Optimal. Leaf size=141 \[ \frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^4}-\frac{x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{6 e^2 (d+e x)^2}-\frac{x \left (6 a+6 b \log \left (c x^n\right )+5 b n\right )}{6 e^3 (d+e x)}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (6 a+6 b \log \left (c x^n\right )+11 b n\right )}{6 e^4}-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3} \]
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Rubi [A] time = 0.254077, antiderivative size = 178, normalized size of antiderivative = 1.26, number of steps used = 12, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {43, 2351, 2319, 44, 2314, 31, 2317, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^4}+\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)^3}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4 (d+e x)^2}-\frac{3 x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{b d^2 n}{6 e^4 (d+e x)^2}+\frac{7 b d n}{6 e^4 (d+e x)}+\frac{11 b n \log (d+e x)}{6 e^4}+\frac{7 b n \log (x)}{6 e^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2351
Rule 2319
Rule 44
Rule 2314
Rule 31
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx &=\int \left (-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)^4}+\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)^3}-\frac{3 d \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)^2}+\frac{a+b \log \left (c x^n\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^3}-\frac{(3 d) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^3}+\frac{\left (3 d^2\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^3}-\frac{d^3 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^3}\\ &=\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)^3}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4 (d+e x)^2}-\frac{3 x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^4}-\frac{(b n) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^4}+\frac{\left (3 b d^2 n\right ) \int \frac{1}{x (d+e x)^2} \, dx}{2 e^4}-\frac{\left (b d^3 n\right ) \int \frac{1}{x (d+e x)^3} \, dx}{3 e^4}+\frac{(3 b n) \int \frac{1}{d+e x} \, dx}{e^3}\\ &=\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)^3}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4 (d+e x)^2}-\frac{3 x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac{3 b n \log (d+e x)}{e^4}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^4}+\frac{b n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^4}+\frac{\left (3 b d^2 n\right ) \int \left (\frac{1}{d^2 x}-\frac{e}{d (d+e x)^2}-\frac{e}{d^2 (d+e x)}\right ) \, dx}{2 e^4}-\frac{\left (b d^3 n\right ) \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 e^4}\\ &=-\frac{b d^2 n}{6 e^4 (d+e x)^2}+\frac{7 b d n}{6 e^4 (d+e x)}+\frac{7 b n \log (x)}{6 e^4}+\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)^3}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4 (d+e x)^2}-\frac{3 x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac{11 b n \log (d+e x)}{6 e^4}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^4}+\frac{b n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^4}\\ \end{align*}
Mathematica [A] time = 0.219061, size = 179, normalized size = 1.27 \[ \frac{6 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}-\frac{9 d^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac{18 d \left (a+b \log \left (c x^n\right )\right )}{d+e x}+6 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac{d (3 d+2 e x)}{(d+e x)^2}-2 \log (d+e x)+2 \log (x)\right )-18 b n (\log (x)-\log (d+e x))+9 b n \left (\frac{d}{d+e x}-\log (d+e x)+\log (x)\right )}{6 e^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.158, size = 801, normalized size = 5.7 \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}{6} \, a{\left (\frac{18 \, d e^{2} x^{2} + 27 \, d^{2} e x + 11 \, d^{3}}{e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}} + \frac{6 \, \log \left (e x + d\right )}{e^{4}}\right )} + b \int \frac{x^{3} \log \left (c\right ) + x^{3} \log \left (x^{n}\right )}{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 x^{3} \log \left (c x^{n}\right ) + a x^{3}}{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 [A] time = 66.2421, size = 500, normalized size = 3.55 \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{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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