Optimal. Leaf size=97 \[ \frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{4 b n \sqrt{e f-d g} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e} g}-\frac{4 b n \sqrt{f+g x}}{g} \]
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Rubi [A] time = 0.0586838, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2395, 50, 63, 208} \[ \frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{4 b n \sqrt{e f-d g} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e} g}-\frac{4 b n \sqrt{f+g x}}{g} \]
Antiderivative was successfully verified.
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Rule 2395
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{f+g x}} \, dx &=\frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac{(2 b e n) \int \frac{\sqrt{f+g x}}{d+e x} \, dx}{g}\\ &=-\frac{4 b n \sqrt{f+g x}}{g}+\frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac{(2 b (e f-d g) n) \int \frac{1}{(d+e x) \sqrt{f+g x}} \, dx}{g}\\ &=-\frac{4 b n \sqrt{f+g x}}{g}+\frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac{(4 b (e f-d g) n) \operatorname{Subst}\left (\int \frac{1}{d-\frac{e f}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{g^2}\\ &=-\frac{4 b n \sqrt{f+g x}}{g}+\frac{4 b \sqrt{e f-d g} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e} g}+\frac{2 \sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}\\ \end{align*}
Mathematica [A] time = 0.0799041, size = 83, normalized size = 0.86 \[ \frac{2 \left (\sqrt{f+g x} \left (a+b \log \left (c (d+e x)^n\right )-2 b n\right )+\frac{2 b n \sqrt{e f-d g} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e}}\right )}{g} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.331, size = 148, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{gx+f}a}{g}}+2\,{\frac{b\sqrt{gx+f}}{g}\ln \left ( c \left ({\frac{ \left ( gx+f \right ) e+dg-fe}{g}} \right ) ^{n} \right ) }-4\,{\frac{bn\sqrt{gx+f}}{g}}+4\,{\frac{bdn}{\sqrt{ \left ( dg-fe \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-fe \right ) e}}} \right ) }-4\,{\frac{befn}{g\sqrt{ \left ( dg-fe \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-fe \right ) e}}} \right ) } \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 [A] time = 1.92591, size = 432, normalized size = 4.45 \begin{align*} \left [\frac{2 \,{\left (b n \sqrt{\frac{e f - d g}{e}} \log \left (\frac{e g x + 2 \, e f - d g + 2 \, \sqrt{g x + f} e \sqrt{\frac{e f - d g}{e}}}{e x + d}\right ) +{\left (b n \log \left (e x + d\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt{g x + f}\right )}}{g}, \frac{2 \,{\left (2 \, b n \sqrt{-\frac{e f - d g}{e}} \arctan \left (-\frac{\sqrt{g x + f} e \sqrt{-\frac{e f - d g}{e}}}{e f - d g}\right ) +{\left (b n \log \left (e x + d\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt{g x + f}\right )}}{g}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.5964, size = 326, normalized size = 3.36 \begin{align*} \begin{cases} - \frac{\frac{2 a f}{\sqrt{f + g x}} + 2 a \left (- \frac{f}{\sqrt{f + g x}} - \sqrt{f + g x}\right ) + 2 b f \left (\frac{2 e n \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{\sqrt{\frac{e}{d g - e f}} \left (d g - e f\right )} + \frac{\log{\left (c \left (d + e x\right )^{n} \right )}}{\sqrt{f + g x}}\right ) + 2 b \left (- \frac{2 e n \left (- \frac{g \sqrt{f + g x}}{e} - \frac{g \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{e \sqrt{\frac{e}{d g - e f}}}\right )}{g} - f \left (\frac{2 e n \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{\sqrt{\frac{e}{d g - e f}} \left (d g - e f\right )} + \frac{\log{\left (c \left (d - \frac{e f}{g} + \frac{e \left (f + g x\right )}{g}\right )^{n} \right )}}{\sqrt{f + g x}}\right ) - \sqrt{f + g x} \log{\left (c \left (d - \frac{e f}{g} + \frac{e \left (f + g x\right )}{g}\right )^{n} \right )}\right )}{g} & \text{for}\: g \neq 0 \\\frac{a x + b \left (- e n \left (- \frac{d \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right )}{e} + \frac{x}{e}\right ) + x \log{\left (c \left (d + e x\right )^{n} \right )}\right )}{\sqrt{f}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26109, size = 149, normalized size = 1.54 \begin{align*} \frac{2 \,{\left ({\left (2 \,{\left (\frac{{\left (d g - f e\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right ) e^{\left (-1\right )}}{\sqrt{d g e - f e^{2}}} - \sqrt{g x + f} e^{\left (-1\right )}\right )} e + \sqrt{g x + f} \log \left (x e + d\right )\right )} b n + \sqrt{g x + f} b \log \left (c\right ) + \sqrt{g x + f} a\right )}}{g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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