Integrand size = 18, antiderivative size = 54 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=2 e \sqrt {c+d x}+\frac {2 f (c+d x)^{3/2}}{3 d}-2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {81, 52, 65, 214} \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=-2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+2 e \sqrt {c+d x}+\frac {2 f (c+d x)^{3/2}}{3 d} \]
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Rule 52
Rule 65
Rule 81
Rule 214
Rubi steps \begin{align*} \text {integral}& = \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) \text {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*}
Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \sqrt {c+d x} (3 d e+c f+d f x)}{3 d}-2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \]
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Time = 5.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {2 f \left (d x +c \right )^{\frac {3}{2}}}{3}+2 d e \sqrt {d x +c}-2 \sqrt {c}\, d e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{d}\) | \(46\) |
default | \(\frac {\frac {2 f \left (d x +c \right )^{\frac {3}{2}}}{3}+2 d e \sqrt {d x +c}-2 \sqrt {c}\, d e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{d}\) | \(46\) |
pseudoelliptic | \(\frac {-6 \sqrt {c}\, d e \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+2 \left (\left (f x +3 e \right ) d +c f \right ) \sqrt {d x +c}}{3 d}\) | \(48\) |
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Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=\left [\frac {3 \, \sqrt {c} d e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (d f x + 3 \, d e + c f\right )} \sqrt {d x + c}}{3 \, d}, \frac {2 \, {\left (3 \, \sqrt {-c} d e \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (d f x + 3 \, d e + c f\right )} \sqrt {d x + c}\right )}}{3 \, d}\right ] \]
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Time = 1.67 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=\begin {cases} \frac {2 c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 e \sqrt {c + d x} + \frac {2 f \left (c + d x\right )^{\frac {3}{2}}}{3 d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (e \log {\left (f x \right )} + f x\right ) & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=\sqrt {c} e \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right ) + \frac {2 \, {\left (3 \, \sqrt {d x + c} d e + {\left (d x + c\right )}^{\frac {3}{2}} f\right )}}{3 \, d} \]
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Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=\frac {2 \, c e \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {2 \, {\left (3 \, \sqrt {d x + c} d^{3} e + {\left (d x + c\right )}^{\frac {3}{2}} d^{2} f\right )}}{3 \, d^{3}} \]
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Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x} \, dx=2\,e\,\sqrt {c+d\,x}+\frac {2\,f\,{\left (c+d\,x\right )}^{3/2}}{3\,d}+\sqrt {c}\,e\,\mathrm {atan}\left (\frac {\sqrt {c+d\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,2{}\mathrm {i} \]
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