3.11.91 \(\int (3-x)^{3/2} (-2+x)^{3/2} \, dx\)

Optimal. Leaf size=91 \[ -\frac {1}{4} (x-2)^{3/2} (3-x)^{5/2}-\frac {1}{8} \sqrt {x-2} (3-x)^{5/2}+\frac {1}{32} \sqrt {x-2} (3-x)^{3/2}+\frac {3}{64} \sqrt {x-2} \sqrt {3-x}-\frac {3}{128} \sin ^{-1}(5-2 x) \]

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Rubi [A]  time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {50, 53, 619, 216} \begin {gather*} -\frac {1}{4} (x-2)^{3/2} (3-x)^{5/2}-\frac {1}{8} \sqrt {x-2} (3-x)^{5/2}+\frac {1}{32} \sqrt {x-2} (3-x)^{3/2}+\frac {3}{64} \sqrt {x-2} \sqrt {3-x}-\frac {3}{128} \sin ^{-1}(5-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - x)^(3/2)*(-2 + x)^(3/2),x]

[Out]

(3*Sqrt[3 - x]*Sqrt[-2 + x])/64 + ((3 - x)^(3/2)*Sqrt[-2 + x])/32 - ((3 - x)^(5/2)*Sqrt[-2 + x])/8 - ((3 - x)^
(5/2)*(-2 + x)^(3/2))/4 - (3*ArcSin[5 - 2*x])/128

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps

\begin {align*} \int (3-x)^{3/2} (-2+x)^{3/2} \, dx &=-\frac {1}{4} (3-x)^{5/2} (-2+x)^{3/2}+\frac {3}{8} \int (3-x)^{3/2} \sqrt {-2+x} \, dx\\ &=-\frac {1}{8} (3-x)^{5/2} \sqrt {-2+x}-\frac {1}{4} (3-x)^{5/2} (-2+x)^{3/2}+\frac {1}{16} \int \frac {(3-x)^{3/2}}{\sqrt {-2+x}} \, dx\\ &=\frac {1}{32} (3-x)^{3/2} \sqrt {-2+x}-\frac {1}{8} (3-x)^{5/2} \sqrt {-2+x}-\frac {1}{4} (3-x)^{5/2} (-2+x)^{3/2}+\frac {3}{64} \int \frac {\sqrt {3-x}}{\sqrt {-2+x}} \, dx\\ &=\frac {3}{64} \sqrt {3-x} \sqrt {-2+x}+\frac {1}{32} (3-x)^{3/2} \sqrt {-2+x}-\frac {1}{8} (3-x)^{5/2} \sqrt {-2+x}-\frac {1}{4} (3-x)^{5/2} (-2+x)^{3/2}+\frac {3}{128} \int \frac {1}{\sqrt {3-x} \sqrt {-2+x}} \, dx\\ &=\frac {3}{64} \sqrt {3-x} \sqrt {-2+x}+\frac {1}{32} (3-x)^{3/2} \sqrt {-2+x}-\frac {1}{8} (3-x)^{5/2} \sqrt {-2+x}-\frac {1}{4} (3-x)^{5/2} (-2+x)^{3/2}+\frac {3}{128} \int \frac {1}{\sqrt {-6+5 x-x^2}} \, dx\\ &=\frac {3}{64} \sqrt {3-x} \sqrt {-2+x}+\frac {1}{32} (3-x)^{3/2} \sqrt {-2+x}-\frac {1}{8} (3-x)^{5/2} \sqrt {-2+x}-\frac {1}{4} (3-x)^{5/2} (-2+x)^{3/2}-\frac {3}{128} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,5-2 x\right )\\ &=\frac {3}{64} \sqrt {3-x} \sqrt {-2+x}+\frac {1}{32} (3-x)^{3/2} \sqrt {-2+x}-\frac {1}{8} (3-x)^{5/2} \sqrt {-2+x}-\frac {1}{4} (3-x)^{5/2} (-2+x)^{3/2}-\frac {3}{128} \sin ^{-1}(5-2 x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 80, normalized size = 0.88 \begin {gather*} \frac {\sqrt {-x^2+5 x-6} \left (\sqrt {x-2} \left (-16 x^4+168 x^3-650 x^2+1095 x-675\right )+3 \sqrt {3-x} \sin ^{-1}\left (\sqrt {3-x}\right )\right )}{64 (x-3) \sqrt {x-2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-6 + 5*x - x^2]*(Sqrt[-2 + x]*(-675 + 1095*x - 650*x^2 + 168*x^3 - 16*x^4) + 3*Sqrt[3 - x]*ArcSin[Sqrt[3
- x]]))/(64*(-3 + x)*Sqrt[-2 + x])

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IntegrateAlgebraic [A]  time = 0.09, size = 115, normalized size = 1.26 \begin {gather*} \frac {-\frac {3 (3-x)^{7/2}}{(x-2)^{7/2}}-\frac {11 (3-x)^{5/2}}{(x-2)^{5/2}}+\frac {11 (3-x)^{3/2}}{(x-2)^{3/2}}+\frac {3 \sqrt {3-x}}{\sqrt {x-2}}}{64 \left (\frac {3-x}{x-2}+1\right )^4}-\frac {3}{64} \tan ^{-1}\left (\frac {\sqrt {3-x}}{\sqrt {x-2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*(3 - x)^(7/2))/(-2 + x)^(7/2) - (11*(3 - x)^(5/2))/(-2 + x)^(5/2) + (11*(3 - x)^(3/2))/(-2 + x)^(3/2) + (
3*Sqrt[3 - x])/Sqrt[-2 + x])/(64*(1 + (3 - x)/(-2 + x))^4) - (3*ArcTan[Sqrt[3 - x]/Sqrt[-2 + x]])/64

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fricas [A]  time = 1.10, size = 62, normalized size = 0.68 \begin {gather*} -\frac {1}{64} \, {\left (16 \, x^{3} - 120 \, x^{2} + 290 \, x - 225\right )} \sqrt {x - 2} \sqrt {-x + 3} - \frac {3}{128} \, \arctan \left (\frac {{\left (2 \, x - 5\right )} \sqrt {x - 2} \sqrt {-x + 3}}{2 \, {\left (x^{2} - 5 \, x + 6\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-x)^(3/2)*(-2+x)^(3/2),x, algorithm="fricas")

[Out]

-1/64*(16*x^3 - 120*x^2 + 290*x - 225)*sqrt(x - 2)*sqrt(-x + 3) - 3/128*arctan(1/2*(2*x - 5)*sqrt(x - 2)*sqrt(
-x + 3)/(x^2 - 5*x + 6))

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giac [A]  time = 0.88, size = 101, normalized size = 1.11 \begin {gather*} -\frac {1}{192} \, {\left (2 \, {\left (4 \, {\left (6 \, x + 35\right )} {\left (x - 2\right )} + 523\right )} {\left (x - 2\right )} + 801\right )} \sqrt {x - 2} \sqrt {-x + 3} + \frac {7}{24} \, {\left (2 \, {\left (4 \, x + 15\right )} {\left (x - 2\right )} + 69\right )} \sqrt {x - 2} \sqrt {-x + 3} - 4 \, {\left (2 \, x + 3\right )} \sqrt {x - 2} \sqrt {-x + 3} + 12 \, \sqrt {x - 2} \sqrt {-x + 3} + \frac {3}{64} \, \arcsin \left (\sqrt {x - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-x)^(3/2)*(-2+x)^(3/2),x, algorithm="giac")

[Out]

-1/192*(2*(4*(6*x + 35)*(x - 2) + 523)*(x - 2) + 801)*sqrt(x - 2)*sqrt(-x + 3) + 7/24*(2*(4*x + 15)*(x - 2) +
69)*sqrt(x - 2)*sqrt(-x + 3) - 4*(2*x + 3)*sqrt(x - 2)*sqrt(-x + 3) + 12*sqrt(x - 2)*sqrt(-x + 3) + 3/64*arcsi
n(sqrt(x - 2))

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maple [A]  time = 0.01, size = 89, normalized size = 0.98 \begin {gather*} \frac {3 \sqrt {\left (x -2\right ) \left (-x +3\right )}\, \arcsin \left (2 x -5\right )}{128 \sqrt {x -2}\, \sqrt {-x +3}}+\frac {\left (-x +3\right )^{\frac {3}{2}} \left (x -2\right )^{\frac {5}{2}}}{4}+\frac {\sqrt {-x +3}\, \left (x -2\right )^{\frac {5}{2}}}{8}-\frac {\sqrt {-x +3}\, \left (x -2\right )^{\frac {3}{2}}}{32}-\frac {3 \sqrt {-x +3}\, \sqrt {x -2}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3-x)^(3/2)*(x-2)^(5/2)+1/8*(3-x)^(1/2)*(x-2)^(5/2)-1/32*(3-x)^(1/2)*(x-2)^(3/2)-3/64*(3-x)^(1/2)*(x-2)^(1
/2)+3/128*((x-2)*(3-x))^(1/2)/(x-2)^(1/2)/(3-x)^(1/2)*arcsin(-5+2*x)

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maxima [A]  time = 2.97, size = 67, normalized size = 0.74 \begin {gather*} \frac {1}{4} \, {\left (-x^{2} + 5 \, x - 6\right )}^{\frac {3}{2}} x - \frac {5}{8} \, {\left (-x^{2} + 5 \, x - 6\right )}^{\frac {3}{2}} + \frac {3}{32} \, \sqrt {-x^{2} + 5 \, x - 6} x - \frac {15}{64} \, \sqrt {-x^{2} + 5 \, x - 6} + \frac {3}{128} \, \arcsin \left (2 \, x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-x)^(3/2)*(-2+x)^(3/2),x, algorithm="maxima")

[Out]

1/4*(-x^2 + 5*x - 6)^(3/2)*x - 5/8*(-x^2 + 5*x - 6)^(3/2) + 3/32*sqrt(-x^2 + 5*x - 6)*x - 15/64*sqrt(-x^2 + 5*
x - 6) + 3/128*arcsin(2*x - 5)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (x-2\right )}^{3/2}\,{\left (3-x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 2)^(3/2)*(3 - x)^(3/2),x)

[Out]

int((x - 2)^(3/2)*(3 - x)^(3/2), x)

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sympy [A]  time = 7.48, size = 199, normalized size = 2.19 \begin {gather*} \begin {cases} - \frac {3 i \operatorname {acosh}{\left (\sqrt {x - 2} \right )}}{64} - \frac {i \left (x - 2\right )^{\frac {9}{2}}}{4 \sqrt {x - 3}} + \frac {5 i \left (x - 2\right )^{\frac {7}{2}}}{8 \sqrt {x - 3}} - \frac {13 i \left (x - 2\right )^{\frac {5}{2}}}{32 \sqrt {x - 3}} - \frac {i \left (x - 2\right )^{\frac {3}{2}}}{64 \sqrt {x - 3}} + \frac {3 i \sqrt {x - 2}}{64 \sqrt {x - 3}} & \text {for}\: \left |{x - 2}\right | > 1 \\\frac {3 \operatorname {asin}{\left (\sqrt {x - 2} \right )}}{64} + \frac {\left (x - 2\right )^{\frac {9}{2}}}{4 \sqrt {3 - x}} - \frac {5 \left (x - 2\right )^{\frac {7}{2}}}{8 \sqrt {3 - x}} + \frac {13 \left (x - 2\right )^{\frac {5}{2}}}{32 \sqrt {3 - x}} + \frac {\left (x - 2\right )^{\frac {3}{2}}}{64 \sqrt {3 - x}} - \frac {3 \sqrt {x - 2}}{64 \sqrt {3 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-3*I*acosh(sqrt(x - 2))/64 - I*(x - 2)**(9/2)/(4*sqrt(x - 3)) + 5*I*(x - 2)**(7/2)/(8*sqrt(x - 3))
- 13*I*(x - 2)**(5/2)/(32*sqrt(x - 3)) - I*(x - 2)**(3/2)/(64*sqrt(x - 3)) + 3*I*sqrt(x - 2)/(64*sqrt(x - 3)),
Abs(x - 2) > 1), (3*asin(sqrt(x - 2))/64 + (x - 2)**(9/2)/(4*sqrt(3 - x)) - 5*(x - 2)**(7/2)/(8*sqrt(3 - x))
+ 13*(x - 2)**(5/2)/(32*sqrt(3 - x)) + (x - 2)**(3/2)/(64*sqrt(3 - x)) - 3*sqrt(x - 2)/(64*sqrt(3 - x)), True)
)

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