Optimal. Leaf size=126 \[ \frac{b^2 n^2 \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d^2 e}-\frac{b n \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac{b^2 n^2 \log (d+e x)}{d^2 e} \]
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Rubi [A] time = 0.201489, antiderivative size = 145, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2319, 2347, 2344, 2301, 2317, 2391, 2314, 31} \[ -\frac{b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^2 e}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}-\frac{b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac{b^2 n^2 \log (d+e x)}{d^2 e} \]
Antiderivative was successfully verified.
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Rule 2319
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d}+\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d e}\\ &=-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2}+\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^2 e}+\frac{\left (b^2 n^2\right ) \int \frac{1}{d+e x} \, dx}{d^2}\\ &=-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac{b^2 n^2 \log (d+e x)}{d^2 e}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^2 e}+\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^2 e}\\ &=-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac{b^2 n^2 \log (d+e x)}{d^2 e}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^2 e}-\frac{b^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^2 e}\\ \end{align*}
Mathematica [A] time = 0.100528, size = 146, normalized size = 1.16 \[ \frac{b n \left (-\frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^2}-\frac{\log \left (\frac{d+e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac{a+b \log \left (c x^n\right )}{d (d+e x)}-\frac{b n \left (\frac{\log (x)}{d}-\frac{\log (d+e x)}{d}\right )}{d}\right )}{e}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.256, size = 990, normalized size = 7.9 \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*} a b n{\left (\frac{1}{d e^{2} x + d^{2} e} - \frac{\log \left (e x + d\right )}{d^{2} e} + \frac{\log \left (x\right )}{d^{2} e}\right )} - \frac{1}{2} \, b^{2}{\left (\frac{\log \left (x^{n}\right )^{2}}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e} - 2 \, \int \frac{e x \log \left (c\right )^{2} +{\left (d n +{\left (e n + 2 \, e \log \left (c\right )\right )} x\right )} \log \left (x^{n}\right )}{e^{4} x^{4} + 3 \, d e^{3} x^{3} + 3 \, d^{2} e^{2} x^{2} + d^{3} e x}\,{d x}\right )} - \frac{a b \log \left (c x^{n}\right )}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e} - \frac{a^{2}}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\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} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{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{\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}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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