Integrand size = 17, antiderivative size = 79 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=\frac {2}{3 (3-x)^{3/2} (-2+x)^{3/2}}+\frac {4}{\sqrt {3-x} (-2+x)^{3/2}}-\frac {16 \sqrt {3-x}}{3 (-2+x)^{3/2}}-\frac {32 \sqrt {3-x}}{3 \sqrt {-2+x}} \] Output:
2/3/(3-x)^(3/2)/(-2+x)^(3/2)+4/(3-x)^(1/2)/(-2+x)^(3/2)-16/3*(3-x)^(1/2)/( -2+x)^(3/2)-32/3*(3-x)^(1/2)/(-2+x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=-\frac {2 \left (-235+294 x-120 x^2+16 x^3\right )}{3 (-((-3+x) (-2+x)))^{3/2}} \] Input:
Integrate[1/((3 - x)^(5/2)*(-2 + x)^(5/2)),x]
Output:
(-2*(-235 + 294*x - 120*x^2 + 16*x^3))/(3*(-((-3 + x)*(-2 + x)))^(3/2))
Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {55, 55, 55, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(3-x)^{5/2} (x-2)^{5/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 2 \int \frac {1}{(3-x)^{3/2} (x-2)^{5/2}}dx+\frac {2}{3 (3-x)^{3/2} (x-2)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 2 \left (4 \int \frac {1}{\sqrt {3-x} (x-2)^{5/2}}dx+\frac {2}{\sqrt {3-x} (x-2)^{3/2}}\right )+\frac {2}{3 (3-x)^{3/2} (x-2)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 2 \left (4 \left (\frac {2}{3} \int \frac {1}{\sqrt {3-x} (x-2)^{3/2}}dx-\frac {2 \sqrt {3-x}}{3 (x-2)^{3/2}}\right )+\frac {2}{\sqrt {3-x} (x-2)^{3/2}}\right )+\frac {2}{3 (3-x)^{3/2} (x-2)^{3/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle 2 \left (4 \left (-\frac {4 \sqrt {3-x}}{3 \sqrt {x-2}}-\frac {2 \sqrt {3-x}}{3 (x-2)^{3/2}}\right )+\frac {2}{\sqrt {3-x} (x-2)^{3/2}}\right )+\frac {2}{3 (3-x)^{3/2} (x-2)^{3/2}}\) |
Input:
Int[1/((3 - x)^(5/2)*(-2 + x)^(5/2)),x]
Output:
2*(4*((-2*Sqrt[3 - x])/(3*(-2 + x)^(3/2)) - (4*Sqrt[3 - x])/(3*Sqrt[-2 + x ])) + 2/(Sqrt[3 - x]*(-2 + x)^(3/2))) + 2/(3*(3 - x)^(3/2)*(-2 + x)^(3/2))
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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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] - Simp[d*(S implify[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] && 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] || !SumSimp lerQ[n, 1])
Time = 0.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38
method | result | size |
gosper | \(-\frac {2 \left (16 x^{3}-120 x^{2}+294 x -235\right )}{3 \left (3-x \right )^{\frac {3}{2}} \left (x -2\right )^{\frac {3}{2}}}\) | \(30\) |
orering | \(\frac {2 \left (-3+x \right ) \left (16 x^{3}-120 x^{2}+294 x -235\right )}{3 \left (x -2\right )^{\frac {3}{2}} \left (3-x \right )^{\frac {5}{2}}}\) | \(33\) |
default | \(\frac {2}{3 \left (3-x \right )^{\frac {3}{2}} \left (x -2\right )^{\frac {3}{2}}}+\frac {4}{\sqrt {3-x}\, \left (x -2\right )^{\frac {3}{2}}}-\frac {16 \sqrt {3-x}}{3 \left (x -2\right )^{\frac {3}{2}}}-\frac {32 \sqrt {3-x}}{3 \sqrt {x -2}}\) | \(58\) |
Input:
int(1/(3-x)^(5/2)/(x-2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3/(3-x)^(3/2)/(x-2)^(3/2)*(16*x^3-120*x^2+294*x-235)
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=-\frac {2 \, {\left (16 \, x^{3} - 120 \, x^{2} + 294 \, x - 235\right )} \sqrt {x - 2} \sqrt {-x + 3}}{3 \, {\left (x^{4} - 10 \, x^{3} + 37 \, x^{2} - 60 \, x + 36\right )}} \] Input:
integrate(1/(3-x)^(5/2)/(-2+x)^(5/2),x, algorithm="fricas")
Output:
-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)
Result contains complex when optimal does not.
Time = 4.23 (sec) , antiderivative size = 282, normalized size of antiderivative = 3.57 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=\begin {cases} - \frac {32 \sqrt {-1 + \frac {1}{x - 2}} \left (x - 2\right )^{3}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} + \frac {48 \sqrt {-1 + \frac {1}{x - 2}} \left (x - 2\right )^{2}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac {12 \sqrt {-1 + \frac {1}{x - 2}} \left (x - 2\right )}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac {2 \sqrt {-1 + \frac {1}{x - 2}}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} & \text {for}\: \frac {1}{\left |{x - 2}\right |} > 1 \\- \frac {32 i \sqrt {1 - \frac {1}{x - 2}} \left (x - 2\right )^{3}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} + \frac {48 i \sqrt {1 - \frac {1}{x - 2}} \left (x - 2\right )^{2}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac {12 i \sqrt {1 - \frac {1}{x - 2}} \left (x - 2\right )}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} - \frac {2 i \sqrt {1 - \frac {1}{x - 2}}}{3 x + 3 \left (x - 2\right )^{3} - 6 \left (x - 2\right )^{2} - 6} & \text {otherwise} \end {cases} \] Input:
integrate(1/(3-x)**(5/2)/(-2+x)**(5/2),x)
Output:
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))
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=\frac {32 \, x}{3 \, \sqrt {-x^{2} + 5 \, x - 6}} - \frac {80}{3 \, \sqrt {-x^{2} + 5 \, x - 6}} + \frac {4 \, x}{3 \, {\left (-x^{2} + 5 \, x - 6\right )}^{\frac {3}{2}}} - \frac {10}{3 \, {\left (-x^{2} + 5 \, x - 6\right )}^{\frac {3}{2}}} \] Input:
integrate(1/(3-x)^(5/2)/(-2+x)^(5/2),x, algorithm="maxima")
Output:
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)
Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=-\frac {{\left (\sqrt {-x + 3} - 1\right )}^{3}}{12 \, {\left (x - 2\right )}^{\frac {3}{2}}} - \frac {11 \, {\left (\sqrt {-x + 3} - 1\right )}}{4 \, \sqrt {x - 2}} - \frac {2 \, {\left (8 \, x - 25\right )} \sqrt {x - 2} \sqrt {-x + 3}}{3 \, {\left (x - 3\right )}^{2}} + \frac {{\left (x - 2\right )}^{\frac {3}{2}} {\left (\frac {33 \, {\left (\sqrt {-x + 3} - 1\right )}^{2}}{x - 2} + 1\right )}}{12 \, {\left (\sqrt {-x + 3} - 1\right )}^{3}} \] Input:
integrate(1/(3-x)^(5/2)/(-2+x)^(5/2),x, algorithm="giac")
Output:
-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*(sqrt(-x + 3) - 1)^2/(x - 2) + 1)/(sqrt(-x + 3) - 1)^3
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=-\frac {32\,{\left (x-2\right )}^3\,\sqrt {3-x}-48\,{\left (x-2\right )}^2\,\sqrt {3-x}+2\,\sqrt {3-x}+12\,\left (x-2\right )\,\sqrt {3-x}}{\left (3\,x-6\right )\,\sqrt {x-2}\,{\left (x-3\right )}^2} \] Input:
int(1/((x - 2)^(5/2)*(3 - x)^(5/2)),x)
Output:
-(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)
Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(3-x)^{5/2} (-2+x)^{5/2}} \, dx=\frac {\frac {32}{3} x^{3}-80 x^{2}+196 x -\frac {470}{3}}{\sqrt {x -2}\, \sqrt {-x +3}\, \left (x^{2}-5 x +6\right )} \] Input:
int(1/(3-x)^(5/2)/(-2+x)^(5/2),x)
Output:
(2*(16*x**3 - 120*x**2 + 294*x - 235))/(3*sqrt(x - 2)*sqrt( - x + 3)*(x**2 - 5*x + 6))