Optimal. Leaf size=54 \[ 2 e \sqrt{c+d x}-2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+\frac{2 f (c+d x)^{3/2}}{3 d} \]
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Rubi [A] time = 0.0165239, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {80, 50, 63, 208} \[ 2 e \sqrt{c+d x}-2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+\frac{2 f (c+d x)^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x} (e+f x)}{x} \, dx &=\frac{2 f (c+d x)^{3/2}}{3 d}+e \int \frac{\sqrt{c+d x}}{x} \, dx\\ &=2 e \sqrt{c+d x}+\frac{2 f (c+d x)^{3/2}}{3 d}+(c e) \int \frac{1}{x \sqrt{c+d x}} \, dx\\ &=2 e \sqrt{c+d x}+\frac{2 f (c+d x)^{3/2}}{3 d}+\frac{(2 c e) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=2 e \sqrt{c+d x}+\frac{2 f (c+d x)^{3/2}}{3 d}-2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0461209, size = 55, normalized size = 1.02 \[ e \left (2 \sqrt{c+d x}-2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )\right )+\frac{2 f (c+d x)^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 46, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{d} \left ( 1/3\,f \left ( dx+c \right ) ^{3/2}+de\sqrt{dx+c}-\sqrt{c}de{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) \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 = 1.32366, size = 279, normalized size = 5.17 \begin{align*} \left [\frac{3 \, \sqrt{c} d e \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \,{\left (d f x + 3 \, d e + c f\right )} \sqrt{d x + c}}{3 \, d}, \frac{2 \,{\left (3 \, \sqrt{-c} d e \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c}}{c}\right ) +{\left (d f x + 3 \, d e + c f\right )} \sqrt{d x + c}\right )}}{3 \, d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.87157, size = 54, normalized size = 1. \begin{align*} \frac{2 c e \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{\sqrt{- c}} + 2 e \sqrt{c + d x} + \frac{2 f \left (c + d x\right )^{\frac{3}{2}}}{3 d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.53136, size = 77, normalized size = 1.43 \begin{align*} \frac{2 \, c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) e}{\sqrt{-c}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} d^{2} f + 3 \, \sqrt{d x + c} d^{3} e\right )}}{3 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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