Optimal. Leaf size=256 \[ -\frac{2 b n \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{\sqrt{e} \sqrt{e f-d g}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{e} \sqrt{e f-d g}}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{\sqrt{e} \sqrt{e f-d g}}-\frac{4 b n \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e} \sqrt{e f-d g}} \]
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Rubi [A] time = 0.693756, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 11, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.355, Rules used = {2411, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315} \[ -\frac{2 b n \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{\sqrt{e} \sqrt{e f-d g}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{e} \sqrt{e f-d g}}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{\sqrt{e} \sqrt{e f-d g}}-\frac{4 b n \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e} \sqrt{e f-d g}} \]
Antiderivative was successfully verified.
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Rule 2411
Rule 63
Rule 208
Rule 2348
Rule 12
Rule 1587
Rule 6741
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt{f+g x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \sqrt{\frac{e f-d g}{e}+\frac{g x}{e}}} \, dx,x,d+e x\right )}{e}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{e} \sqrt{e f-d g}}-\frac{(b n) \operatorname{Subst}\left (\int -\frac{2 \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f-\frac{d g}{e}+\frac{g x}{e}}}{\sqrt{e f-d g}}\right )}{\sqrt{e f-d g} x} \, dx,x,d+e x\right )}{e}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{e} \sqrt{e f-d g}}+\frac{(2 b n) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f-\frac{d g}{e}+\frac{g x}{e}}}{\sqrt{e f-d g}}\right )}{x} \, dx,x,d+e x\right )}{\sqrt{e} \sqrt{e f-d g}}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{e} \sqrt{e f-d g}}+\frac{\left (4 b \sqrt{e} n\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{e f-d g}}\right )}{d g+e \left (-f+x^2\right )} \, dx,x,\sqrt{f+g x}\right )}{\sqrt{e f-d g}}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{e} \sqrt{e f-d g}}+\frac{\left (4 b \sqrt{e} n\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{e f-d g}}\right )}{-e f+d g+e x^2} \, dx,x,\sqrt{f+g x}\right )}{\sqrt{e f-d g}}\\ &=\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{\sqrt{e} \sqrt{e f-d g}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{e} \sqrt{e f-d g}}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{e f-d g}}\right )}{1-\frac{\sqrt{e} x}{\sqrt{e f-d g}}} \, dx,x,\sqrt{f+g x}\right )}{e f-d g}\\ &=\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{\sqrt{e} \sqrt{e f-d g}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{e} \sqrt{e f-d g}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{\sqrt{e} \sqrt{e f-d g}}+\frac{(4 b n) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-\frac{\sqrt{e} x}{\sqrt{e f-d g}}}\right )}{1-\frac{e x^2}{e f-d g}} \, dx,x,\sqrt{f+g x}\right )}{e f-d g}\\ &=\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{\sqrt{e} \sqrt{e f-d g}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{e} \sqrt{e f-d g}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{\sqrt{e} \sqrt{e f-d g}}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{\sqrt{e} \sqrt{e f-d g}}\\ &=\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{\sqrt{e} \sqrt{e f-d g}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{e} \sqrt{e f-d g}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{\sqrt{e} \sqrt{e f-d g}}-\frac{2 b n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{\sqrt{e} \sqrt{e f-d g}}\\ \end{align*}
Mathematica [A] time = 0.219524, size = 457, normalized size = 1.79 \[ \frac{-2 b n \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{e} \sqrt{f+g x}}{2 \sqrt{e f-d g}}\right )+2 b n \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}+1\right )\right )+2 a \log \left (\sqrt{e f-d g}-\sqrt{e} \sqrt{f+g x}\right )-2 a \log \left (\sqrt{e f-d g}+\sqrt{e} \sqrt{f+g x}\right )+2 b \log \left (c (d+e x)^n\right ) \log \left (\sqrt{e f-d g}-\sqrt{e} \sqrt{f+g x}\right )-2 b \log \left (c (d+e x)^n\right ) \log \left (\sqrt{e f-d g}+\sqrt{e} \sqrt{f+g x}\right )-b n \log ^2\left (\sqrt{e f-d g}-\sqrt{e} \sqrt{f+g x}\right )+b n \log ^2\left (\sqrt{e f-d g}+\sqrt{e} \sqrt{f+g x}\right )-2 b n \log \left (\frac{1}{2} \left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}+1\right )\right ) \log \left (\sqrt{e f-d g}-\sqrt{e} \sqrt{f+g x}\right )+2 b n \log \left (\sqrt{e f-d g}+\sqrt{e} \sqrt{f+g x}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{e} \sqrt{f+g x}}{2 \sqrt{e f-d g}}\right )}{2 \sqrt{e} \sqrt{e f-d g}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.168, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) }{ex+d}{\frac{1}{\sqrt{gx+f}}}}\, dx \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{\sqrt{g x + f} b \log \left ({\left (e x + d\right )}^{n} c\right ) + \sqrt{g x + f} a}{e g x^{2} + d f +{\left (e f + d g\right )} x}, 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 (e x + d\right )} \sqrt{g x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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