Optimal. Leaf size=36 \[ \frac {1}{2} (2+x) \sqrt {5-4 x-x^2}-\frac {9}{2} \sin ^{-1}\left (\frac {1}{3} (-2-x)\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {626, 633, 222}
\begin {gather*} \frac {1}{2} (x+2) \sqrt {-x^2-4 x+5}-\frac {9}{2} \sin ^{-1}\left (\frac {1}{3} (-x-2)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 626
Rule 633
Rubi steps
\begin {align*} \int \sqrt {5-4 x-x^2} \, dx &=\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} \sin ^{-1}\left (\frac {1}{3} (-2-x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 45, normalized size = 1.25 \begin {gather*} \frac {1}{2} (2+x) \sqrt {5-4 x-x^2}-9 \tan ^{-1}\left (\frac {\sqrt {5-4 x-x^2}}{5+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.09, size = 29, normalized size = 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 | \(\left (1+\frac {x}{2}\right ) \sqrt {-x^{2}-4 x +5}+\frac {9 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-4 x +5}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{2}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 36, normalized size = 1.00 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 47, normalized size = 1.31 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- x^{2} - 4 x + 5}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 35, normalized size = 0.97 \begin {gather*} 2 \left (\frac {x}{4}+\frac 1{2}\right ) \sqrt {-x^{2}-4 x+5}+\frac {9}{2} \arcsin \left (\frac {x+2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 27, normalized size = 0.75 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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