Integrand size = 17, antiderivative size = 21 \[ \int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx=4 \sqrt {x}+x+4 \log \left (1-\sqrt {x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {383, 78} \[ \int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx=x+4 \sqrt {x}+4 \log \left (1-\sqrt {x}\right ) \]
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Rule 78
Rule 383
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x (1+x)}{-1+x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (2+\frac {2}{-1+x}+x\right ) \, dx,x,\sqrt {x}\right ) \\ & = 4 \sqrt {x}+x+4 \log \left (1-\sqrt {x}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx=4 \sqrt {x}+x+4 \log \left (-1+\sqrt {x}\right ) \]
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Time = 0.93 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(x +4 \sqrt {x}+4 \ln \left (-1+\sqrt {x}\right )\) | \(16\) |
default | \(x +4 \sqrt {x}+4 \ln \left (-1+\sqrt {x}\right )\) | \(16\) |
trager | \(x -2+4 \sqrt {x}+2 \ln \left (2 \sqrt {x}-1-x \right )\) | \(22\) |
meijerg | \(2 \sqrt {x}+4 \ln \left (1-\sqrt {x}\right )+\frac {\sqrt {x}\, \left (6+3 \sqrt {x}\right )}{3}\) | \(29\) |
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none
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx=x + 4 \, \sqrt {x} + 4 \, \log \left (\sqrt {x} - 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx=4 \sqrt {x} + x + 4 \log {\left (\sqrt {x} - 1 \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx=x + 4 \, \sqrt {x} + 4 \, \log \left (\sqrt {x} - 1\right ) \]
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none
Time = 0.43 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx=x + 4 \, \sqrt {x} + 4 \, \log \left ({\left | \sqrt {x} - 1 \right |}\right ) \]
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Time = 17.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx=x+4\,\ln \left (\sqrt {x}-1\right )+4\,\sqrt {x} \]
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