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 = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 50
Rule 63
Rule 80
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 1.27, size = 111, normalized size = 2.06 \[ \left [\frac {3 \, \sqrt {c} d e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (d f x + 3 \, d e + c f\right )} \sqrt {d x + c}}{3 \, d}, \frac {2 \, {\left (3 \, \sqrt {-c} d e \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (d f x + 3 \, d e + c f\right )} \sqrt {d x + c}\right )}}{3 \, d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 57, normalized size = 1.06 \[ \frac {2 \, c \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right ) e}{\sqrt {-c}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} d^{2} f + 3 \, \sqrt {d x + c} d^{3} e\right )}}{3 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.85 \[ \frac {-2 \sqrt {c}\, d e \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+2 \sqrt {d x +c}\, d e +\frac {2 \left (d x +c \right )^{\frac {3}{2}} f}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 60, normalized size = 1.11 \[ \sqrt {c} e \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right ) + \frac {2 \, {\left (3 \, \sqrt {d x + c} d e + {\left (d x + c\right )}^{\frac {3}{2}} f\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 45, normalized size = 0.83 \[ 2\,e\,\sqrt {c+d\,x}+\frac {2\,f\,{\left (c+d\,x\right )}^{3/2}}{3\,d}+\sqrt {c}\,e\,\mathrm {atan}\left (\frac {\sqrt {c+d\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.98, size = 54, normalized size = 1.00 \[ \frac {2 c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 e \sqrt {c + d x} + \frac {2 f \left (c + d x\right )^{\frac {3}{2}}}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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