Optimal. Leaf size=152 \[ \frac{3 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^4}+\frac{d^2 \log \left (\frac{e x}{d}+1\right ) \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{e^4}-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac{x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{2 e^2}-\frac{d x (3 a+b n)}{e^3}-\frac{3 b d x \log \left (c x^n\right )}{e^3}+\frac{3 b d n x}{e^3}-\frac{3 b n x^2}{4 e^2} \]
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Rubi [A] time = 0.180966, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {43, 2351, 2295, 2304, 2314, 31, 2317, 2391} \[ \frac{3 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^4}-\frac{d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac{3 d^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{2 a d x}{e^3}-\frac{2 b d x \log \left (c x^n\right )}{e^3}+\frac{b d^2 n \log (d+e x)}{e^4}+\frac{2 b d n x}{e^3}-\frac{b n x^2}{4 e^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2351
Rule 2295
Rule 2304
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)^2} \, dx &=\int \left (-\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)^2}+\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac{(2 d) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^3}+\frac{\left (3 d^2\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^3}-\frac{d^3 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^3}+\frac{\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}\\ &=-\frac{2 a d x}{e^3}-\frac{b n x^2}{4 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^4}-\frac{(2 b d) \int \log \left (c x^n\right ) \, dx}{e^3}-\frac{\left (3 b d^2 n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^4}+\frac{\left (b d^2 n\right ) \int \frac{1}{d+e x} \, dx}{e^3}\\ &=-\frac{2 a d x}{e^3}+\frac{2 b d n x}{e^3}-\frac{b n x^2}{4 e^2}-\frac{2 b d x \log \left (c x^n\right )}{e^3}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac{b d^2 n \log (d+e x)}{e^4}+\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^4}+\frac{3 b d^2 n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^4}\\ \end{align*}
Mathematica [A] time = 0.116074, size = 141, normalized size = 0.93 \[ \frac{12 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{4 d^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+12 d^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-8 a d e x-8 b d e x \log \left (c x^n\right )-4 b d^2 n (\log (x)-\log (d+e x))+8 b d e n x-b e^2 n x^2}{4 e^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.205, size = 739, normalized size = 4.9 \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*} \frac{1}{2} \,{\left (\frac{2 \, d^{3}}{e^{5} x + d e^{4}} + \frac{6 \, d^{2} \log \left (e x + d\right )}{e^{4}} + \frac{e x^{2} - 4 \, d x}{e^{3}}\right )} a + b \int \frac{x^{3} \log \left (c\right ) + x^{3} \log \left (x^{n}\right )}{e^{2} x^{2} + 2 \, d e x + d^{2}}\,{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^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 131.855, size = 304, normalized size = 2. \begin{align*} - \frac{a d^{3} \left (\begin{cases} \frac{x}{d^{2}} & \text{for}\: e = 0 \\- \frac{1}{d e + e^{2} x} & \text{otherwise} \end{cases}\right )}{e^{3}} + \frac{3 a d^{2} \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right )}{e^{3}} - \frac{2 a d x}{e^{3}} + \frac{a x^{2}}{2 e^{2}} + \frac{b d^{3} n \left (\begin{cases} \frac{x}{d^{2}} & \text{for}\: e = 0 \\- \frac{\log{\left (x \right )}}{d e} + \frac{\log{\left (\frac{d}{e} + x \right )}}{d e} & \text{otherwise} \end{cases}\right )}{e^{3}} - \frac{b d^{3} \left (\begin{cases} \frac{x}{d^{2}} & \text{for}\: e = 0 \\- \frac{1}{d e + e^{2} x} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )}}{e^{3}} - \frac{3 b d^{2} n \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\begin{cases} \log{\left (d \right )} \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (d \right )} \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\right ) & \text{for}\: \frac{1}{\left |{x}\right |} < 1 \\-{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x} \right )} \log{\left (d \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (d \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\right ) & \text{otherwise} \end{cases}}{e} & \text{otherwise} \end{cases}\right )}{e^{3}} + \frac{3 b d^{2} \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )}}{e^{3}} + \frac{2 b d n x}{e^{3}} - \frac{2 b d x \log{\left (c x^{n} \right )}}{e^{3}} - \frac{b n x^{2}}{4 e^{2}} + \frac{b x^{2} \log{\left (c x^{n} \right )}}{2 e^{2}} \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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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