Optimal. Leaf size=183 \[ \frac{3 i b \sqrt{e} n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{5/2}}-\frac{3 i b \sqrt{e} n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{5/2}}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 a+3 b \log \left (c x^n\right )-b n\right )}{2 d^{5/2}}-\frac{3 a+3 b \log \left (c x^n\right )-b n}{2 d^2 x}+\frac{a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac{3 b n}{2 d^2 x} \]
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Rubi [A] time = 0.283844, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2340, 325, 205, 2351, 2304, 2324, 12, 4848, 2391} \[ \frac{3 i b \sqrt{e} n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{5/2}}-\frac{3 i b \sqrt{e} n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{5/2}}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 a+3 b \log \left (c x^n\right )-b n\right )}{2 d^{5/2}}-\frac{3 a+3 b \log \left (c x^n\right )-b n}{2 d^2 x}+\frac{a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac{3 b n}{2 d^2 x} \]
Antiderivative was successfully verified.
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Rule 2340
Rule 325
Rule 205
Rule 2351
Rule 2304
Rule 2324
Rule 12
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^2} \, dx &=\frac{a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac{\int \frac{-3 a+b n-3 b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )} \, dx}{2 d}\\ &=\frac{a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac{\int \left (\frac{-3 a+b n-3 b \log \left (c x^n\right )}{d x^2}-\frac{e \left (-3 a+b n-3 b \log \left (c x^n\right )\right )}{d \left (d+e x^2\right )}\right ) \, dx}{2 d}\\ &=\frac{a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac{\int \frac{-3 a+b n-3 b \log \left (c x^n\right )}{x^2} \, dx}{2 d^2}+\frac{e \int \frac{-3 a+b n-3 b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 d^2}\\ &=-\frac{3 b n}{2 d^2 x}+\frac{a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac{3 a-b n+3 b \log \left (c x^n\right )}{2 d^2 x}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )}{2 d^{5/2}}+\frac{(3 b e n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} x} \, dx}{2 d^2}\\ &=-\frac{3 b n}{2 d^2 x}+\frac{a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac{3 a-b n+3 b \log \left (c x^n\right )}{2 d^2 x}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )}{2 d^{5/2}}+\frac{\left (3 b \sqrt{e} n\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 d^{5/2}}\\ &=-\frac{3 b n}{2 d^2 x}+\frac{a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac{3 a-b n+3 b \log \left (c x^n\right )}{2 d^2 x}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )}{2 d^{5/2}}+\frac{\left (3 i b \sqrt{e} n\right ) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{4 d^{5/2}}-\frac{\left (3 i b \sqrt{e} n\right ) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{4 d^{5/2}}\\ &=-\frac{3 b n}{2 d^2 x}+\frac{a+b \log \left (c x^n\right )}{2 d x \left (d+e x^2\right )}-\frac{3 a-b n+3 b \log \left (c x^n\right )}{2 d^2 x}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )}{2 d^{5/2}}+\frac{3 i b \sqrt{e} n \text{Li}_2\left (-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{5/2}}-\frac{3 i b \sqrt{e} n \text{Li}_2\left (\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.78284, size = 328, normalized size = 1.79 \[ \frac{1}{4} \left (-\frac{3 b \sqrt{e} n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{5/2}}+\frac{3 b \sqrt{e} n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2}}+\frac{\sqrt{e} \left (a+b \log \left (c x^n\right )\right )}{d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{e} \left (a+b \log \left (c x^n\right )\right )}{d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{4 \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{3 \sqrt{e} \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2}}-\frac{3 \sqrt{e} \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2}}-\frac{4 b n}{d^2 x}+\frac{b \sqrt{e} n \left (\log \left (\sqrt{-d}-\sqrt{e} x\right )-\log (x)\right )}{(-d)^{5/2}}+\frac{b \sqrt{e} n \left (\log (x)-\log \left (\sqrt{-d}+\sqrt{e} x\right )\right )}{(-d)^{5/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.35, size = 933, normalized size = 5.1 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{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{b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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