Integrand size = 16, antiderivative size = 54 \[ \int \frac {x}{\sqrt {-2+4 x+3 x^2}} \, dx=\frac {1}{3} \sqrt {-2+4 x+3 x^2}-\frac {2 \text {arctanh}\left (\frac {2+3 x}{\sqrt {3} \sqrt {-2+4 x+3 x^2}}\right )}{3 \sqrt {3}} \]
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Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {654, 635, 212} \[ \int \frac {x}{\sqrt {-2+4 x+3 x^2}} \, dx=\frac {1}{3} \sqrt {3 x^2+4 x-2}-\frac {2 \text {arctanh}\left (\frac {3 x+2}{\sqrt {3} \sqrt {3 x^2+4 x-2}}\right )}{3 \sqrt {3}} \]
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Rule 212
Rule 635
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \sqrt {-2+4 x+3 x^2}-\frac {2}{3} \int \frac {1}{\sqrt {-2+4 x+3 x^2}} \, dx \\ & = \frac {1}{3} \sqrt {-2+4 x+3 x^2}-\frac {4}{3} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {4+6 x}{\sqrt {-2+4 x+3 x^2}}\right ) \\ & = \frac {1}{3} \sqrt {-2+4 x+3 x^2}-\frac {2 \tanh ^{-1}\left (\frac {2+3 x}{\sqrt {3} \sqrt {-2+4 x+3 x^2}}\right )}{3 \sqrt {3}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\sqrt {-2+4 x+3 x^2}} \, dx=\frac {1}{9} \left (3 \sqrt {-2+4 x+3 x^2}+2 \sqrt {3} \log \left (-2-3 x+\sqrt {-6+12 x+9 x^2}\right )\right ) \]
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Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\sqrt {3 x^{2}+4 x -2}}{3}-\frac {2 \ln \left (\frac {\left (2+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+4 x -2}\right ) \sqrt {3}}{9}\) | \(45\) |
risch | \(\frac {\sqrt {3 x^{2}+4 x -2}}{3}-\frac {2 \ln \left (\frac {\left (2+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+4 x -2}\right ) \sqrt {3}}{9}\) | \(45\) |
trager | \(\frac {\sqrt {3 x^{2}+4 x -2}}{3}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +3 \sqrt {3 x^{2}+4 x -2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{9}\) | \(57\) |
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Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {x}{\sqrt {-2+4 x+3 x^2}} \, dx=\frac {1}{9} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 4 \, x - 2} {\left (3 \, x + 2\right )} + 9 \, x^{2} + 12 \, x - 1\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 4 \, x - 2} \]
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Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\sqrt {-2+4 x+3 x^2}} \, dx=\frac {\sqrt {3 x^{2} + 4 x - 2}}{3} - \frac {2 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 4 x - 2} + 4 \right )}}{9} \]
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\sqrt {-2+4 x+3 x^2}} \, dx=-\frac {2}{9} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 4 \, x - 2} + 6 \, x + 4\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 4 \, x - 2} \]
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\sqrt {-2+4 x+3 x^2}} \, dx=\frac {2}{9} \, \sqrt {3} \log \left ({\left | -\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 4 \, x - 2}\right )} - 2 \right |}\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 4 \, x - 2} \]
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Time = 10.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\sqrt {-2+4 x+3 x^2}} \, dx=\frac {\sqrt {3\,x^2+4\,x-2}}{3}-\frac {2\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+4\,x-2}+\frac {\sqrt {3}\,\left (3\,x+2\right )}{3}\right )}{9} \]
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