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

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

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Rubi [A]  time = 0.01, antiderivative size = 79, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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 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])

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*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.42 \begin {gather*} \frac {-32 x^3+240 x^2-588 x+470}{3 \left (-x^2+5 x-6\right )^{3/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.06, size = 61, normalized size = 0.77 \begin {gather*} -\frac {2 \left (\frac {(3-x)^3}{(x-2)^3}+\frac {9 (3-x)^2}{(x-2)^2}-\frac {9 (3-x)}{x-2}-1\right ) (x-2)^{3/2}}{3 (3-x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-1 + (3 - x)^3/(-2 + x)^3 + (9*(3 - x)^2)/(-2 + x)^2 - (9*(3 - x))/(-2 + x))*(-2 + x)^(3/2))/(3*(3 - x)^(
3/2))

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fricas [A]  time = 1.22, size = 49, normalized size = 0.62 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.10, size = 97, normalized size = 1.23 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.00, size = 30, normalized size = 0.38 \begin {gather*} -\frac {2 \left (16 x^{3}-120 x^{2}+294 x -235\right )}{3 \left (x -2\right )^{\frac {3}{2}} \left (-x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 59, normalized size = 0.75 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.37, size = 69, normalized size = 0.87 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 9.85, size = 282, normalized size = 3.57 \begin {gather*} \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 {gather*}

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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