Optimal. Leaf size=331 \[ \frac{b g n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}+\frac{b g n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 f^2}-\frac{b g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f^2}-\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac{g \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f^2}+\frac{g \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f^2}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac{b e^2 n \log (x)}{2 d^2 f}+\frac{b e^2 n \log (d+e x)}{2 d^2 f}-\frac{b e n}{2 d f x} \]
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Rubi [A] time = 0.368075, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 44, 2416, 2395, 2394, 2315, 260, 2393, 2391} \[ \frac{b g n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}+\frac{b g n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 f^2}-\frac{b g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f^2}-\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac{g \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f^2}+\frac{g \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f^2}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac{b e^2 n \log (x)}{2 d^2 f}+\frac{b e^2 n \log (d+e x)}{2 d^2 f}-\frac{b e n}{2 d f x} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rule 2416
Rule 2395
Rule 2394
Rule 2315
Rule 260
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{x^3 \left (f+g x^2\right )} \, dx &=\int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{f x^3}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x}+\frac{g^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^3} \, dx}{f}-\frac{g \int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{f^2}+\frac{g^2 \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{f^2}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac{g^2 \int \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{a+b \log \left (c (d+e x)^n\right )}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{f^2}+\frac{(b e n) \int \frac{1}{x^2 (d+e x)} \, dx}{2 f}+\frac{(b e g n) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{f^2}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}-\frac{b g n \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^2}-\frac{g^{3/2} \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 f^2}+\frac{g^{3/2} \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 f^2}+\frac{(b e n) \int \left (\frac{1}{d x^2}-\frac{e}{d^2 x}+\frac{e^2}{d^2 (d+e x)}\right ) \, dx}{2 f}\\ &=-\frac{b e n}{2 d f x}-\frac{b e^2 n \log (x)}{2 d^2 f}+\frac{b e^2 n \log (d+e x)}{2 d^2 f}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}-\frac{b g n \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^2}-\frac{(b e g n) \int \frac{\log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{d+e x} \, dx}{2 f^2}-\frac{(b e g n) \int \frac{\log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{d+e x} \, dx}{2 f^2}\\ &=-\frac{b e n}{2 d f x}-\frac{b e^2 n \log (x)}{2 d^2 f}+\frac{b e^2 n \log (d+e x)}{2 d^2 f}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}-\frac{b g n \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^2}-\frac{(b g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{2 f^2}-\frac{(b g n) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{2 f^2}\\ &=-\frac{b e n}{2 d f x}-\frac{b e^2 n \log (x)}{2 d^2 f}+\frac{b e^2 n \log (d+e x)}{2 d^2 f}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}+\frac{b g n \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 f^2}+\frac{b g n \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 f^2}-\frac{b g n \text{Li}_2\left (1+\frac{e x}{d}\right )}{f^2}\\ \end{align*}
Mathematica [A] time = 0.178629, size = 279, normalized size = 0.84 \[ \frac{b g n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )+b g n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )-2 b g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )+g \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+g \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^2}-2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b e f n (-e x \log (d+e x)+d+e x \log (x))}{d^2 x}}{2 f^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.417, size = 841, normalized size = 2.5 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{g \log \left (g x^{2} + f\right )}{f^{2}} - \frac{2 \, g \log \left (x\right )}{f^{2}} - \frac{1}{f x^{2}}\right )} + b \int \frac{\log \left ({\left (e x + d\right )}^{n}\right ) + \log \left (c\right )}{g x^{5} + f x^{3}}\,{d x} \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 ({\left (e x + d\right )}^{n} c\right ) + a}{g x^{5} + f x^{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{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x^{2} + f\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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