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

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

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Rubi [A]  time = 0.00, antiderivative size = 37, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2}{\sqrt {3-x} \sqrt {x-2}}-\frac {4 \sqrt {3-x}}{\sqrt {x-2}} \end {gather*}

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 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]

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])

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*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.57 \begin {gather*} \frac {2 (2 x-5)}{\sqrt {-x^2+5 x-6}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.05, size = 31, normalized size = 0.84 \begin {gather*} -\frac {2 \left (\frac {3-x}{x-2}-1\right ) \sqrt {x-2}}{\sqrt {3-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-1 + (3 - x)/(-2 + x))*Sqrt[-2 + x])/Sqrt[3 - x]

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fricas [A]  time = 1.27, size = 29, normalized size = 0.78 \begin {gather*} -\frac {2 \, {\left (2 \, x - 5\right )} \sqrt {x - 2} \sqrt {-x + 3}}{x^{2} - 5 \, x + 6} \end {gather*}

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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giac [A]  time = 0.85, size = 53, normalized size = 1.43 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.00, size = 20, normalized size = 0.54 \begin {gather*} \frac {-10+4 x}{\sqrt {-x +3}\, \sqrt {x -2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 30, normalized size = 0.81 \begin {gather*} \frac {4 \, x}{\sqrt {-x^{2} + 5 \, x - 6}} - \frac {10}{\sqrt {-x^{2} + 5 \, x - 6}} \end {gather*}

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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mupad [B]  time = 0.25, size = 32, normalized size = 0.86 \begin {gather*} -\frac {4\,x\,\sqrt {3-x}-10\,\sqrt {3-x}}{\sqrt {x-2}\,\left (x-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*x*(3 - x)^(1/2) - 10*(3 - x)^(1/2))/((x - 2)^(1/2)*(x - 3))

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sympy [A]  time = 2.30, size = 100, normalized size = 2.70 \begin {gather*} \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 {gather*}

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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