Optimal. Leaf size=506 \[ -\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \text{PolyLog}\left (2,-\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+e^2 f}}\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \text{PolyLog}\left (2,-\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{\sqrt{d^2 g+e^2 f}+d \sqrt{g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\sqrt{f} \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+e^2 f}}+1\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{\sqrt{d^2 g+e^2 f}+d \sqrt{g}}+1\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )^2}{2 \sqrt{g} \sqrt{f+g x^2}} \]
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Rubi [A] time = 0.556819, antiderivative size = 506, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {2406, 215, 2404, 12, 5799, 5561, 2190, 2279, 2391} \[ -\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \text{PolyLog}\left (2,-\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+e^2 f}}\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \text{PolyLog}\left (2,-\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{\sqrt{d^2 g+e^2 f}+d \sqrt{g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\sqrt{f} \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+e^2 f}}+1\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{\sqrt{d^2 g+e^2 f}+d \sqrt{g}}+1\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )^2}{2 \sqrt{g} \sqrt{f+g x^2}} \]
Antiderivative was successfully verified.
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Rule 2406
Rule 215
Rule 2404
Rule 12
Rule 5799
Rule 5561
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{f+g x^2}} \, dx &=\frac{\sqrt{1+\frac{g x^2}{f}} \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{1+\frac{g x^2}{f}}} \, dx}{\sqrt{f+g x^2}}\\ &=\frac{\sqrt{f} \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{\left (b e n \sqrt{1+\frac{g x^2}{f}}\right ) \int \frac{\sqrt{f} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g} (d+e x)} \, dx}{\sqrt{f+g x^2}}\\ &=\frac{\sqrt{f} \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{\left (b e \sqrt{f} n \sqrt{1+\frac{g x^2}{f}}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{d+e x} \, dx}{\sqrt{g} \sqrt{f+g x^2}}\\ &=\frac{\sqrt{f} \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{\left (b e \sqrt{f} n \sqrt{1+\frac{g x^2}{f}}\right ) \operatorname{Subst}\left (\int \frac{x \cosh (x)}{\frac{d \sqrt{g}}{\sqrt{f}}+e \sinh (x)} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}\\ &=\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )^2}{2 \sqrt{g} \sqrt{f+g x^2}}+\frac{\sqrt{f} \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{\left (b e \sqrt{f} n \sqrt{1+\frac{g x^2}{f}}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{e e^x+\frac{d \sqrt{g}}{\sqrt{f}}-\frac{\sqrt{e^2 f+d^2 g}}{\sqrt{f}}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{\left (b e \sqrt{f} n \sqrt{1+\frac{g x^2}{f}}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{e e^x+\frac{d \sqrt{g}}{\sqrt{f}}+\frac{\sqrt{e^2 f+d^2 g}}{\sqrt{f}}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}\\ &=\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )^2}{2 \sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )} \sqrt{f}}{d \sqrt{g}-\sqrt{e^2 f+d^2 g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )} \sqrt{f}}{d \sqrt{g}+\sqrt{e^2 f+d^2 g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\sqrt{f} \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\left (b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{e e^x}{\frac{d \sqrt{g}}{\sqrt{f}}-\frac{\sqrt{e^2 f+d^2 g}}{\sqrt{f}}}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\left (b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{e e^x}{\frac{d \sqrt{g}}{\sqrt{f}}+\frac{\sqrt{e^2 f+d^2 g}}{\sqrt{f}}}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}\\ &=\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )^2}{2 \sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )} \sqrt{f}}{d \sqrt{g}-\sqrt{e^2 f+d^2 g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )} \sqrt{f}}{d \sqrt{g}+\sqrt{e^2 f+d^2 g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\sqrt{f} \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\left (b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{\frac{d \sqrt{g}}{\sqrt{f}}-\frac{\sqrt{e^2 f+d^2 g}}{\sqrt{f}}}\right )}{x} \, dx,x,e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\left (b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{\frac{d \sqrt{g}}{\sqrt{f}}+\frac{\sqrt{e^2 f+d^2 g}}{\sqrt{f}}}\right )}{x} \, dx,x,e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}\right )}{\sqrt{g} \sqrt{f+g x^2}}\\ &=\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )^2}{2 \sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )} \sqrt{f}}{d \sqrt{g}-\sqrt{e^2 f+d^2 g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )} \sqrt{f}}{d \sqrt{g}+\sqrt{e^2 f+d^2 g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\sqrt{f} \sqrt{1+\frac{g x^2}{f}} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \text{Li}_2\left (-\frac{e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )} \sqrt{f}}{d \sqrt{g}-\sqrt{e^2 f+d^2 g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{1+\frac{g x^2}{f}} \text{Li}_2\left (-\frac{e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )} \sqrt{f}}{d \sqrt{g}+\sqrt{e^2 f+d^2 g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}\\ \end{align*}
Mathematica [F] time = 3.60508, size = 0, normalized size = 0. \[ \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{f+g x^2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.822, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) ){\frac{1}{\sqrt{g{x}^{2}+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^{2} + f} b \log \left ({\left (e x + d\right )}^{n} c\right ) + \sqrt{g x^{2} + f} a}{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{a + b \log{\left (c \left (d + e x\right )^{n} \right )}}{\sqrt{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{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{\sqrt{g x^{2} + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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