Integrand size = 13, antiderivative size = 8 \[ \int \frac {\sqrt {x}}{x+x^2} \, dx=2 \arctan \left (\sqrt {x}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {661, 65, 209} \[ \int \frac {\sqrt {x}}{x+x^2} \, dx=2 \arctan \left (\sqrt {x}\right ) \]
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Rule 65
Rule 209
Rule 661
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {x} (1+x)} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \tan ^{-1}\left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x}}{x+x^2} \, dx=2 \arctan \left (\sqrt {x}\right ) \]
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Time = 1.01 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(2 \arctan \left (\sqrt {x}\right )\) | \(7\) |
default | \(2 \arctan \left (\sqrt {x}\right )\) | \(7\) |
meijerg | \(2 \arctan \left (\sqrt {x}\right )\) | \(7\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x}+x -1}{x +1}\right )\) | \(29\) |
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none
Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {x}}{x+x^2} \, dx=2 \, \arctan \left (\sqrt {x}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {x}}{x+x^2} \, dx=2 \operatorname {atan}{\left (\sqrt {x} \right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {x}}{x+x^2} \, dx=2 \, \arctan \left (\sqrt {x}\right ) \]
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none
Time = 0.35 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {x}}{x+x^2} \, dx=2 \, \arctan \left (\sqrt {x}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {x}}{x+x^2} \, dx=2\,\mathrm {atan}\left (\sqrt {x}\right ) \]
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