Optimal. Leaf size=340 \[ -\frac{2 b \sqrt{e} n \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{(e f-d g)^{3/2}}+\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{f+g x} (e f-d g)}-\frac{2 \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^{3/2}}+\frac{2 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{(e f-d g)^{3/2}}+\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{(e f-d g)^{3/2}}-\frac{4 b \sqrt{e} n \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{(e f-d g)^{3/2}} \]
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Rubi [A] time = 1.02555, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 13, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.419, Rules used = {2411, 2347, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319} \[ -\frac{2 b \sqrt{e} n \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{(e f-d g)^{3/2}}+\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{f+g x} (e f-d g)}-\frac{2 \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^{3/2}}+\frac{2 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{(e f-d g)^{3/2}}+\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{(e f-d g)^{3/2}}-\frac{4 b \sqrt{e} n \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{(e f-d g)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2411
Rule 2347
Rule 63
Rule 208
Rule 2348
Rule 12
Rule 1587
Rule 6741
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 2319
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{e}\\ &=\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 f-d g}-\frac{g \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{e (e f-d g)}\\ &=\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) \sqrt{f+g x}}-\frac{2 \sqrt{e} \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 )}{(e f-d g)^{3/2}}-\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 f-d g}-\frac{(2 b n) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{\frac{e f-d g}{e}+\frac{g x}{e}}} \, dx,x,d+e x\right )}{e f-d g}\\ &=\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) \sqrt{f+g x}}-\frac{2 \sqrt{e} \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 )}{(e f-d g)^{3/2}}+\frac{\left (2 b \sqrt{e} n\right ) \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 )}{(e f-d g)^{3/2}}-\frac{(4 b e n) \operatorname{Subst}\left (\int \frac{1}{-\frac{e f-d g}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{g (e f-d g)}\\ &=\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{(e f-d g)^{3/2}}+\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) \sqrt{f+g x}}-\frac{2 \sqrt{e} \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 )}{(e f-d g)^{3/2}}+\frac{\left (4 b e^{3/2} 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 )}{(e f-d g)^{3/2}}\\ &=\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{(e f-d g)^{3/2}}+\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) \sqrt{f+g x}}-\frac{2 \sqrt{e} \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 )}{(e f-d g)^{3/2}}+\frac{\left (4 b e^{3/2} 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 )}{(e f-d g)^{3/2}}\\ &=\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{(e f-d g)^{3/2}}+\frac{2 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{(e f-d g)^{3/2}}+\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) \sqrt{f+g x}}-\frac{2 \sqrt{e} \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 )}{(e f-d g)^{3/2}}-\frac{(4 b e 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)^2}\\ &=\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{(e f-d g)^{3/2}}+\frac{2 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{(e f-d g)^{3/2}}+\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) \sqrt{f+g x}}-\frac{2 \sqrt{e} \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 )}{(e f-d g)^{3/2}}-\frac{4 b \sqrt{e} 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 )}{(e f-d g)^{3/2}}+\frac{(4 b e 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)^2}\\ &=\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{(e f-d g)^{3/2}}+\frac{2 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{(e f-d g)^{3/2}}+\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) \sqrt{f+g x}}-\frac{2 \sqrt{e} \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 )}{(e f-d g)^{3/2}}-\frac{4 b \sqrt{e} 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 )}{(e f-d g)^{3/2}}-\frac{\left (4 b \sqrt{e} n\right ) \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 )}{(e f-d g)^{3/2}}\\ &=\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{(e f-d g)^{3/2}}+\frac{2 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{(e f-d g)^{3/2}}+\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) \sqrt{f+g x}}-\frac{2 \sqrt{e} \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 )}{(e f-d g)^{3/2}}-\frac{4 b \sqrt{e} 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 )}{(e f-d g)^{3/2}}-\frac{2 b \sqrt{e} n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{(e f-d g)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.519943, size = 526, normalized size = 1.55 \[ \frac{-b \sqrt{e} n \sqrt{f+g x} \left (2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{e} \sqrt{f+g x}}{2 \sqrt{e f-d g}}\right )+\log \left (\sqrt{e f-d g}-\sqrt{e} \sqrt{f+g x}\right ) \left (\log \left (\sqrt{e f-d g}-\sqrt{e} \sqrt{f+g x}\right )+2 \log \left (\frac{1}{2} \left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}+1\right )\right )\right )\right )+b \sqrt{e} n \sqrt{f+g x} \left (2 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}+1\right )\right )+\log \left (\sqrt{e f-d g}+\sqrt{e} \sqrt{f+g x}\right ) \left (\log \left (\sqrt{e f-d g}+\sqrt{e} \sqrt{f+g x}\right )+2 \log \left (\frac{1}{2}-\frac{\sqrt{e} \sqrt{f+g x}}{2 \sqrt{e f-d g}}\right )\right )\right )+4 \sqrt{e f-d g} \left (a+b \log \left (c (d+e x)^n\right )\right )+2 \sqrt{e} \sqrt{f+g x} \log \left (\sqrt{e f-d g}-\sqrt{e} \sqrt{f+g x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-2 \sqrt{e} \sqrt{f+g x} \log \left (\sqrt{e f-d g}+\sqrt{e} \sqrt{f+g x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+8 b \sqrt{e} n \sqrt{f+g x} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{2 \sqrt{f+g x} (e f-d g)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.152, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) }{ex+d} \left ( gx+f \right ) ^{-{\frac{3}{2}}}}\, 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^{2} x^{3} + d f^{2} +{\left (2 \, e f g + d g^{2}\right )} x^{2} +{\left (e f^{2} + 2 \, d f 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 )}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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