3.1164 \(\int \frac{1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{32 \sqrt{3-x}}{3 \sqrt{x-2}}-\frac{16 \sqrt{3-x}}{3 (x-2)^{3/2}}+\frac{4}{(x-2)^{3/2} \sqrt{3-x}}+\frac{2}{3 (x-2)^{3/2} (3-x)^{3/2}} \]

[Out]

2/(3*(3 - x)^(3/2)*(-2 + x)^(3/2)) + 4/(Sqrt[3 - x]*(-2 + x)^(3/2)) - (16*Sqrt[3
- x])/(3*(-2 + x)^(3/2)) - (32*Sqrt[3 - x])/(3*Sqrt[-2 + x])

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Rubi [A]  time = 0.0535757, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{32 \sqrt{3-x}}{3 \sqrt{x-2}}-\frac{16 \sqrt{3-x}}{3 (x-2)^{3/2}}+\frac{4}{(x-2)^{3/2} \sqrt{3-x}}+\frac{2}{3 (x-2)^{3/2} (3-x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((3 - x)^(5/2)*(-2 + x)^(5/2)),x]

[Out]

2/(3*(3 - x)^(3/2)*(-2 + x)^(3/2)) + 4/(Sqrt[3 - x]*(-2 + x)^(3/2)) - (16*Sqrt[3
- x])/(3*(-2 + x)^(3/2)) - (32*Sqrt[3 - x])/(3*Sqrt[-2 + x])

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Rubi in Sympy [A]  time = 6.68575, size = 65, normalized size = 0.82 \[ \frac{32 \sqrt{x - 2}}{3 \sqrt{- x + 3}} + \frac{16 \sqrt{x - 2}}{3 \left (- x + 3\right )^{\frac{3}{2}}} - \frac{4}{\left (- x + 3\right )^{\frac{3}{2}} \sqrt{x - 2}} - \frac{2}{3 \left (- x + 3\right )^{\frac{3}{2}} \left (x - 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3-x)**(5/2)/(-2+x)**(5/2),x)

[Out]

32*sqrt(x - 2)/(3*sqrt(-x + 3)) + 16*sqrt(x - 2)/(3*(-x + 3)**(3/2)) - 4/((-x +
3)**(3/2)*sqrt(x - 2)) - 2/(3*(-x + 3)**(3/2)*(x - 2)**(3/2))

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Mathematica [A]  time = 0.024145, size = 43, normalized size = 0.54 \[ \frac{2 \left (16 x^3-120 x^2+294 x-235\right )}{3 (x-3) (x-2) \sqrt{-x^2+5 x-6}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((3 - x)^(5/2)*(-2 + x)^(5/2)),x]

[Out]

(2*(-235 + 294*x - 120*x^2 + 16*x^3))/(3*(-3 + x)*(-2 + x)*Sqrt[-6 + 5*x - x^2])

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Maple [A]  time = 0.004, size = 30, normalized size = 0.4 \[ -{\frac{32\,{x}^{3}-240\,{x}^{2}+588\,x-470}{3} \left ( 3-x \right ) ^{-{\frac{3}{2}}} \left ( -2+x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3-x)^(5/2)/(-2+x)^(5/2),x)

[Out]

-2/3*(16*x^3-120*x^2+294*x-235)/(-2+x)^(3/2)/(3-x)^(3/2)

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Maxima [A]  time = 1.48786, size = 80, normalized size = 1.01 \[ \frac{32 \, x}{3 \, \sqrt{-x^{2} + 5 \, x - 6}} - \frac{80}{3 \, \sqrt{-x^{2} + 5 \, x - 6}} + \frac{4 \, x}{3 \,{\left (-x^{2} + 5 \, x - 6\right )}^{\frac{3}{2}}} - \frac{10}{3 \,{\left (-x^{2} + 5 \, x - 6\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x - 2)^(5/2)*(-x + 3)^(5/2)),x, algorithm="maxima")

[Out]

32/3*x/sqrt(-x^2 + 5*x - 6) - 80/3/sqrt(-x^2 + 5*x - 6) + 4/3*x/(-x^2 + 5*x - 6)
^(3/2) - 10/3/(-x^2 + 5*x - 6)^(3/2)

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Fricas [A]  time = 0.210469, size = 66, normalized size = 0.84 \[ -\frac{2 \,{\left (16 \, x^{3} - 120 \, x^{2} + 294 \, x - 235\right )} \sqrt{x - 2} \sqrt{-x + 3}}{3 \,{\left (x^{4} - 10 \, x^{3} + 37 \, x^{2} - 60 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x - 2)^(5/2)*(-x + 3)^(5/2)),x, algorithm="fricas")

[Out]

-2/3*(16*x^3 - 120*x^2 + 294*x - 235)*sqrt(x - 2)*sqrt(-x + 3)/(x^4 - 10*x^3 + 3
7*x^2 - 60*x + 36)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3-x)**(5/2)/(-2+x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218095, size = 131, normalized size = 1.66 \[ -\frac{{\left (\sqrt{-x + 3} - 1\right )}^{3}}{12 \,{\left (x - 2\right )}^{\frac{3}{2}}} - \frac{11 \,{\left (\sqrt{-x + 3} - 1\right )}}{4 \, \sqrt{x - 2}} - \frac{2 \,{\left (8 \, x - 25\right )} \sqrt{x - 2} \sqrt{-x + 3}}{3 \,{\left (x - 3\right )}^{2}} + \frac{{\left (x - 2\right )}^{\frac{3}{2}}{\left (\frac{33 \,{\left (\sqrt{-x + 3} - 1\right )}^{2}}{x - 2} + 1\right )}}{12 \,{\left (\sqrt{-x + 3} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x - 2)^(5/2)*(-x + 3)^(5/2)),x, algorithm="giac")

[Out]

-1/12*(sqrt(-x + 3) - 1)^3/(x - 2)^(3/2) - 11/4*(sqrt(-x + 3) - 1)/sqrt(x - 2) -
2/3*(8*x - 25)*sqrt(x - 2)*sqrt(-x + 3)/(x - 3)^2 + 1/12*(x - 2)^(3/2)*(33*(sqr
t(-x + 3) - 1)^2/(x - 2) + 1)/(sqrt(-x + 3) - 1)^3