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

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Rubi [A]  time = 0.0127822, 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, Rules used = {45, 37} \[ -\frac{32 \sqrt{3-x}}{3 \sqrt{x-2}}-\frac{16 \sqrt{3-x}}{3 (x-2)^{3/2}}+\frac{4}{(x-2)^{3/2} \sqrt{3-x}}+\frac{2}{3 (x-2)^{3/2} (3-x)^{3/2}} \]

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)) - (3
2*Sqrt[3 - x])/(3*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)^{5/2} (-2+x)^{5/2}} \, dx &=\frac{2}{3 (3-x)^{3/2} (-2+x)^{3/2}}+2 \int \frac{1}{(3-x)^{3/2} (-2+x)^{5/2}} \, dx\\ &=\frac{2}{3 (3-x)^{3/2} (-2+x)^{3/2}}+\frac{4}{\sqrt{3-x} (-2+x)^{3/2}}+8 \int \frac{1}{\sqrt{3-x} (-2+x)^{5/2}} \, dx\\ &=\frac{2}{3 (3-x)^{3/2} (-2+x)^{3/2}}+\frac{4}{\sqrt{3-x} (-2+x)^{3/2}}-\frac{16 \sqrt{3-x}}{3 (-2+x)^{3/2}}+\frac{16}{3} \int \frac{1}{\sqrt{3-x} (-2+x)^{3/2}} \, dx\\ &=\frac{2}{3 (3-x)^{3/2} (-2+x)^{3/2}}+\frac{4}{\sqrt{3-x} (-2+x)^{3/2}}-\frac{16 \sqrt{3-x}}{3 (-2+x)^{3/2}}-\frac{32 \sqrt{3-x}}{3 \sqrt{-2+x}}\\ \end{align*}

Mathematica [A]  time = 0.0130121, size = 33, normalized size = 0.42 \[ \frac{-32 x^3+240 x^2-588 x+470}{3 \left (-x^2+5 x-6\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(470 - 588*x + 240*x^2 - 32*x^3)/(3*(-6 + 5*x - x^2)^(3/2))

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Maple [A]  time = 0.003, size = 30, normalized size = 0.4 \begin{align*} -{\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}}}} \end{align*}

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 = 0.990339, size = 80, normalized size = 1.01 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-x)^(5/2)/(-2+x)^(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 = 1.58898, size = 135, normalized size = 1.71 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-x)^(5/2)/(-2+x)^(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 + 37*x^2 - 60*x + 36)

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Sympy [B]  time = 39.3784, size = 282, normalized size = 3.57 \begin{align*} \begin{cases} - \frac{32 \sqrt{-1 + \frac{1}{x - 2}} \left (x - 2\right )^{3}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} + \frac{48 \sqrt{-1 + \frac{1}{x - 2}} \left (x - 2\right )^{2}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac{12 \sqrt{-1 + \frac{1}{x - 2}} \left (x - 2\right )}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac{2 \sqrt{-1 + \frac{1}{x - 2}}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} & \text{for}\: \frac{1}{\left |{x - 2}\right |} > 1 \\- \frac{32 i \sqrt{1 - \frac{1}{x - 2}} \left (x - 2\right )^{3}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} + \frac{48 i \sqrt{1 - \frac{1}{x - 2}} \left (x - 2\right )^{2}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac{12 i \sqrt{1 - \frac{1}{x - 2}} \left (x - 2\right )}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac{2 i \sqrt{1 - \frac{1}{x - 2}}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-32*sqrt(-1 + 1/(x - 2))*(x - 2)**3/(3*x + 3*(x - 2)**3 - 6*(x - 2)**2 - 6) + 48*sqrt(-1 + 1/(x - 2
))*(x - 2)**2/(3*x + 3*(x - 2)**3 - 6*(x - 2)**2 - 6) - 12*sqrt(-1 + 1/(x - 2))*(x - 2)/(3*x + 3*(x - 2)**3 -
6*(x - 2)**2 - 6) - 2*sqrt(-1 + 1/(x - 2))/(3*x + 3*(x - 2)**3 - 6*(x - 2)**2 - 6), 1/Abs(x - 2) > 1), (-32*I*
sqrt(1 - 1/(x - 2))*(x - 2)**3/(3*x + 3*(x - 2)**3 - 6*(x - 2)**2 - 6) + 48*I*sqrt(1 - 1/(x - 2))*(x - 2)**2/(
3*x + 3*(x - 2)**3 - 6*(x - 2)**2 - 6) - 12*I*sqrt(1 - 1/(x - 2))*(x - 2)/(3*x + 3*(x - 2)**3 - 6*(x - 2)**2 -
6) - 2*I*sqrt(1 - 1/(x - 2))/(3*x + 3*(x - 2)**3 - 6*(x - 2)**2 - 6), True))

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Giac [A]  time = 1.08445, size = 131, normalized size = 1.66 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-x)^(5/2)/(-2+x)^(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)*sq
rt(-x + 3)/(x - 3)^2 + 1/12*(x - 2)^(3/2)*(33*(sqrt(-x + 3) - 1)^2/(x - 2) + 1)/(sqrt(-x + 3) - 1)^3