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 = 0.0044166, 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, Rules used = {45, 37} \[ \frac{2}{\sqrt{3-x} \sqrt{x-2}}-\frac{4 \sqrt{3-x}}{\sqrt{x-2}} \]

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]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx &=\frac{2}{\sqrt{3-x} \sqrt{-2+x}}+2 \int \frac{1}{\sqrt{3-x} (-2+x)^{3/2}} \, dx\\ &=\frac{2}{\sqrt{3-x} \sqrt{-2+x}}-\frac{4 \sqrt{3-x}}{\sqrt{-2+x}}\\ \end{align*}

Mathematica [A]  time = 0.0069226, 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 = 0.003, size = 20, normalized size = 0.5 \begin{align*} 2\,{\frac{2\,x-5}{\sqrt{3-x}\sqrt{-2+x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.978491, size = 41, normalized size = 1.11 \begin{align*} \frac{4 \, x}{\sqrt{-x^{2} + 5 \, x - 6}} - \frac{10}{\sqrt{-x^{2} + 5 \, x - 6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-x)^(3/2)/(-2+x)^(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 = 1.52193, size = 74, normalized size = 2. \begin{align*} -\frac{2 \,{\left (2 \, x - 5\right )} \sqrt{x - 2} \sqrt{-x + 3}}{x^{2} - 5 \, x + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-x)^(3/2)/(-2+x)^(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 = 3.25889, size = 100, normalized size = 2.7 \begin{align*} \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{3 - x} \left (x - 2\right )}{\left (x - 2\right )^{\frac{3}{2}} - \sqrt{x - 2}} + \frac{2 \sqrt{3 - x}}{\left (x - 2\right )^{\frac{3}{2}} - \sqrt{x - 2}} & \text{otherwise} \end{cases} \end{align*}

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(3 - x)*(x - 2)/((x - 2)**(3/2) - sqrt(x - 2)) + 2*sqrt(3 - x)/((x - 2)**(3/2)
- sqrt(x - 2)), True))

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Giac [A]  time = 1.0689, size = 72, normalized size = 1.95 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-x)^(3/2)/(-2+x)^(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)