right )-2 a b e^3 n x^3 \log \left (\frac{e x}{d}+1\right )+b^2 e^2 x^2 (3 d+e x) \log ^2\left (c x^n\right )+4 b^2 d^2 e n^2 x+2 b^2 d^3 n^2+2 b^2 d e^2 n^2 x^2}{6 d^2 e^2 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.276, size = 1400, normalized size = 6.7 \begin{align*} \text{result too large to display} \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 b n{\left (\frac{x}{d e^{3} x^{2} + 2 \, d^{2} e^{2} x + d^{3} e} - \frac{\log \left (e x + d\right )}{d^{2} e^{2}} + \frac{\log \left (x\right )}{d^{2} e^{2}}\right )} - \frac{1}{6} \,{\left (\frac{{\left (3 \, e x + d\right )} \log \left (x^{n}\right )^{2}}{e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}} - 6 \, \int \frac{3 \, e^{2} x^{2} \log \left (c\right )^{2} +{\left (4 \, d e n x + d^{2} n + 3 \,{\left (e^{2} n + 2 \, e^{2} \log \left (c\right )\right )} x^{2}\right )} \log \left (x^{n}\right )}{3 \,{\left (e^{6} x^{5} + 4 \, d e^{5} x^{4} + 6 \, d^{2} e^{4} x^{3} + 4 \, d^{3} e^{3} x^{2} + d^{4} e^{2} x\right )}}\,{d x}\right )} b^{2} - \frac{{\left (3 \, e x + d\right )} a b \log \left (c x^{n}\right )}{3 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac{{\left (3 \, e x + d\right )} a^{2}}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \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 \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} x}{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 \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}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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