3.1163 \(\int \frac{1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{2}{\sqrt{3-x} \sqrt{x-2}}-\frac{4 \sqrt{3-x}}{\sqrt{x-2}} \]

[Out]

2/(Sqrt[3 - x]*Sqrt[-2 + x]) - (4*Sqrt[3 - x])/Sqrt[-2 + x]

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Rubi [A]  time = 0.0257071, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2}{\sqrt{3-x} \sqrt{x-2}}-\frac{4 \sqrt{3-x}}{\sqrt{x-2}} \]

Antiderivative was successfully verified.

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

[Out]

2/(Sqrt[3 - x]*Sqrt[-2 + x]) - (4*Sqrt[3 - x])/Sqrt[-2 + x]

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Rubi in Sympy [A]  time = 3.64084, size = 29, normalized size = 0.78 \[ \frac{4 \sqrt{x - 2}}{\sqrt{- x + 3}} - \frac{2}{\sqrt{- x + 3} \sqrt{x - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(x - 2)/sqrt(-x + 3) - 2/(sqrt(-x + 3)*sqrt(x - 2))

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Mathematica [A]  time = 0.013086, size = 21, normalized size = 0.57 \[ \frac{2 (2 x-5)}{\sqrt{-x^2+5 x-6}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-5 + 2*x))/Sqrt[-6 + 5*x - x^2]

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Maple [A]  time = 0.004, size = 20, normalized size = 0.5 \[ 2\,{\frac{-5+2\,x}{\sqrt{3-x}\sqrt{-2+x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3-x)^(3/2)/(-2+x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.33223, size = 41, normalized size = 1.11 \[ \frac{4 \, x}{\sqrt{-x^{2} + 5 \, x - 6}} - \frac{10}{\sqrt{-x^{2} + 5 \, x - 6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x - 2)^(3/2)*(-x + 3)^(3/2)),x, algorithm="maxima")

[Out]

4*x/sqrt(-x^2 + 5*x - 6) - 10/sqrt(-x^2 + 5*x - 6)

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Fricas [A]  time = 0.207504, size = 39, normalized size = 1.05 \[ -\frac{2 \,{\left (2 \, x - 5\right )} \sqrt{x - 2} \sqrt{-x + 3}}{x^{2} - 5 \, x + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x - 2)^(3/2)*(-x + 3)^(3/2)),x, algorithm="fricas")

[Out]

-2*(2*x - 5)*sqrt(x - 2)*sqrt(-x + 3)/(x^2 - 5*x + 6)

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Sympy [A]  time = 13.3961, size = 100, normalized size = 2.7 \[ \begin{cases} - \frac{4 i \sqrt{x - 3} \left (x - 2\right )}{\left (x - 2\right )^{\frac{3}{2}} - \sqrt{x - 2}} + \frac{2 i \sqrt{x - 3}}{\left (x - 2\right )^{\frac{3}{2}} - \sqrt{x - 2}} & \text{for}\: \left |{x - 2}\right | > 1 \\- \frac{4 \sqrt{- x + 3} \left (x - 2\right )}{\left (x - 2\right )^{\frac{3}{2}} - \sqrt{x - 2}} + \frac{2 \sqrt{- x + 3}}{\left (x - 2\right )^{\frac{3}{2}} - \sqrt{x - 2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-4*I*sqrt(x - 3)*(x - 2)/((x - 2)**(3/2) - sqrt(x - 2)) + 2*I*sqrt(x
- 3)/((x - 2)**(3/2) - sqrt(x - 2)), Abs(x - 2) > 1), (-4*sqrt(-x + 3)*(x - 2)/(
(x - 2)**(3/2) - sqrt(x - 2)) + 2*sqrt(-x + 3)/((x - 2)**(3/2) - sqrt(x - 2)), T
rue))

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GIAC/XCAS [A]  time = 0.220804, size = 72, normalized size = 1.95 \[ -\frac{\sqrt{-x + 3} - 1}{\sqrt{x - 2}} - \frac{2 \, \sqrt{x - 2} \sqrt{-x + 3}}{x - 3} + \frac{\sqrt{x - 2}}{\sqrt{-x + 3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x - 2)^(3/2)*(-x + 3)^(3/2)),x, algorithm="giac")

[Out]

-(sqrt(-x + 3) - 1)/sqrt(x - 2) - 2*sqrt(x - 2)*sqrt(-x + 3)/(x - 3) + sqrt(x -
2)/(sqrt(-x + 3) - 1)