Optimal. Leaf size=165 \[ -\frac{i b e^{3/2} n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{i b e^{3/2} n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3}+\frac{b e n}{d^2 x}-\frac{b n}{9 d x^3} \]
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Rubi [A] time = 0.197425, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {325, 205, 2351, 2304, 2324, 12, 4848, 2391} \[ -\frac{i b e^{3/2} n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{i b e^{3/2} n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3}+\frac{b e n}{d^2 x}-\frac{b n}{9 d x^3} \]
Antiderivative was successfully verified.
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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^4 \left (d+e x^2\right )} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{d x^4}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x^2}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x^4} \, dx}{d}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{d^2}+\frac{e^2 \int \frac{a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{d^2}\\ &=-\frac{b n}{9 d x^3}+\frac{b e n}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac{\left (b e^2 n\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} x} \, dx}{d^2}\\ &=-\frac{b n}{9 d x^3}+\frac{b e n}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac{\left (b e^{3/2} n\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{d^{5/2}}\\ &=-\frac{b n}{9 d x^3}+\frac{b e n}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac{\left (i b e^{3/2} n\right ) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 d^{5/2}}+\frac{\left (i b e^{3/2} n\right ) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 d^{5/2}}\\ &=-\frac{b n}{9 d x^3}+\frac{b e n}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{3 d x^3}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac{i b e^{3/2} n \text{Li}_2\left (-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{i b e^{3/2} n \text{Li}_2\left (\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.182793, size = 211, normalized size = 1.28 \[ \frac{1}{18} \left (\frac{9 b e^{3/2} n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{5/2}}-\frac{9 b e^{3/2} n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2}}+\frac{18 e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac{9 e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2}}+\frac{9 e^{3/2} \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2}}-\frac{6 \left (a+b \log \left (c x^n\right )\right )}{d x^3}+\frac{18 b e n}{d^2 x}-\frac{2 b n}{d x^3}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.208, size = 706, normalized size = 4.3 \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(-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 x^{6} + d x^{4}}, 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 )}}{x^{4} \left (d + e x^{2}\right )}\, 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 )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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