Optimal. Leaf size=52 \[ -\frac {1}{2} (1+x) \sqrt {3-2 x-x^2}-\frac {1}{3} \left (3-2 x-x^2\right )^{3/2}+2 \sin ^{-1}\left (\frac {1}{2} (-1-x)\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {654, 626, 633,
222} \begin {gather*} 2 \text {ArcSin}\left (\frac {1}{2} (-x-1)\right )-\frac {1}{3} \left (-x^2-2 x+3\right )^{3/2}-\frac {1}{2} (x+1) \sqrt {-x^2-2 x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 626
Rule 633
Rule 654
Rubi steps
\begin {align*} \int x \sqrt {3-2 x-x^2} \, dx &=-\frac {1}{3} \left (3-2 x-x^2\right )^{3/2}-\int \sqrt {3-2 x-x^2} \, dx\\ &=-\frac {1}{2} (1+x) \sqrt {3-2 x-x^2}-\frac {1}{3} \left (3-2 x-x^2\right )^{3/2}-2 \int \frac {1}{\sqrt {3-2 x-x^2}} \, dx\\ &=-\frac {1}{2} (1+x) \sqrt {3-2 x-x^2}-\frac {1}{3} \left (3-2 x-x^2\right )^{3/2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{16}}} \, dx,x,-2-2 x\right )\\ &=-\frac {1}{2} (1+x) \sqrt {3-2 x-x^2}-\frac {1}{3} \left (3-2 x-x^2\right )^{3/2}+2 \sin ^{-1}\left (\frac {1}{2} (-1-x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 50, normalized size = 0.96 \begin {gather*} \frac {1}{6} \sqrt {3-2 x-x^2} \left (-9+x+2 x^2\right )+4 \tan ^{-1}\left (\frac {\sqrt {3-2 x-x^2}}{3+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 43, normalized size = 0.83
method | result | size |
risch | \(-\frac {\left (2 x^{2}+x -9\right ) \left (x^{2}+2 x -3\right )}{6 \sqrt {-x^{2}-2 x +3}}-2 \arcsin \left (\frac {x}{2}+\frac {1}{2}\right )\) | \(40\) |
default | \(-\frac {\left (-x^{2}-2 x +3\right )^{\frac {3}{2}}}{3}+\frac {\left (-2-2 x \right ) \sqrt {-x^{2}-2 x +3}}{4}-2 \arcsin \left (\frac {x}{2}+\frac {1}{2}\right )\) | \(43\) |
trager | \(\left (\frac {1}{3} x^{2}+\frac {1}{6} x -\frac {3}{2}\right ) \sqrt {-x^{2}-2 x +3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (x \RootOf \left (\textit {\_Z}^{2}+1\right )+\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-2 x +3}\right )\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 52, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \, {\left (-x^{2} - 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {-x^{2} - 2 \, x + 3} x - \frac {1}{2} \, \sqrt {-x^{2} - 2 \, x + 3} + 2 \, \arcsin \left (-\frac {1}{2} \, x - \frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.41, size = 52, normalized size = 1.00 \begin {gather*} \frac {1}{6} \, {\left (2 \, x^{2} + x - 9\right )} \sqrt {-x^{2} - 2 \, x + 3} + 2 \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 3} {\left (x + 1\right )}}{x^{2} + 2 \, x - 3}\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 x \sqrt {- \left (x - 1\right ) \left (x + 3\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.52, size = 32, normalized size = 0.62 \begin {gather*} \frac {1}{6} \, {\left ({\left (2 \, x + 1\right )} x - 9\right )} \sqrt {-x^{2} - 2 \, x + 3} - 2 \, \arcsin \left (\frac {1}{2} \, x + \frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 47, normalized size = 0.90 \begin {gather*} \frac {\sqrt {-x^2-2\,x+3}\,\left (8\,x^2+4\,x-36\right )}{24}+\ln \left (x+1-\sqrt {-x^2-2\,x+3}\,1{}\mathrm {i}\right )\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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