Optimal. Leaf size=51 \[ -\frac {1}{2} \sqrt {x-2} (3-x)^{3/2}+\frac {1}{4} \sqrt {x-2} \sqrt {3-x}-\frac {1}{8} \sin ^{-1}(5-2 x) \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {50, 53, 619, 216} \[ -\frac {1}{2} \sqrt {x-2} (3-x)^{3/2}+\frac {1}{4} \sqrt {x-2} \sqrt {3-x}-\frac {1}{8} \sin ^{-1}(5-2 x) \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 216
Rule 619
Rubi steps
\begin {align*} \int \sqrt {3-x} \sqrt {-2+x} \, dx &=-\frac {1}{2} (3-x)^{3/2} \sqrt {-2+x}+\frac {1}{4} \int \frac {\sqrt {3-x}}{\sqrt {-2+x}} \, dx\\ &=\frac {1}{4} \sqrt {3-x} \sqrt {-2+x}-\frac {1}{2} (3-x)^{3/2} \sqrt {-2+x}+\frac {1}{8} \int \frac {1}{\sqrt {3-x} \sqrt {-2+x}} \, dx\\ &=\frac {1}{4} \sqrt {3-x} \sqrt {-2+x}-\frac {1}{2} (3-x)^{3/2} \sqrt {-2+x}+\frac {1}{8} \int \frac {1}{\sqrt {-6+5 x-x^2}} \, dx\\ &=\frac {1}{4} \sqrt {3-x} \sqrt {-2+x}-\frac {1}{2} (3-x)^{3/2} \sqrt {-2+x}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,5-2 x\right )\\ &=\frac {1}{4} \sqrt {3-x} \sqrt {-2+x}-\frac {1}{2} (3-x)^{3/2} \sqrt {-2+x}-\frac {1}{8} \sin ^{-1}(5-2 x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 69, normalized size = 1.35 \[ \frac {\sqrt {-x^2+5 x-6} \left (\sqrt {x-2} \left (2 x^2-11 x+15\right )+\sqrt {3-x} \sin ^{-1}\left (\sqrt {3-x}\right )\right )}{4 (x-3) \sqrt {x-2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 52, normalized size = 1.02 \[ \frac {1}{4} \, {\left (2 \, x - 5\right )} \sqrt {x - 2} \sqrt {-x + 3} - \frac {1}{8} \, \arctan \left (\frac {{\left (2 \, x - 5\right )} \sqrt {x - 2} \sqrt {-x + 3}}{2 \, {\left (x^{2} - 5 \, x + 6\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 42, normalized size = 0.82 \[ \frac {1}{4} \, {\left (2 \, x + 3\right )} \sqrt {x - 2} \sqrt {-x + 3} - 2 \, \sqrt {x - 2} \sqrt {-x + 3} + \frac {1}{4} \, \arcsin \left (\sqrt {x - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 1.20 \[ \frac {\sqrt {\left (x -2\right ) \left (-x +3\right )}\, \arcsin \left (2 x -5\right )}{8 \sqrt {x -2}\, \sqrt {-x +3}}-\frac {\left (-x +3\right )^{\frac {3}{2}} \sqrt {x -2}}{2}+\frac {\sqrt {-x +3}\, \sqrt {x -2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 38, normalized size = 0.75 \[ \frac {1}{2} \, \sqrt {-x^{2} + 5 \, x - 6} x - \frac {5}{4} \, \sqrt {-x^{2} + 5 \, x - 6} + \frac {1}{8} \, \arcsin \left (2 \, x - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 41, normalized size = 0.80 \[ \left (\frac {x}{2}-\frac {5}{4}\right )\,\sqrt {x-2}\,\sqrt {3-x}-\frac {\ln \left (x-\frac {5}{2}-\sqrt {x-2}\,\sqrt {3-x}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.01, size = 124, normalized size = 2.43 \[ \begin {cases} - \frac {i \operatorname {acosh}{\left (\sqrt {x - 2} \right )}}{4} + \frac {i \left (x - 2\right )^{\frac {5}{2}}}{2 \sqrt {x - 3}} - \frac {3 i \left (x - 2\right )^{\frac {3}{2}}}{4 \sqrt {x - 3}} + \frac {i \sqrt {x - 2}}{4 \sqrt {x - 3}} & \text {for}\: \left |{x - 2}\right | > 1 \\\frac {\operatorname {asin}{\left (\sqrt {x - 2} \right )}}{4} - \frac {\left (x - 2\right )^{\frac {5}{2}}}{2 \sqrt {3 - x}} + \frac {3 \left (x - 2\right )^{\frac {3}{2}}}{4 \sqrt {3 - x}} - \frac {\sqrt {x - 2}}{4 \sqrt {3 - x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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