Optimal. Leaf size=174 \[ \frac{b n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d^4}-\frac{\log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac{e x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}+\frac{a+b \log \left (c x^n\right )}{2 d^2 (d+e x)^2}+\frac{a+b \log \left (c x^n\right )}{3 d (d+e x)^3}-\frac{5 b n}{6 d^3 (d+e x)}-\frac{b n}{6 d^2 (d+e x)^2}+\frac{11 b n \log (d+e x)}{6 d^4}-\frac{5 b n \log (x)}{6 d^4} \]
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Rubi [A] time = 0.357489, antiderivative size = 196, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^4}-\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac{e x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}+\frac{a+b \log \left (c x^n\right )}{2 d^2 (d+e x)^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^4 n}+\frac{a+b \log \left (c x^n\right )}{3 d (d+e x)^3}-\frac{5 b n}{6 d^3 (d+e x)}-\frac{b n}{6 d^2 (d+e x)^2}+\frac{11 b n \log (d+e x)}{6 d^4}-\frac{5 b n \log (x)}{6 d^4} \]
Antiderivative was successfully verified.
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Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 2319
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{d}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d}\\ &=\frac{a+b \log \left (c x^n\right )}{3 d (d+e x)^3}+\frac{\int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{d^2}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^2}-\frac{(b n) \int \frac{1}{x (d+e x)^3} \, dx}{3 d}\\ &=\frac{a+b \log \left (c x^n\right )}{3 d (d+e x)^3}+\frac{a+b \log \left (c x^n\right )}{2 d^2 (d+e x)^2}+\frac{\int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^3}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^3}-\frac{(b n) \int \frac{1}{x (d+e x)^2} \, dx}{2 d^2}-\frac{(b n) \int \left (\frac{1}{d^3 x}-\frac{e}{d (d+e x)^3}-\frac{e}{d^2 (d+e x)^2}-\frac{e}{d^3 (d+e x)}\right ) \, dx}{3 d}\\ &=-\frac{b n}{6 d^2 (d+e x)^2}-\frac{b n}{3 d^3 (d+e x)}-\frac{b n \log (x)}{3 d^4}+\frac{a+b \log \left (c x^n\right )}{3 d (d+e x)^3}+\frac{a+b \log \left (c x^n\right )}{2 d^2 (d+e x)^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}+\frac{b n \log (d+e x)}{3 d^4}+\frac{\int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^4}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^4}-\frac{(b n) \int \left (\frac{1}{d^2 x}-\frac{e}{d (d+e x)^2}-\frac{e}{d^2 (d+e x)}\right ) \, dx}{2 d^2}+\frac{(b e n) \int \frac{1}{d+e x} \, dx}{d^4}\\ &=-\frac{b n}{6 d^2 (d+e x)^2}-\frac{5 b n}{6 d^3 (d+e x)}-\frac{5 b n \log (x)}{6 d^4}+\frac{a+b \log \left (c x^n\right )}{3 d (d+e x)^3}+\frac{a+b \log \left (c x^n\right )}{2 d^2 (d+e x)^2}-\frac{e 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}{2 b d^4 n}+\frac{11 b n \log (d+e x)}{6 d^4}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^4}+\frac{(b n) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^4}\\ &=-\frac{b n}{6 d^2 (d+e x)^2}-\frac{5 b n}{6 d^3 (d+e x)}-\frac{5 b n \log (x)}{6 d^4}+\frac{a+b \log \left (c x^n\right )}{3 d (d+e x)^3}+\frac{a+b \log \left (c x^n\right )}{2 d^2 (d+e x)^2}-\frac{e 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}{2 b d^4 n}+\frac{11 b n \log (d+e x)}{6 d^4}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^4}-\frac{b n \text{Li}_2\left (-\frac{e x}{d}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 0.170912, size = 222, normalized size = 1.28 \[ \frac{-6 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{3 a^2}{b n}+\frac{6 a \log \left (c x^n\right )}{n}+\frac{2 a d^3}{(d+e x)^3}+\frac{3 a d^2}{(d+e x)^2}+\frac{6 a d}{d+e x}-6 a \log \left (\frac{e x}{d}+1\right )+\frac{2 b d^3 \log \left (c x^n\right )}{(d+e x)^3}+\frac{3 b d^2 \log \left (c x^n\right )}{(d+e x)^2}+\frac{6 b d \log \left (c x^n\right )}{d+e x}-6 b \log \left (c x^n\right ) \log \left (\frac{e x}{d}+1\right )+\frac{3 b \log ^2\left (c x^n\right )}{n}-\frac{b d^2 n}{(d+e x)^2}-\frac{5 b d n}{d+e x}+11 b n \log (d+e x)-11 b n \log (x)}{6 d^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.164, size = 884, normalized size = 5.1 \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}{6} \, a{\left (\frac{6 \, e^{2} x^{2} + 15 \, d e x + 11 \, d^{2}}{d^{3} e^{3} x^{3} + 3 \, d^{4} e^{2} x^{2} + 3 \, d^{5} e x + d^{6}} - \frac{6 \, \log \left (e x + d\right )}{d^{4}} + \frac{6 \, \log \left (x\right )}{d^{4}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{4} x^{5} + 4 \, d e^{3} x^{4} + 6 \, d^{2} e^{2} x^{3} + 4 \, d^{3} e x^{2} + d^{4} x}\,{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 \log \left (c x^{n}\right ) + a}{e^{4} x^{5} + 4 \, d e^{3} x^{4} + 6 \, d^{2} e^{2} x^{3} + 4 \, d^{3} e x^{2} + d^{4} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 130.542, size = 493, normalized size = 2.83 \begin{align*} \text{result too large to display} \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{b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{4} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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