Optimal. Leaf size=135 \[ -\frac{b d^2 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{e^3}+\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^3 n}-\frac{d^2 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 x}-\frac{a+b \log \left (c x^n\right )}{2 e x^2}+\frac{b d n}{e^2 x}-\frac{b n}{4 e x^2} \]
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Rubi [A] time = 0.176522, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {263, 44, 2351, 2304, 2301, 2317, 2391} \[ -\frac{b d^2 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{e^3}+\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^3 n}-\frac{d^2 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 x}-\frac{a+b \log \left (c x^n\right )}{2 e x^2}+\frac{b d n}{e^2 x}-\frac{b n}{4 e x^2} \]
Antiderivative was successfully verified.
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Rule 263
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 )}{\left (d+\frac{e}{x}\right ) x^4} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{e x^3}-\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 x^2}+\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 x}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (e+d x)}\right ) \, dx\\ &=\frac{d^2 \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{e^3}-\frac{d^3 \int \frac{a+b \log \left (c x^n\right )}{e+d x} \, dx}{e^3}-\frac{d \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{e^2}+\frac{\int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{e}\\ &=-\frac{b n}{4 e x^2}+\frac{b d n}{e^2 x}-\frac{a+b \log \left (c x^n\right )}{2 e x^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^3 n}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x}{e}\right )}{e^3}+\frac{\left (b d^2 n\right ) \int \frac{\log \left (1+\frac{d x}{e}\right )}{x} \, dx}{e^3}\\ &=-\frac{b n}{4 e x^2}+\frac{b d n}{e^2 x}-\frac{a+b \log \left (c x^n\right )}{2 e x^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^3 n}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x}{e}\right )}{e^3}-\frac{b d^2 n \text{Li}_2\left (-\frac{d x}{e}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.200894, size = 124, normalized size = 0.92 \[ -\frac{4 b d^2 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )+4 d^2 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}-\frac{4 d e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{2 e^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac{4 b d e n}{x}+\frac{b e^2 n}{x^2}}{4 e^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.151, size = 689, 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}{2} \, a{\left (\frac{2 \, d^{2} \log \left (d x + e\right )}{e^{3}} - \frac{2 \, d^{2} \log \left (x\right )}{e^{3}} - \frac{2 \, d x - e}{e^{2} x^{2}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{d x^{4} + e x^{3}}\,{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}{d x^{4} + e x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 72.1526, size = 246, normalized size = 1.82 \begin{align*} - \frac{a d^{3} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right )}{e^{3}} + \frac{a d^{2} \log{\left (x \right )}}{e^{3}} + \frac{a d}{e^{2} x} - \frac{a}{2 e x^{2}} + \frac{b d^{3} 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 )}{e^{3}} - \frac{b d^{3} \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 )}}{e^{3}} - \frac{b d^{2} n \log{\left (x \right )}^{2}}{2 e^{3}} + \frac{b d^{2} \log{\left (x \right )} \log{\left (c x^{n} \right )}}{e^{3}} + \frac{b d n}{e^{2} x} + \frac{b d \log{\left (c x^{n} \right )}}{e^{2} x} - \frac{b n}{4 e x^{2}} - \frac{b \log{\left (c x^{n} \right )}}{2 e x^{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{b \log \left (c x^{n}\right ) + a}{{\left (d + \frac{e}{x}\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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