Integrand size = 14, antiderivative size = 45 \[ \int \sqrt {2+4 x-3 x^2} \, dx=-\frac {1}{6} (2-3 x) \sqrt {2+4 x-3 x^2}-\frac {5 \arcsin \left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {626, 633, 222} \[ \int \sqrt {2+4 x-3 x^2} \, dx=-\frac {5 \arcsin \left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}}-\frac {1}{6} \sqrt {-3 x^2+4 x+2} (2-3 x) \]
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Rule 222
Rule 626
Rule 633
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{6} (2-3 x) \sqrt {2+4 x-3 x^2}+\frac {5}{3} \int \frac {1}{\sqrt {2+4 x-3 x^2}} \, dx \\ & = -\frac {1}{6} (2-3 x) \sqrt {2+4 x-3 x^2}-\frac {1}{6} \sqrt {\frac {5}{6}} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{40}}} \, dx,x,4-6 x\right ) \\ & = -\frac {1}{6} (2-3 x) \sqrt {2+4 x-3 x^2}-\frac {5 \sin ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.36 \[ \int \sqrt {2+4 x-3 x^2} \, dx=\frac {1}{6} (-2+3 x) \sqrt {2+4 x-3 x^2}+\frac {10 \arctan \left (\frac {-2-\sqrt {10}+3 x}{\sqrt {6+12 x-9 x^2}}\right )}{3 \sqrt {3}} \]
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Time = 2.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {\left (4-6 x \right ) \sqrt {-3 x^{2}+4 x +2}}{12}+\frac {5 \sqrt {3}\, \arcsin \left (\frac {3 \sqrt {10}\, \left (-\frac {2}{3}+x \right )}{10}\right )}{9}\) | \(35\) |
risch | \(-\frac {\left (3 x^{2}-4 x -2\right ) \left (-2+3 x \right )}{6 \sqrt {-3 x^{2}+4 x +2}}+\frac {5 \sqrt {3}\, \arcsin \left (\frac {3 \sqrt {10}\, \left (-\frac {2}{3}+x \right )}{10}\right )}{9}\) | \(45\) |
trager | \(\left (-\frac {1}{3}+\frac {x}{2}\right ) \sqrt {-3 x^{2}+4 x +2}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (3 x \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )+3 \sqrt {-3 x^{2}+4 x +2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )\right )}{9}\) | \(61\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.33 \[ \int \sqrt {2+4 x-3 x^2} \, dx=\frac {1}{6} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (3 \, x - 2\right )} - \frac {5}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (3 \, x - 2\right )}}{3 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int \sqrt {2+4 x-3 x^2} \, dx=\left (\frac {x}{2} - \frac {1}{3}\right ) \sqrt {- 3 x^{2} + 4 x + 2} + \frac {5 \sqrt {3} \operatorname {asin}{\left (\frac {3 \sqrt {10} \left (x - \frac {2}{3}\right )}{10} \right )}}{9} \]
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Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int \sqrt {2+4 x-3 x^2} \, dx=\frac {1}{2} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} x - \frac {5}{9} \, \sqrt {3} \arcsin \left (-\frac {1}{10} \, \sqrt {10} {\left (3 \, x - 2\right )}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \]
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80 \[ \int \sqrt {2+4 x-3 x^2} \, dx=\frac {1}{6} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (3 \, x - 2\right )} + \frac {5}{9} \, \sqrt {3} \arcsin \left (\frac {1}{10} \, \sqrt {10} {\left (3 \, x - 2\right )}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \sqrt {2+4 x-3 x^2} \, dx=\frac {5\,\sqrt {3}\,\mathrm {asin}\left (\frac {\sqrt {10}\,\left (3\,x-2\right )}{10}\right )}{9}+\left (\frac {x}{2}-\frac {1}{3}\right )\,\sqrt {-3\,x^2+4\,x+2} \]
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