Optimal. Leaf size=150 \[ -\frac{b e^3 n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d^4}+\frac{e^3 \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac{a+b \log \left (c x^n\right )}{3 d x^3}-\frac{b e^2 n}{d^3 x}+\frac{b e n}{4 d^2 x^2}-\frac{b n}{9 d x^3} \]
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Rubi [A] time = 0.212092, antiderivative size = 173, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {44, 2351, 2304, 2301, 2317, 2391} \[ \frac{b e^3 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^4}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^4 n}+\frac{e^3 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac{a+b \log \left (c x^n\right )}{3 d x^3}-\frac{b e^2 n}{d^3 x}+\frac{b e n}{4 d^2 x^2}-\frac{b n}{9 d x^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2351
Rule 2304
Rule 2301
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^4 (d+e x)} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{d x^4}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x^3}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x^2}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac{e^4 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x^4} \, dx}{d}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{d^2}+\frac{e^2 \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{d^3}-\frac{e^3 \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^4}+\frac{e^4 \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^4}\\ &=-\frac{b n}{9 d x^3}+\frac{b e n}{4 d^2 x^2}-\frac{b e^2 n}{d^3 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^4 n}+\frac{e^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^4}-\frac{\left (b e^3 n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^4}\\ &=-\frac{b n}{9 d x^3}+\frac{b e n}{4 d^2 x^2}-\frac{b e^2 n}{d^3 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^4 n}+\frac{e^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^4}+\frac{b e^3 n \text{Li}_2\left (-\frac{e x}{d}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 0.17413, size = 159, normalized size = 1.06 \[ \frac{36 b e^3 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{18 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac{12 d^3 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac{36 d e^2 \left (a+b \log \left (c x^n\right )\right )}{x}+36 e^3 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{18 e^3 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+\frac{9 b d^2 e n}{x^2}-\frac{4 b d^3 n}{x^3}-\frac{36 b d e^2 n}{x}}{36 d^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.163, size = 868, normalized size = 5.8 \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^{3} \log \left (e x + d\right )}{d^{4}} - \frac{6 \, e^{3} \log \left (x\right )}{d^{4}} - \frac{6 \, e^{2} x^{2} - 3 \, d e x + 2 \, d^{2}}{d^{3} x^{3}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e x^{5} + d x^{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 \log \left (c x^{n}\right ) + a}{e x^{5} + d x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 150.42, size = 296, normalized size = 1.97 \begin{align*} - \frac{a}{3 d x^{3}} + \frac{a e}{2 d^{2} x^{2}} - \frac{a e^{2}}{d^{3} x} + \frac{a e^{4} \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right )}{d^{4}} - \frac{a e^{3} \log{\left (x \right )}}{d^{4}} - \frac{b n}{9 d x^{3}} - \frac{b \log{\left (c x^{n} \right )}}{3 d x^{3}} + \frac{b e n}{4 d^{2} x^{2}} + \frac{b e \log{\left (c x^{n} \right )}}{2 d^{2} x^{2}} - \frac{b e^{2} n}{d^{3} x} - \frac{b e^{2} \log{\left (c x^{n} \right )}}{d^{3} x} - \frac{b e^{4} 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 )}{d^{4}} + \frac{b e^{4} \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 )}}{d^{4}} + \frac{b e^{3} n \log{\left (x \right )}^{2}}{2 d^{4}} - \frac{b e^{3} \log{\left (x \right )} \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{b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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