Integrand size = 17, antiderivative size = 51 \[ \int \sqrt {3-x} \sqrt {-2+x} \, dx=\frac {1}{4} \sqrt {3-x} \sqrt {-2+x}-\frac {1}{2} (3-x)^{3/2} \sqrt {-2+x}-\frac {1}{8} \arcsin (5-2 x) \]
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Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {52, 55, 633, 222} \[ \int \sqrt {3-x} \sqrt {-2+x} \, dx=-\frac {1}{8} \arcsin (5-2 x)-\frac {1}{2} \sqrt {x-2} (3-x)^{3/2}+\frac {1}{4} \sqrt {x-2} \sqrt {3-x} \]
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Rule 52
Rule 55
Rule 222
Rule 633
Rubi steps \begin{align*} \text {integral}& = -\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} \text {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*}
Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.04 \[ \int \sqrt {3-x} \sqrt {-2+x} \, dx=\frac {1}{4} \sqrt {-6+5 x-x^2} \left (-5+2 x-\frac {\text {arctanh}\left (\frac {1}{\sqrt {\frac {-3+x}{-2+x}}}\right )}{\sqrt {-3+x} \sqrt {-2+x}}\right ) \]
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Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {\sqrt {3-x}\, \left (-2+x \right )^{\frac {3}{2}}}{2}-\frac {\sqrt {3-x}\, \sqrt {-2+x}}{4}+\frac {\sqrt {\left (-2+x \right ) \left (3-x \right )}\, \arcsin \left (-5+2 x \right )}{8 \sqrt {-2+x}\, \sqrt {3-x}}\) | \(61\) |
risch | \(-\frac {\left (-5+2 x \right ) \left (-3+x \right ) \sqrt {-2+x}\, \sqrt {\left (-2+x \right ) \left (3-x \right )}}{4 \sqrt {-\left (-3+x \right ) \left (-2+x \right )}\, \sqrt {3-x}}+\frac {\sqrt {\left (-2+x \right ) \left (3-x \right )}\, \arcsin \left (-5+2 x \right )}{8 \sqrt {-2+x}\, \sqrt {3-x}}\) | \(76\) |
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Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.02 \[ \int \sqrt {3-x} \sqrt {-2+x} \, dx=\frac {1}{4} \, {\left (2 \, x - 5\right )} \sqrt {x - 2} \sqrt {-x + 3} - \frac {1}{8} \, \arctan \left (\frac {{\left (2 \, x - 5\right )} \sqrt {x - 2} \sqrt {-x + 3}}{2 \, {\left (x^{2} - 5 \, x + 6\right )}}\right ) \]
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Result contains complex when optimal does not.
Time = 2.57 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.43 \[ \int \sqrt {3-x} \sqrt {-2+x} \, dx=\begin {cases} - \frac {i \operatorname {acosh}{\left (\sqrt {x - 2} \right )}}{4} + \frac {i \left (x - 2\right )^{\frac {5}{2}}}{2 \sqrt {x - 3}} - \frac {3 i \left (x - 2\right )^{\frac {3}{2}}}{4 \sqrt {x - 3}} + \frac {i \sqrt {x - 2}}{4 \sqrt {x - 3}} & \text {for}\: \left |{x - 2}\right | > 1 \\\frac {\operatorname {asin}{\left (\sqrt {x - 2} \right )}}{4} - \frac {\left (x - 2\right )^{\frac {5}{2}}}{2 \sqrt {3 - x}} + \frac {3 \left (x - 2\right )^{\frac {3}{2}}}{4 \sqrt {3 - x}} - \frac {\sqrt {x - 2}}{4 \sqrt {3 - x}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.75 \[ \int \sqrt {3-x} \sqrt {-2+x} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 5 \, x - 6} x - \frac {5}{4} \, \sqrt {-x^{2} + 5 \, x - 6} + \frac {1}{8} \, \arcsin \left (2 \, x - 5\right ) \]
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Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int \sqrt {3-x} \sqrt {-2+x} \, dx=\frac {1}{4} \, {\left (2 \, x + 3\right )} \sqrt {x - 2} \sqrt {-x + 3} - 2 \, \sqrt {x - 2} \sqrt {-x + 3} + \frac {1}{4} \, \arcsin \left (\sqrt {x - 2}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int \sqrt {3-x} \sqrt {-2+x} \, dx=\left (\frac {x}{2}-\frac {5}{4}\right )\,\sqrt {x-2}\,\sqrt {3-x}-\frac {\ln \left (x-\frac {5}{2}-\sqrt {x-2}\,\sqrt {3-x}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8} \]
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