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 = 0.0097638, antiderivative size = 51, normalized size of antiderivative = 1., 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 619
Rule 216
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*}
Mathematica [A] time = 0.0226581, 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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Maple [A] time = 0.004, size = 61, normalized size = 1.2 \begin{align*} -{\frac{1}{2} \left ( 3-x \right ) ^{{\frac{3}{2}}}\sqrt{-2+x}}+{\frac{1}{4}\sqrt{3-x}\sqrt{-2+x}}+{\frac{\arcsin \left ( 2\,x-5 \right ) }{8}\sqrt{ \left ( -2+x \right ) \left ( 3-x \right ) }{\frac{1}{\sqrt{3-x}}}{\frac{1}{\sqrt{-2+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.423, size = 51, normalized size = 1. \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49894, size = 147, normalized size = 2.88 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.0069, size = 124, normalized size = 2.43 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09726, size = 38, normalized size = 0.75 \begin{align*} \frac{1}{4} \,{\left (2 \, x - 5\right )} \sqrt{x - 2} \sqrt{-x + 3} + \frac{1}{4} \, \arcsin \left (\sqrt{x - 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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