Optimal. Leaf size=74 \[ -\frac{b e n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d^2}+\frac{e \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac{a+b \log \left (c x^n\right )}{d x}-\frac{b n}{d x} \]
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Rubi [A] time = 0.14466, antiderivative size = 95, normalized size of antiderivative = 1.28, number of steps used = 6, 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} \[ \frac{b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac{e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac{a+b \log \left (c x^n\right )}{d x}-\frac{b n}{d x} \]
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^2 (d+e x)} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{d x^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{d}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^2}+\frac{e^2 \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2}\\ &=-\frac{b n}{d x}-\frac{a+b \log \left (c x^n\right )}{d x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac{e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^2}-\frac{(b e n) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^2}\\ &=-\frac{b n}{d x}-\frac{a+b \log \left (c x^n\right )}{d x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac{e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^2}+\frac{b e n \text{Li}_2\left (-\frac{e x}{d}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.082567, size = 88, normalized size = 1.19 \[ -\frac{-2 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right )-2 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{b n}+\frac{2 b d n}{x}}{2 d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.178, size = 504, normalized size = 6.8 \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*} a{\left (\frac{e \log \left (e x + d\right )}{d^{2}} - \frac{e \log \left (x\right )}{d^{2}} - \frac{1}{d x}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e x^{3} + d x^{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 \log \left (c x^{n}\right ) + a}{e x^{3} + d x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 127.667, size = 197, normalized size = 2.66 \begin{align*} - \frac{a}{d x} + \frac{a e^{2} \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right )}{d^{2}} - \frac{a e \log{\left (x \right )}}{d^{2}} - \frac{b n}{d x} - \frac{b \log{\left (c x^{n} \right )}}{d x} - \frac{b e^{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 )}{d^{2}} + \frac{b e^{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 )}}{d^{2}} + \frac{b e n \log{\left (x \right )}^{2}}{2 d^{2}} - \frac{b e \log{\left (x \right )} \log{\left (c x^{n} \right )}}{d^{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 (e x + d\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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