Optimal. Leaf size=129 \[ -\frac{b d n \text{PolyLog}\left (2,-\frac{e x^2}{d}\right )}{2 e^3}+\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac{d \log \left (\frac{e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{b d n \log \left (d+e x^2\right )}{4 e^3}-\frac{b n x^2}{4 e^2} \]
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Rubi [A] time = 0.220363, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {266, 43, 2351, 2304, 2335, 260, 2337, 2391} \[ -\frac{b d n \text{PolyLog}\left (2,-\frac{e x^2}{d}\right )}{2 e^3}+\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac{d \log \left (\frac{e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{b d n \log \left (d+e x^2\right )}{4 e^3}-\frac{b n x^2}{4 e^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2351
Rule 2304
Rule 2335
Rule 260
Rule 2337
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )^2}-\frac{2 d x \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}-\frac{(2 d) \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{e^2}+\frac{d^2 \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx}{e^2}\\ &=-\frac{b n x^2}{4 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac{d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^2}{d}\right )}{e^3}+\frac{(b d n) \int \frac{\log \left (1+\frac{e x^2}{d}\right )}{x} \, dx}{e^3}-\frac{(b d n) \int \frac{x}{d+e x^2} \, dx}{2 e^2}\\ &=-\frac{b n x^2}{4 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac{b d n \log \left (d+e x^2\right )}{4 e^3}-\frac{d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^2}{d}\right )}{e^3}-\frac{b d n \text{Li}_2\left (-\frac{e x^2}{d}\right )}{2 e^3}\\ \end{align*}
Mathematica [C] time = 0.467416, size = 287, normalized size = 2.22 \[ \frac{b n \left (-4 d \left (\text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )+\log (x) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )-4 d \left (\text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )+\log (x) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )+\frac{d \sqrt{e} x \log (x)}{\sqrt{e} x-i \sqrt{d}}+\frac{d \sqrt{e} x \log (x)}{\sqrt{e} x+i \sqrt{d}}-d \log \left (-\sqrt{e} x+i \sqrt{d}\right )-d \log \left (\sqrt{e} x+i \sqrt{d}\right )+e x^2 (2 \log (x)-1)\right )-\frac{2 d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d+e x^2}-4 d \log \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+2 e x^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{4 e^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.19, size = 687, normalized size = 5.3 \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^{2}}{e^{4} x^{2} + d e^{3}} - \frac{x^{2}}{e^{2}} + \frac{2 \, d \log \left (e x^{2} + d\right )}{e^{3}}\right )} + b \int \frac{x^{5} \log \left (c\right ) + x^{5} \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^{5} \log \left (c x^{n}\right ) + a x^{5}}{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^{5}}{{\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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