\(\int \frac {1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx\) [1163]

Optimal result

Integrand size = 17, antiderivative size = 37 \[ \int \frac {1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx=\frac {2}{\sqrt {3-x} \sqrt {-2+x}}-\frac {4 \sqrt {3-x}}{\sqrt {-2+x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx=\frac {2}{\sqrt {3-x} \sqrt {x-2}}-\frac {4 \sqrt {3-x}}{\sqrt {x-2}} \]

[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 47

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*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx=\frac {2 (-5+2 x)}{\sqrt {-6+5 x-x^2}} \]

[In]

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

[Out]

(2*(-5 + 2*x))/Sqrt[-6 + 5*x - x^2]

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.54

method	result	size
gosper	\(\frac {-10+4 x}{\sqrt {3-x}\, \sqrt {-2+x}}\)	\(20\)
default	\(\frac {2}{\sqrt {3-x}\, \sqrt {-2+x}}-\frac {4 \sqrt {3-x}}{\sqrt {-2+x}}\)	\(30\)
risch	\(\frac {2 \sqrt {\left (-2+x \right ) \left (3-x \right )}\, \left (-5+2 x \right )}{\sqrt {3-x}\, \sqrt {-2+x}\, \sqrt {-\left (-3+x \right ) \left (-2+x \right )}}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx=-\frac {2 \, {\left (2 \, x - 5\right )} \sqrt {x - 2} \sqrt {-x + 3}}{x^{2} - 5 \, x + 6} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.77 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx=\begin {cases} - \frac {4 i \sqrt {x - 3} \left (x - 2\right )}{\left (x - 2\right )^{\frac {3}{2}} - \sqrt {x - 2}} + \frac {2 i \sqrt {x - 3}}{\left (x - 2\right )^{\frac {3}{2}} - \sqrt {x - 2}} & \text {for}\: \left |{x - 2}\right | > 1 \\- \frac {4 \sqrt {3 - x} \left (x - 2\right )}{\left (x - 2\right )^{\frac {3}{2}} - \sqrt {x - 2}} + \frac {2 \sqrt {3 - x}}{\left (x - 2\right )^{\frac {3}{2}} - \sqrt {x - 2}} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx=\frac {4 \, x}{\sqrt {-x^{2} + 5 \, x - 6}} - \frac {10}{\sqrt {-x^{2} + 5 \, x - 6}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx=-\frac {\sqrt {-x + 3} - 1}{\sqrt {x - 2}} - \frac {2 \, \sqrt {x - 2} \sqrt {-x + 3}}{x - 3} + \frac {\sqrt {x - 2}}{\sqrt {-x + 3} - 1} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(3-x)^{3/2} (-2+x)^{3/2}} \, dx=-\frac {4\,x\,\sqrt {3-x}-10\,\sqrt {3-x}}{\sqrt {x-2}\,\left (x-3\right )} \]

[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))