Optimal. Leaf size=148 \[ -\frac{b e^3 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^4}-\frac{e^3 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac{e x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d}+\frac{a e^2 x}{d^3}+\frac{b e^2 x \log \left (c x^n\right )}{d^3}-\frac{b e^2 n x}{d^3}+\frac{b e n x^2}{4 d^2}-\frac{b n x^3}{9 d} \]
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Rubi [A] time = 0.168242, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{b e^3 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^4}-\frac{e^3 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac{e x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d}+\frac{a e^2 x}{d^3}+\frac{b e^2 x \log \left (c x^n\right )}{d^3}-\frac{b e^2 n x}{d^3}+\frac{b e n x^2}{4 d^2}-\frac{b n x^3}{9 d} \]
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^2 \left (a+b \log \left (c x^n\right )\right )}{d+\frac{e}{x}} \, dx &=\int \left (\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{e x \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{d}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{d^3 (e+d x)}\right ) \, dx\\ &=\frac{\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{d}-\frac{e \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^2}+\frac{e^2 \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^3}-\frac{e^3 \int \frac{a+b \log \left (c x^n\right )}{e+d x} \, dx}{d^3}\\ &=\frac{a e^2 x}{d^3}+\frac{b e n x^2}{4 d^2}-\frac{b n x^3}{9 d}-\frac{e x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x}{e}\right )}{d^4}+\frac{\left (b e^2\right ) \int \log \left (c x^n\right ) \, dx}{d^3}+\frac{\left (b e^3 n\right ) \int \frac{\log \left (1+\frac{d x}{e}\right )}{x} \, dx}{d^4}\\ &=\frac{a e^2 x}{d^3}-\frac{b e^2 n x}{d^3}+\frac{b e n x^2}{4 d^2}-\frac{b n x^3}{9 d}+\frac{b e^2 x \log \left (c x^n\right )}{d^3}-\frac{e x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x}{e}\right )}{d^4}-\frac{b e^3 n \text{Li}_2\left (-\frac{d x}{e}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 0.0694653, size = 142, normalized size = 0.96 \[ \frac{-36 b e^3 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )-18 a d^2 e x^2+12 a d^3 x^3+36 a d e^2 x-36 a e^3 \log \left (\frac{d x}{e}+1\right )+6 b \log \left (c x^n\right ) \left (d x \left (2 d^2 x^2-3 d e x+6 e^2\right )-6 e^3 \log \left (\frac{d x}{e}+1\right )\right )+9 b d^2 e n x^2-4 b d^3 n x^3-36 b d e^2 n x}{36 d^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.174, size = 693, 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}{6} \, a{\left (\frac{6 \, e^{3} \log \left (d x + e\right )}{d^{4}} - \frac{2 \, d^{2} x^{3} - 3 \, d e x^{2} + 6 \, e^{2} x}{d^{3}}\right )} + b \int \frac{x^{3} \log \left (c\right ) + x^{3} \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^{3} \log \left (c x^{n}\right ) + a x^{3}}{d x + e}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 109.785, size = 248, normalized size = 1.68 \begin{align*} \frac{a x^{3}}{3 d} - \frac{a e x^{2}}{2 d^{2}} - \frac{a e^{3} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right )}{d^{3}} + \frac{a e^{2} x}{d^{3}} - \frac{b n x^{3}}{9 d} + \frac{b x^{3} \log{\left (c x^{n} \right )}}{3 d} + \frac{b e n x^{2}}{4 d^{2}} - \frac{b e x^{2} \log{\left (c x^{n} \right )}}{2 d^{2}} + \frac{b e^{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 )}{d^{3}} - \frac{b e^{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 )}}{d^{3}} - \frac{b e^{2} n x}{d^{3}} + \frac{b e^{2} x \log{\left (c x^{n} \right )}}{d^{3}} \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^{2}}{d + \frac{e}{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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