Integrand size = 14, antiderivative size = 36 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} (2+x) \sqrt {5-4 x-x^2}-\frac {9}{2} \arcsin \left (\frac {1}{3} (-2-x)\right ) \]
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Time = 0.01 (sec) , antiderivative size = 36, 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} \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} (x+2) \sqrt {-x^2-4 x+5}-\frac {9}{2} \arcsin \left (\frac {1}{3} (-x-2)\right ) \]
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Rule 222
Rule 626
Rule 633
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} (2+x) \sqrt {5-4 x-x^2}+\frac {9}{2} \int \frac {1}{\sqrt {5-4 x-x^2}} \, dx \\ & = \frac {1}{2} (2+x) \sqrt {5-4 x-x^2}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{36}}} \, dx,x,-4-2 x\right ) \\ & = \frac {1}{2} (2+x) \sqrt {5-4 x-x^2}-\frac {9}{2} \arcsin \left (\frac {1}{3} (-2-x)\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.25 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} (2+x) \sqrt {5-4 x-x^2}-9 \arctan \left (\frac {\sqrt {5-4 x-x^2}}{5+x}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\left (-2 x -4\right ) \sqrt {-x^{2}-4 x +5}}{4}+\frac {9 \arcsin \left (\frac {2}{3}+\frac {x}{3}\right )}{2}\) | \(29\) |
risch | \(-\frac {\left (2+x \right ) \left (x^{2}+4 x -5\right )}{2 \sqrt {-x^{2}-4 x +5}}+\frac {9 \arcsin \left (\frac {2}{3}+\frac {x}{3}\right )}{2}\) | \(35\) |
trager | \(\left (1+\frac {x}{2}\right ) \sqrt {-x^{2}-4 x +5}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-4 x +5}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{2}\) | \(59\) |
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Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )} - \frac {9}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )}}{x^{2} + 4 \, x - 5}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \sqrt {5-4 x-x^2} \, dx=\left (\frac {x}{2} + 1\right ) \sqrt {- x^{2} - 4 x + 5} + \frac {9 \operatorname {asin}{\left (\frac {x}{3} + \frac {2}{3} \right )}}{2} \]
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Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} - 4 \, x + 5} x + \sqrt {-x^{2} - 4 \, x + 5} - \frac {9}{2} \, \arcsin \left (-\frac {1}{3} \, x - \frac {2}{3}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, x + \frac {2}{3}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {9\,\mathrm {asin}\left (\frac {x}{3}+\frac {2}{3}\right )}{2}+\left (\frac {x}{2}+1\right )\,\sqrt {-x^2-4\,x+5} \]
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