Integrand size = 22, antiderivative size = 19 \[ \int \frac {9+6 \sqrt {x}+x}{4 \sqrt {x}+x} \, dx=4 \sqrt {x}+x+2 \log \left (4+\sqrt {x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {28, 1411, 785} \[ \int \frac {9+6 \sqrt {x}+x}{4 \sqrt {x}+x} \, dx=x+4 \sqrt {x}+2 \log \left (\sqrt {x}+4\right ) \]
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Rule 28
Rule 785
Rule 1411
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (3+\sqrt {x}\right )^2}{4 \sqrt {x}+x} \, dx \\ & = 2 \text {Subst}\left (\int \frac {x (3+x)^2}{4 x+x^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (2+x+\frac {1}{4+x}\right ) \, dx,x,\sqrt {x}\right ) \\ & = 4 \sqrt {x}+x+2 \log \left (4+\sqrt {x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {9+6 \sqrt {x}+x}{4 \sqrt {x}+x} \, dx=4 \sqrt {x}+x+2 \log \left (4+\sqrt {x}\right ) \]
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Time = 1.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(x +2 \ln \left (4+\sqrt {x}\right )+4 \sqrt {x}\) | \(16\) |
default | \(x +2 \ln \left (4+\sqrt {x}\right )+4 \sqrt {x}\) | \(16\) |
trager | \(x -1+4 \sqrt {x}+\ln \left (8 \sqrt {x}+16+x \right )\) | \(18\) |
meijerg | \(2 \ln \left (1+\frac {\sqrt {x}}{4}\right )-\frac {4 \sqrt {x}\, \left (-\frac {3 \sqrt {x}}{4}+6\right )}{3}+12 \sqrt {x}\) | \(29\) |
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Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {9+6 \sqrt {x}+x}{4 \sqrt {x}+x} \, dx=x + 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 4\right ) \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {9+6 \sqrt {x}+x}{4 \sqrt {x}+x} \, dx=4 \sqrt {x} + x + 2 \log {\left (\sqrt {x} + 4 \right )} \]
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Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {9+6 \sqrt {x}+x}{4 \sqrt {x}+x} \, dx=x + 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 4\right ) \]
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Time = 0.34 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {9+6 \sqrt {x}+x}{4 \sqrt {x}+x} \, dx=x + 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 4\right ) \]
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Time = 18.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {9+6 \sqrt {x}+x}{4 \sqrt {x}+x} \, dx=x+2\,\ln \left (\sqrt {x}+4\right )+4\,\sqrt {x} \]
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