Optimal. Leaf size=164 \[ -\frac{i b n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 \sqrt{d} e^{3/2}}+\frac{i b n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 \sqrt{d} e^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d} e^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}} \]
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Rubi [A] time = 0.269079, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {288, 205, 2351, 2323, 2324, 12, 4848, 2391} \[ -\frac{i b n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 \sqrt{d} e^{3/2}}+\frac{i b n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 \sqrt{d} e^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d} e^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 205
Rule 2351
Rule 2323
Rule 2324
Rule 12
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{d \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )^2}+\frac{a+b \log \left (c x^n\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{e}-\frac{d \int \frac{a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=-\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 e \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 )}{\sqrt{d} e^{3/2}}-\frac{\int \frac{a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 e}+\frac{(b n) \int \frac{1}{d+e x^2} \, dx}{2 e}-\frac{(b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} x} \, dx}{e}\\ &=\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 e \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 \sqrt{d} e^{3/2}}-\frac{(b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{\sqrt{d} e^{3/2}}+\frac{(b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} x} \, dx}{2 e}\\ &=\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 e \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 \sqrt{d} e^{3/2}}-\frac{(i b n) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 \sqrt{d} e^{3/2}}+\frac{(i b n) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 \sqrt{d} e^{3/2}}+\frac{(b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 \sqrt{d} e^{3/2}}\\ &=\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 e \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 \sqrt{d} e^{3/2}}-\frac{i b n \text{Li}_2\left (-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}+\frac{i b n \text{Li}_2\left (\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}+\frac{(i b n) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{4 \sqrt{d} e^{3/2}}-\frac{(i b n) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{4 \sqrt{d} e^{3/2}}\\ &=\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 e \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 \sqrt{d} e^{3/2}}-\frac{i b n \text{Li}_2\left (-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 \sqrt{d} e^{3/2}}+\frac{i b n \text{Li}_2\left (\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 \sqrt{d} e^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.526648, size = 258, normalized size = 1.57 \[ \frac{\frac{b n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{\sqrt{-d}}+\frac{b d n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}+\frac{d \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac{\log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{-d}}+\frac{a+b \log \left (c x^n\right )}{\sqrt{-d}-\sqrt{e} x}-\frac{a+b \log \left (c x^n\right )}{\sqrt{-d}+\sqrt{e} x}+\frac{b d n \left (\log (x)-\log \left (\sqrt{-d}-\sqrt{e} x\right )\right )}{(-d)^{3/2}}+\frac{b n \left (\log (x)-\log \left (\sqrt{-d}+\sqrt{e} x\right )\right )}{\sqrt{-d}}}{4 e^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.327, size = 752, normalized size = 4.6 \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 x^{2} \log \left (c x^{n}\right ) + a x^{2}}{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{x^{2} \left (a + b \log{\left (c x^{n} \right )}\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{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\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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