Optimal. Leaf size=95 \[ \frac{b n \text{PolyLog}\left (2,-\frac{e x^2}{d}\right )}{4 e^2}+\frac{\log \left (\frac{e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac{b n \log \left (d+e x^2\right )}{4 e^2} \]
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Rubi [A] time = 0.187815, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {266, 43, 2351, 2335, 260, 2337, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{e x^2}{d}\right )}{4 e^2}+\frac{\log \left (\frac{e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac{b n \log \left (d+e x^2\right )}{4 e^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2351
Rule 2335
Rule 260
Rule 2337
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{d x \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{e}-\frac{d \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^2}{d}\right )}{2 e^2}-\frac{(b n) \int \frac{\log \left (1+\frac{e x^2}{d}\right )}{x} \, dx}{2 e^2}+\frac{(b n) \int \frac{x}{d+e x^2} \, dx}{2 e}\\ &=-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac{b n \log \left (d+e x^2\right )}{4 e^2}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^2}{d}\right )}{2 e^2}+\frac{b n \text{Li}_2\left (-\frac{e x^2}{d}\right )}{4 e^2}\\ \end{align*}
Mathematica [C] time = 0.233394, size = 321, normalized size = 3.38 \[ \frac{\frac{b n \left (2 \left (d+e x^2\right ) \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )+2 \left (d+e x^2\right ) \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )+e x^2 \log \left (-\sqrt{e} x+i \sqrt{d}\right )+e x^2 \log \left (\sqrt{e} x+i \sqrt{d}\right )+2 e x^2 \log (x) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )+2 e x^2 \log (x) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )+d \log \left (-\sqrt{e} x+i \sqrt{d}\right )+d \log \left (\sqrt{e} x+i \sqrt{d}\right )+2 d \log (x) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )+2 d \log (x) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )-2 e x^2 \log (x)\right )}{d+e x^2}+2 \log \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+\frac{2 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d+e x^2}}{4 e^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.174, size = 511, normalized size = 5.4 \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{d}{e^{3} x^{2} + d e^{2}} + \frac{\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^{2} x^{4} + 2 \, d e x^{2} + d^{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 x^{3} \log \left (c x^{n}\right ) + a x^{3}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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