\(\int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx\) [1164]

Optimal result

Integrand size = 17, antiderivative size = 79 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=\frac {2}{3 (3-x)^{3/2} (-2+x)^{3/2}}+\frac {4}{\sqrt {3-x} (-2+x)^{3/2}}-\frac {16 \sqrt {3-x}}{3 (-2+x)^{3/2}}-\frac {32 \sqrt {3-x}}{3 \sqrt {-2+x}} \]

[Out]

2/3/(3-x)^(3/2)/(-2+x)^(3/2)+4/(-2+x)^(3/2)/(3-x)^(1/2)-16/3*(3-x)^(1/2)/(-2+x)^(3/2)-32/3*(3-x)^(1/2)/(-2+x)^
(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=-\frac {32 \sqrt {3-x}}{3 \sqrt {x-2}}-\frac {16 \sqrt {3-x}}{3 (x-2)^{3/2}}+\frac {4}{(x-2)^{3/2} \sqrt {3-x}}+\frac {2}{3 (x-2)^{3/2} (3-x)^{3/2}} \]

[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 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}{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] (verified)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=-\frac {2 \left (-235+294 x-120 x^2+16 x^3\right )}{3 (-((-3+x) (-2+x)))^{3/2}} \]

[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)))^(3/2))

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38

method	result	size
gosper	\(-\frac {2 \left (16 x^{3}-120 x^{2}+294 x -235\right )}{3 \left (-2+x \right )^{\frac {3}{2}} \left (3-x \right )^{\frac {3}{2}}}\)	\(30\)
default	\(\frac {2}{3 \left (3-x \right )^{\frac {3}{2}} \left (-2+x \right )^{\frac {3}{2}}}+\frac {4}{\left (-2+x \right )^{\frac {3}{2}} \sqrt {3-x}}-\frac {16 \sqrt {3-x}}{3 \left (-2+x \right )^{\frac {3}{2}}}-\frac {32 \sqrt {3-x}}{3 \sqrt {-2+x}}\)	\(58\)

[In]

int(1/(3-x)^(5/2)/(-2+x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=-\frac {2 \, {\left (16 \, x^{3} - 120 \, x^{2} + 294 \, x - 235\right )} \sqrt {x - 2} \sqrt {-x + 3}}{3 \, {\left (x^{4} - 10 \, x^{3} + 37 \, x^{2} - 60 \, x + 36\right )}} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.82 (sec) , antiderivative size = 282, normalized size of antiderivative = 3.57 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=\begin {cases} - \frac {32 \sqrt {-1 + \frac {1}{x - 2}} \left (x - 2\right )^{3}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} + \frac {48 \sqrt {-1 + \frac {1}{x - 2}} \left (x - 2\right )^{2}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac {12 \sqrt {-1 + \frac {1}{x - 2}} \left (x - 2\right )}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac {2 \sqrt {-1 + \frac {1}{x - 2}}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} & \text {for}\: \frac {1}{\left |{x - 2}\right |} > 1 \\- \frac {32 i \sqrt {1 - \frac {1}{x - 2}} \left (x - 2\right )^{3}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} + \frac {48 i \sqrt {1 - \frac {1}{x - 2}} \left (x - 2\right )^{2}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac {12 i \sqrt {1 - \frac {1}{x - 2}} \left (x - 2\right )}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac {2 i \sqrt {1 - \frac {1}{x - 2}}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=\frac {32 \, x}{3 \, \sqrt {-x^{2} + 5 \, x - 6}} - \frac {80}{3 \, \sqrt {-x^{2} + 5 \, x - 6}} + \frac {4 \, x}{3 \, {\left (-x^{2} + 5 \, x - 6\right )}^{\frac {3}{2}}} - \frac {10}{3 \, {\left (-x^{2} + 5 \, x - 6\right )}^{\frac {3}{2}}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=-\frac {{\left (\sqrt {-x + 3} - 1\right )}^{3}}{12 \, {\left (x - 2\right )}^{\frac {3}{2}}} - \frac {11 \, {\left (\sqrt {-x + 3} - 1\right )}}{4 \, \sqrt {x - 2}} - \frac {2 \, {\left (8 \, x - 25\right )} \sqrt {x - 2} \sqrt {-x + 3}}{3 \, {\left (x - 3\right )}^{2}} + \frac {{\left (x - 2\right )}^{\frac {3}{2}} {\left (\frac {33 \, {\left (\sqrt {-x + 3} - 1\right )}^{2}}{x - 2} + 1\right )}}{12 \, {\left (\sqrt {-x + 3} - 1\right )}^{3}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=-\frac {32\,{\left (x-2\right )}^3\,\sqrt {3-x}-48\,{\left (x-2\right )}^2\,\sqrt {3-x}+2\,\sqrt {3-x}+12\,\left (x-2\right )\,\sqrt {3-x}}{\left (3\,x-6\right )\,\sqrt {x-2}\,{\left (x-3\right )}^2} \]

[In]

int(1/((x - 2)^(5/2)*(3 - x)^(5/2)),x)

[Out]

-(32*(x - 2)^3*(3 - x)^(1/2) - 48*(x - 2)^2*(3 - x)^(1/2) + 2*(3 - x)^(1/2) + 12*(x - 2)*(3 - x)^(1/2))/((3*x
- 6)*(x - 2)^(1/2)*(x - 3)^2)