Optimal. Leaf size=185 \[ \frac{b e^4 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^5}+\frac{e^4 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac{e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac{e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}-\frac{a e^3 x}{d^4}-\frac{b e^3 x \log \left (c x^n\right )}{d^4}-\frac{b e^2 n x^2}{4 d^3}+\frac{b e^3 n x}{d^4}+\frac{b e n x^3}{9 d^2}-\frac{b n x^4}{16 d} \]
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Rubi [A] time = 0.199915, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {263, 43, 2351, 2295, 2304, 2317, 2391} \[ \frac{b e^4 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^5}+\frac{e^4 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac{e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac{e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}-\frac{a e^3 x}{d^4}-\frac{b e^3 x \log \left (c x^n\right )}{d^4}-\frac{b e^2 n x^2}{4 d^3}+\frac{b e^3 n x}{d^4}+\frac{b e n x^3}{9 d^2}-\frac{b n x^4}{16 d} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rule 2351
Rule 2295
Rule 2304
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{d+\frac{e}{x}} \, dx &=\int \left (-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{e x^2 \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{d}+\frac{e^4 \left (a+b \log \left (c x^n\right )\right )}{d^4 (e+d x)}\right ) \, dx\\ &=\frac{\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx}{d}-\frac{e \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^2}+\frac{e^2 \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^3}-\frac{e^3 \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^4}+\frac{e^4 \int \frac{a+b \log \left (c x^n\right )}{e+d x} \, dx}{d^4}\\ &=-\frac{a e^3 x}{d^4}-\frac{b e^2 n x^2}{4 d^3}+\frac{b e n x^3}{9 d^2}-\frac{b n x^4}{16 d}+\frac{e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac{e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}+\frac{e^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x}{e}\right )}{d^5}-\frac{\left (b e^3\right ) \int \log \left (c x^n\right ) \, dx}{d^4}-\frac{\left (b e^4 n\right ) \int \frac{\log \left (1+\frac{d x}{e}\right )}{x} \, dx}{d^5}\\ &=-\frac{a e^3 x}{d^4}+\frac{b e^3 n x}{d^4}-\frac{b e^2 n x^2}{4 d^3}+\frac{b e n x^3}{9 d^2}-\frac{b n x^4}{16 d}-\frac{b e^3 x \log \left (c x^n\right )}{d^4}+\frac{e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac{e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}+\frac{e^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x}{e}\right )}{d^5}+\frac{b e^4 n \text{Li}_2\left (-\frac{d x}{e}\right )}{d^5}\\ \end{align*}
Mathematica [A] time = 0.101681, size = 171, normalized size = 0.92 \[ \frac{144 b e^4 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )+72 d^2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-48 d^3 e x^3 \left (a+b \log \left (c x^n\right )\right )+36 d^4 x^4 \left (a+b \log \left (c x^n\right )\right )+144 e^4 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )-144 a d e^3 x-144 b d e^3 x \log \left (c x^n\right )-36 b d^2 e^2 n x^2+16 b d^3 e n x^3-9 b d^4 n x^4+144 b d e^3 n x}{144 d^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.183, size = 867, normalized size = 4.7 \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}{12} \, a{\left (\frac{12 \, e^{4} \log \left (d x + e\right )}{d^{5}} + \frac{3 \, d^{3} x^{4} - 4 \, d^{2} e x^{3} + 6 \, d e^{2} x^{2} - 12 \, e^{3} x}{d^{4}}\right )} + b \int \frac{x^{4} \log \left (c\right ) + x^{4} \log \left (x^{n}\right )}{d x + e}\,{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^{4} \log \left (c x^{n}\right ) + a x^{4}}{d x + e}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 135.23, size = 298, normalized size = 1.61 \begin{align*} \frac{a x^{4}}{4 d} - \frac{a e x^{3}}{3 d^{2}} + \frac{a e^{2} x^{2}}{2 d^{3}} + \frac{a e^{4} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right )}{d^{4}} - \frac{a e^{3} x}{d^{4}} - \frac{b n x^{4}}{16 d} + \frac{b x^{4} \log{\left (c x^{n} \right )}}{4 d} + \frac{b e n x^{3}}{9 d^{2}} - \frac{b e x^{3} \log{\left (c x^{n} \right )}}{3 d^{2}} - \frac{b e^{2} n x^{2}}{4 d^{3}} + \frac{b e^{2} x^{2} \log{\left (c x^{n} \right )}}{2 d^{3}} - \frac{b e^{4} n \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\begin{cases} \log{\left (e \right )} \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (e \right )} \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\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 (e \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (e \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\right ) & \text{otherwise} \end{cases}}{d} & \text{otherwise} \end{cases}\right )}{d^{4}} + \frac{b e^{4} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )}}{d^{4}} + \frac{b e^{3} n x}{d^{4}} - \frac{b e^{3} x \log{\left (c x^{n} \right )}}{d^{4}} \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}}{d + \frac{e}{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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