Optimal. Leaf size=187 \[ -\frac{i b n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{3/2} e^{3/2}}+\frac{i b n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{3/2} e^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac{b n x}{8 d e \left (d+e x^2\right )} \]
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Rubi [A] time = 0.368746, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {288, 199, 205, 2351, 2323, 2324, 12, 4848, 2391} \[ -\frac{i b n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{3/2} e^{3/2}}+\frac{i b n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{3/2} e^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac{b n x}{8 d e \left (d+e x^2\right )} \]
Antiderivative was successfully verified.
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Rule 288
Rule 199
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 )^3} \, dx &=\int \left (-\frac{d \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )^3}+\frac{a+b \log \left (c x^n\right )}{e \left (d+e x^2\right )^2}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{e}-\frac{d \int \frac{a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx}{e}\\ &=-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{2 d e \left (d+e x^2\right )}-\frac{3 \int \frac{a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{4 e}+\frac{\int \frac{a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 d e}+\frac{(b n) \int \frac{1}{\left (d+e x^2\right )^2} \, dx}{4 e}-\frac{(b n) \int \frac{1}{d+e x^2} \, dx}{2 d e}\\ &=\frac{b n x}{8 d e \left (d+e x^2\right )}-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} e^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{8 d 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 d^{3/2} e^{3/2}}-\frac{3 \int \frac{a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{8 d e}+\frac{(b n) \int \frac{1}{d+e x^2} \, dx}{8 d e}+\frac{(3 b n) \int \frac{1}{d+e x^2} \, dx}{8 d e}-\frac{(b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} x} \, dx}{2 d e}\\ &=\frac{b n x}{8 d e \left (d+e x^2\right )}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{8 d 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 )}{8 d^{3/2} e^{3/2}}-\frac{(b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 d^{3/2} e^{3/2}}+\frac{(3 b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} x} \, dx}{8 d e}\\ &=\frac{b n x}{8 d e \left (d+e x^2\right )}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{8 d 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 )}{8 d^{3/2} e^{3/2}}-\frac{(i b n) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{4 d^{3/2} e^{3/2}}+\frac{(i b n) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{4 d^{3/2} e^{3/2}}+\frac{(3 b n) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{8 d^{3/2} e^{3/2}}\\ &=\frac{b n x}{8 d e \left (d+e x^2\right )}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{8 d 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 )}{8 d^{3/2} e^{3/2}}-\frac{i b n \text{Li}_2\left (-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} e^{3/2}}+\frac{i b n \text{Li}_2\left (\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{4 d^{3/2} e^{3/2}}+\frac{(3 i b n) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{16 d^{3/2} e^{3/2}}-\frac{(3 i b n) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{16 d^{3/2} e^{3/2}}\\ &=\frac{b n x}{8 d e \left (d+e x^2\right )}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{8 d 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 )}{8 d^{3/2} e^{3/2}}-\frac{i b n \text{Li}_2\left (-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{3/2} e^{3/2}}+\frac{i b n \text{Li}_2\left (\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{3/2} e^{3/2}}\\ \end{align*}
Mathematica [B] time = 1.04301, size = 497, normalized size = 2.66 \[ \frac{\frac{b d n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{5/2}}+\frac{b n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac{d \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2}}-\frac{a+b \log \left (c x^n\right )}{\sqrt{-d} d-d \sqrt{e} x}+\frac{a+b \log \left (c x^n\right )}{d \sqrt{e} x+\sqrt{-d} d}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{a+b \log \left (c x^n\right )}{\sqrt{-d} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{b n \left (\log (x) \left (d-\sqrt{-d} \sqrt{e} x\right )+\left (\sqrt{-d} \sqrt{e} x-d\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )+d\right )}{d^2 \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{b n \left (\log (x) \left (\sqrt{-d} \sqrt{e} x+d\right )-\left (\sqrt{-d} \sqrt{e} x+d\right ) \log \left (d \sqrt{e} x+(-d)^{3/2}\right )+d\right )}{d^2 \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b d n \left (\log (x)-\log \left (\sqrt{-d}-\sqrt{e} x\right )\right )}{(-d)^{5/2}}+\frac{b n \left (\log (x)-\log \left (\sqrt{-d}+\sqrt{e} x\right )\right )}{(-d)^{3/2}}}{16 e^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.326, size = 1247, 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(-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^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, 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^{2}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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