Optimal. Leaf size=83 \[ -\frac{b d n \text{PolyLog}\left (2,-\frac{e x^2}{d}\right )}{4 e^2}-\frac{d \log \left (\frac{e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{b n x^2}{4 e} \]
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Rubi [A] time = 0.142518, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {266, 43, 2351, 2304, 2337, 2391} \[ -\frac{b d n \text{PolyLog}\left (2,-\frac{e x^2}{d}\right )}{4 e^2}-\frac{d \log \left (\frac{e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{b n x^2}{4 e} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2351
Rule 2304
Rule 2337
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx &=\int \left (\frac{x \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{d x \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}-\frac{d \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{e}\\ &=-\frac{b n x^2}{4 e}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^2}{d}\right )}{2 e^2}+\frac{(b d n) \int \frac{\log \left (1+\frac{e x^2}{d}\right )}{x} \, dx}{2 e^2}\\ &=-\frac{b n x^2}{4 e}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^2}{d}\right )}{2 e^2}-\frac{b d n \text{Li}_2\left (-\frac{e x^2}{d}\right )}{4 e^2}\\ \end{align*}
Mathematica [A] time = 0.0706448, size = 135, normalized size = 1.63 \[ -\frac{2 b d n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )+2 b d n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )+2 d \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )+2 d \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )-2 e x^2 \left (a+b \log \left (c x^n\right )\right )+b e n x^2}{4 e^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.174, size = 460, normalized size = 5.5 \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{x^{2}}{e} - \frac{d \log \left (e x^{2} + d\right )}{e^{2}}\right )} + b \int \frac{x^{3} \log \left (c\right ) + x^{3} \log \left (x^{n}\right )}{e x^{2} + d}\,{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}}{e x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 36.34, size = 180, normalized size = 2.17 \begin{align*} - \frac{a d \left (\begin{cases} \frac{x^{2}}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x^{2} \right )}}{e} & \text{otherwise} \end{cases}\right )}{2 e} + \frac{a x^{2}}{2 e} + \frac{b d n \left (\begin{cases} \frac{x^{2}}{2 d} & \text{for}\: e = 0 \\\frac{\begin{cases} \log{\left (d \right )} \log{\left (x \right )} - \frac{\operatorname{Li}_{2}\left (\frac{e x^{2} e^{i \pi }}{d}\right )}{2} & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (d \right )} \log{\left (\frac{1}{x} \right )} - \frac{\operatorname{Li}_{2}\left (\frac{e x^{2} e^{i \pi }}{d}\right )}{2} & \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 )} - \frac{\operatorname{Li}_{2}\left (\frac{e x^{2} e^{i \pi }}{d}\right )}{2} & \text{otherwise} \end{cases}}{e} & \text{otherwise} \end{cases}\right )}{2 e} - \frac{b d \left (\begin{cases} \frac{x^{2}}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x^{2} \right )}}{e} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )}}{2 e} - \frac{b n x^{2}}{4 e} + \frac{b x^{2} \log{\left (c x^{n} \right )}}{2 e} \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}}{e x^{2} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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