Optimal. Leaf size=164 \[ -\frac{i b n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}+\frac{i b n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}} \]
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Rubi [A] time = 0.0990367, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2323, 205, 2324, 12, 4848, 2391} \[ -\frac{i b n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}+\frac{i b n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 2323
Rule 205
Rule 2324
Rule 12
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx &=\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac{\int \frac{a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 d}-\frac{(b n) \int \frac{1}{d+e x^2} \, dx}{2 d}\\ &=-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}-\frac{(b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} x} \, dx}{2 d}\\ &=-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}-\frac{(b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 d^{3/2} \sqrt{e}}\\ &=-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}-\frac{(i b n) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{4 d^{3/2} \sqrt{e}}+\frac{(i b n) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{4 d^{3/2} \sqrt{e}}\\ &=-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt{e}}-\frac{i b n \text{Li}_2\left (-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}+\frac{i b n \text{Li}_2\left (\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.537206, size = 289, normalized size = 1.76 \[ \frac{1}{4} \left (\frac{b d n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{5/2} \sqrt{e}}+\frac{b n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2} \sqrt{e}}+\frac{d \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2} \sqrt{e}}+\frac{a+b \log \left (c x^n\right )}{d \left (\sqrt{-d} \sqrt{e}+e x\right )}+\frac{a+b \log \left (c x^n\right )}{d e x+(-d)^{3/2} \sqrt{e}}+\frac{b d n \left (\log (x)-\log \left (\sqrt{-d}-\sqrt{e} x\right )\right )}{(-d)^{5/2} \sqrt{e}}+\frac{b n \left (\log (x)-\log \left (\sqrt{-d}+\sqrt{e} x\right )\right )}{(-d)^{3/2} \sqrt{e}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.301, size = 685, normalized size = 4.2 \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^{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \log{\left (c x^{n} \right )}}{\left (d + e x^{2}\right )^{2}}\, dx \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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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