3.1161 \(\int \sqrt{3-x} \sqrt{-2+x} \, dx\)

Optimal. Leaf size=51 \[ -\frac{1}{2} \sqrt{x-2} (3-x)^{3/2}+\frac{1}{4} \sqrt{x-2} \sqrt{3-x}-\frac{1}{8} \sin ^{-1}(5-2 x) \]

[Out]

(Sqrt[3 - x]*Sqrt[-2 + x])/4 - ((3 - x)^(3/2)*Sqrt[-2 + x])/2 - ArcSin[5 - 2*x]/
8

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Rubi [A]  time = 0.040138, 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 \[ -\frac{1}{2} \sqrt{x-2} (3-x)^{3/2}+\frac{1}{4} \sqrt{x-2} \sqrt{3-x}-\frac{1}{8} \sin ^{-1}(5-2 x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3 - x]*Sqrt[-2 + x],x]

[Out]

(Sqrt[3 - x]*Sqrt[-2 + x])/4 - ((3 - x)^(3/2)*Sqrt[-2 + x])/2 - ArcSin[5 - 2*x]/
8

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Rubi in Sympy [A]  time = 6.16512, size = 51, normalized size = 1. \[ - \frac{\left (- x + 3\right )^{\frac{3}{2}} \sqrt{x - 2}}{2} + \frac{\sqrt{- x + 3} \sqrt{x - 2}}{4} - \frac{\operatorname{atan}{\left (\frac{- 2 x + 5}{2 \sqrt{- x^{2} + 5 x - 6}} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3-x)**(1/2)*(-2+x)**(1/2),x)

[Out]

-(-x + 3)**(3/2)*sqrt(x - 2)/2 + sqrt(-x + 3)*sqrt(x - 2)/4 - atan((-2*x + 5)/(2
*sqrt(-x**2 + 5*x - 6)))/8

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Mathematica [A]  time = 0.036225, size = 63, normalized size = 1.24 \[ \frac{\sqrt{-x^2+5 x-6} \left (\sqrt{x-3} \sqrt{x-2} (2 x-5)-\sinh ^{-1}\left (\sqrt{x-3}\right )\right )}{4 \sqrt{x-3} \sqrt{x-2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3 - x]*Sqrt[-2 + x],x]

[Out]

(Sqrt[-6 + 5*x - x^2]*(Sqrt[-3 + x]*Sqrt[-2 + x]*(-5 + 2*x) - ArcSinh[Sqrt[-3 +
x]]))/(4*Sqrt[-3 + x]*Sqrt[-2 + x])

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Maple [A]  time = 0.008, size = 61, normalized size = 1.2 \[ -{\frac{1}{2} \left ( 3-x \right ) ^{{\frac{3}{2}}}\sqrt{-2+x}}+{\frac{1}{4}\sqrt{3-x}\sqrt{-2+x}}+{\frac{\arcsin \left ( -5+2\,x \right ) }{8}\sqrt{ \left ( -2+x \right ) \left ( 3-x \right ) }{\frac{1}{\sqrt{3-x}}}{\frac{1}{\sqrt{-2+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3-x)^(1/2)*(-2+x)^(1/2),x)

[Out]

-1/2*(3-x)^(3/2)*(-2+x)^(1/2)+1/4*(3-x)^(1/2)*(-2+x)^(1/2)+1/8*((-2+x)*(3-x))^(1
/2)/(-2+x)^(1/2)/(3-x)^(1/2)*arcsin(-5+2*x)

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Maxima [A]  time = 1.48902, size = 51, normalized size = 1. \[ \frac{1}{2} \, \sqrt{-x^{2} + 5 \, x - 6} x - \frac{5}{4} \, \sqrt{-x^{2} + 5 \, x - 6} + \frac{1}{8} \, \arcsin \left (2 \, x - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x - 2)*sqrt(-x + 3),x, algorithm="maxima")

[Out]

1/2*sqrt(-x^2 + 5*x - 6)*x - 5/4*sqrt(-x^2 + 5*x - 6) + 1/8*arcsin(2*x - 5)

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Fricas [A]  time = 0.216452, size = 57, normalized size = 1.12 \[ \frac{1}{4} \,{\left (2 \, x - 5\right )} \sqrt{x - 2} \sqrt{-x + 3} + \frac{1}{8} \, \arctan \left (\frac{2 \, x - 5}{2 \, \sqrt{x - 2} \sqrt{-x + 3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x - 2)*sqrt(-x + 3),x, algorithm="fricas")

[Out]

1/4*(2*x - 5)*sqrt(x - 2)*sqrt(-x + 3) + 1/8*arctan(1/2*(2*x - 5)/(sqrt(x - 2)*s
qrt(-x + 3)))

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Sympy [A]  time = 9.57374, size = 124, normalized size = 2.43 \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (\sqrt{x - 2} \right )}}{4} + \frac{i \left (x - 2\right )^{\frac{5}{2}}}{2 \sqrt{x - 3}} - \frac{3 i \left (x - 2\right )^{\frac{3}{2}}}{4 \sqrt{x - 3}} + \frac{i \sqrt{x - 2}}{4 \sqrt{x - 3}} & \text{for}\: \left |{x - 2}\right | > 1 \\\frac{\operatorname{asin}{\left (\sqrt{x - 2} \right )}}{4} - \frac{\left (x - 2\right )^{\frac{5}{2}}}{2 \sqrt{- x + 3}} + \frac{3 \left (x - 2\right )^{\frac{3}{2}}}{4 \sqrt{- x + 3}} - \frac{\sqrt{x - 2}}{4 \sqrt{- x + 3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3-x)**(1/2)*(-2+x)**(1/2),x)

[Out]

Piecewise((-I*acosh(sqrt(x - 2))/4 + I*(x - 2)**(5/2)/(2*sqrt(x - 3)) - 3*I*(x -
2)**(3/2)/(4*sqrt(x - 3)) + I*sqrt(x - 2)/(4*sqrt(x - 3)), Abs(x - 2) > 1), (as
in(sqrt(x - 2))/4 - (x - 2)**(5/2)/(2*sqrt(-x + 3)) + 3*(x - 2)**(3/2)/(4*sqrt(-
x + 3)) - sqrt(x - 2)/(4*sqrt(-x + 3)), True))

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GIAC/XCAS [A]  time = 0.218295, size = 38, normalized size = 0.75 \[ \frac{1}{4} \,{\left (2 \, x - 5\right )} \sqrt{x - 2} \sqrt{-x + 3} + \frac{1}{4} \, \arcsin \left (\sqrt{x - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x - 2)*sqrt(-x + 3),x, algorithm="giac")

[Out]

1/4*(2*x - 5)*sqrt(x - 2)*sqrt(-x + 3) + 1/4*arcsin(sqrt(x - 2))