Integrand size = 27, antiderivative size = 33 \[ \int \frac {1-\sqrt {2+3 x}}{1+\sqrt {2+3 x}} \, dx=-x+\frac {4}{3} \sqrt {2+3 x}-\frac {4}{3} \log \left (1+\sqrt {2+3 x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {442, 383, 78} \[ \int \frac {1-\sqrt {2+3 x}}{1+\sqrt {2+3 x}} \, dx=-x+\frac {4}{3} \sqrt {3 x+2}-\frac {4}{3} \log \left (\sqrt {3 x+2}+1\right ) \]
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Rule 78
Rule 383
Rule 442
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1-\sqrt {x}}{1+\sqrt {x}} \, dx,x,2+3 x\right ) \\ & = \frac {2}{3} \text {Subst}\left (\int \frac {(1-x) x}{1+x} \, dx,x,\sqrt {2+3 x}\right ) \\ & = \frac {2}{3} \text {Subst}\left (\int \left (2-x-\frac {2}{1+x}\right ) \, dx,x,\sqrt {2+3 x}\right ) \\ & = -x+\frac {4}{3} \sqrt {2+3 x}-\frac {4}{3} \log \left (1+\sqrt {2+3 x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {1-\sqrt {2+3 x}}{1+\sqrt {2+3 x}} \, dx=\frac {1}{3} \left (-2-3 x+4 \sqrt {2+3 x}-4 \log \left (1+\sqrt {2+3 x}\right )\right ) \]
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Time = 0.94 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(-x -\frac {2}{3}+\frac {4 \sqrt {3 x +2}}{3}-\frac {4 \ln \left (1+\sqrt {3 x +2}\right )}{3}\) | \(27\) |
default | \(-x -\frac {2}{3}+\frac {4 \sqrt {3 x +2}}{3}-\frac {4 \ln \left (1+\sqrt {3 x +2}\right )}{3}\) | \(27\) |
trager | \(-x +\frac {4 \sqrt {3 x +2}}{3}-\frac {2 \ln \left (2 \sqrt {3 x +2}+3+3 x \right )}{3}\) | \(31\) |
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Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1-\sqrt {2+3 x}}{1+\sqrt {2+3 x}} \, dx=-x + \frac {4}{3} \, \sqrt {3 \, x + 2} - \frac {4}{3} \, \log \left (\sqrt {3 \, x + 2} + 1\right ) \]
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Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {1-\sqrt {2+3 x}}{1+\sqrt {2+3 x}} \, dx=- x + \frac {4 \sqrt {3 x + 2}}{3} - \frac {4 \log {\left (\sqrt {3 x + 2} + 1 \right )}}{3} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {1-\sqrt {2+3 x}}{1+\sqrt {2+3 x}} \, dx=-x + \frac {4}{3} \, \sqrt {3 \, x + 2} - \frac {4}{3} \, \log \left (\sqrt {3 \, x + 2} + 1\right ) - \frac {2}{3} \]
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Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {1-\sqrt {2+3 x}}{1+\sqrt {2+3 x}} \, dx=-x + \frac {4}{3} \, \sqrt {3 \, x + 2} - \frac {4}{3} \, \log \left (\sqrt {3 \, x + 2} + 1\right ) - \frac {2}{3} \]
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Time = 17.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1-\sqrt {2+3 x}}{1+\sqrt {2+3 x}} \, dx=\frac {4\,\sqrt {3\,x+2}}{3}-\frac {4\,\ln \left (\sqrt {3\,x+2}+1\right )}{3}-x \]
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