Optimal. Leaf size=121 \[ \frac{b e^2 n \text{PolyLog}\left (2,-\frac{d}{e x^2}\right )}{4 d^3}-\frac{e^2 \log \left (\frac{d}{e x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^3}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac{a+b \log \left (c x^n\right )}{4 d x^4}+\frac{b e n}{4 d^2 x^2}-\frac{b n}{16 d x^4} \]
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Rubi [A] time = 0.214049, antiderivative size = 149, normalized size of antiderivative = 1.23, 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 = {266, 44, 2351, 2304, 2301, 2337, 2391} \[ -\frac{b e^2 n \text{PolyLog}\left (2,-\frac{e x^2}{d}\right )}{4 d^3}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac{e^2 \log \left (\frac{e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^3}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac{a+b \log \left (c x^n\right )}{4 d x^4}+\frac{b e n}{4 d^2 x^2}-\frac{b n}{16 d x^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rule 2351
Rule 2304
Rule 2301
Rule 2337
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^5 \left (d+e x^2\right )} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{d x^5}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x^3}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac{e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^3 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x^5} \, dx}{d}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{d^2}+\frac{e^2 \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^3}-\frac{e^3 \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{d^3}\\ &=-\frac{b n}{16 d x^4}+\frac{b e n}{4 d^2 x^2}-\frac{a+b \log \left (c x^n\right )}{4 d x^4}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^2}{d}\right )}{2 d^3}+\frac{\left (b e^2 n\right ) \int \frac{\log \left (1+\frac{e x^2}{d}\right )}{x} \, dx}{2 d^3}\\ &=-\frac{b n}{16 d x^4}+\frac{b e n}{4 d^2 x^2}-\frac{a+b \log \left (c x^n\right )}{4 d x^4}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^2}{d}\right )}{2 d^3}-\frac{b e^2 n \text{Li}_2\left (-\frac{e x^2}{d}\right )}{4 d^3}\\ \end{align*}
Mathematica [A] time = 0.184593, size = 196, normalized size = 1.62 \[ -\frac{8 b e^2 n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )+8 b e^2 n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )+\frac{4 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^4}+8 e^2 \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )+8 e^2 \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{8 d e \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac{8 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+\frac{b d^2 n}{x^4}-\frac{4 b d e n}{x^2}}{16 d^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.168, size = 805, normalized size = 6.7 \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}{4} \, a{\left (\frac{2 \, e^{2} \log \left (e x^{2} + d\right )}{d^{3}} - \frac{4 \, e^{2} \log \left (x\right )}{d^{3}} - \frac{2 \, e x^{2} - d}{d^{2} x^{4}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e x^{7} + d x^{5}}\,{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^{7} + d x^{5}}, 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{b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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