Optimal. Leaf size=203 \[ \frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g}+\frac{b n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 g}+\frac{\log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac{\log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g} \]
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Rubi [A] time = 0.179626, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {260, 2416, 2394, 2393, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g}+\frac{b n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 g}+\frac{\log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac{\log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx &=\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\\ &=-\frac{\int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 \sqrt{g}}+\frac{\int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 \sqrt{g}}\\ &=\frac{\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 g}+\frac{\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 g}-\frac{(b e 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 g}-\frac{(b e 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 g}\\ &=\frac{\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 g}+\frac{\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 g}-\frac{(b 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 g}-\frac{(b 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 g}\\ &=\frac{\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 g}+\frac{\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 g}+\frac{b n \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g}+\frac{b n \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g}\\ \end{align*}
Mathematica [A] time = 0.0370277, size = 172, normalized size = 0.85 \[ \frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )+b n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )+\left (\log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right )+\log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.395, size = 411, normalized size = 2. \begin{align*}{\frac{b\ln \left ( g{x}^{2}+f \right ) \ln \left ( \left ( ex+d \right ) ^{n} \right ) }{2\,g}}-{\frac{bn\ln \left ( ex+d \right ) \ln \left ( g{x}^{2}+f \right ) }{2\,g}}+{\frac{bn\ln \left ( ex+d \right ) }{2\,g}\ln \left ({ \left ( e\sqrt{-fg}-g \left ( ex+d \right ) +dg \right ) \left ( e\sqrt{-fg}+dg \right ) ^{-1}} \right ) }+{\frac{bn\ln \left ( ex+d \right ) }{2\,g}\ln \left ({ \left ( e\sqrt{-fg}+g \left ( ex+d \right ) -dg \right ) \left ( e\sqrt{-fg}-dg \right ) ^{-1}} \right ) }+{\frac{bn}{2\,g}{\it dilog} \left ({ \left ( e\sqrt{-fg}-g \left ( ex+d \right ) +dg \right ) \left ( e\sqrt{-fg}+dg \right ) ^{-1}} \right ) }+{\frac{bn}{2\,g}{\it dilog} \left ({ \left ( e\sqrt{-fg}+g \left ( ex+d \right ) -dg \right ) \left ( e\sqrt{-fg}-dg \right ) ^{-1}} \right ) }-{\frac{{\frac{i}{4}}\ln \left ( g{x}^{2}+f \right ) b\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ){\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) }{g}}+{\frac{{\frac{i}{4}}\ln \left ( g{x}^{2}+f \right ) b\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{g}}+{\frac{{\frac{i}{4}}\ln \left ( g{x}^{2}+f \right ) b\pi \,{\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{g}}-{\frac{{\frac{i}{4}}\ln \left ( g{x}^{2}+f \right ) b\pi \, \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}}{g}}+{\frac{b\ln \left ( g{x}^{2}+f \right ) \ln \left ( c \right ) }{2\,g}}+{\frac{a\ln \left ( g{x}^{2}+f \right ) }{2\,g}} \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*} b \int \frac{x \log \left ({\left (e x + d\right )}^{n}\right ) + x \log \left (c\right )}{g x^{2} + f}\,{d x} + \frac{a \log \left (g x^{2} + f\right )}{2 \, g} \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 \log \left ({\left (e x + d\right )}^{n} c\right ) + a x}{g x^{2} + f}, 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 \left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )}{f + g x^{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 ({\left (e x + d\right )}^{n} c\right ) + a\right )} x}{g x^{2} + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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