∫ 1_______ (3−x)5∕2(−2+x)5∕2 dx [1164]

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

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)^

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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 = {47, 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*}

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

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

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]))) && (

\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 = 0.04, size = 43, normalized size = 0.54 \begin {gather*} \frac {2 \left (-235+294 x-120 x^2+16 x^3\right )}{3 (-3+x) (-2+x) \sqrt {-6+5 x-x^2}} \end {gather*}

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 8.75, size = 206, normalized size = 2.61 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (235-294 x+120 x^2-16 x^3\right ) \sqrt {\frac {3-x}{-2+x}}}{3 \left (-18+21 x-8 x^2+x^3\right )},\frac {1}{\text {Abs}\left [-2+x\right ]}>1\right \}\right \},\frac {-32 I \left (-2+x\right )^3 \sqrt {1-\frac {1}{-2+x}}}{-6+3 x-6 \left (-2+x\right )^2+3 \left (-2+x\right )^3}-\frac {12 I \left (-2+x\right ) \sqrt {1-\frac {1}{-2+x}}}{-6+3 x-6 \left (-2+x\right )^2+3 \left (-2+x\right )^3}-\frac {2 I \sqrt {1-\frac {1}{-2+x}}}{-6+3 x-6 \left (-2+x\right )^2+3 \left (-2+x\right )^3}+\frac {I 48 \left (-2+x\right )^2 \sqrt {1-\frac {1}{-2+x}}}{-6+3 x-6 \left (-2+x\right )^2+3 \left (-2+x\right )^3}\right ] \end {gather*}

Piecewise[{{2 (235 - 294 x + 120 x ^ 2 - 16 x ^ 3) Sqrt[(3 - x) / (-2 + x)] / (3 (-18 + 21 x - 8 x ^ 2 + x ^ 3

)), 1 / Abs[-2 + x] > 1}}, -32 I (-2 + x) ^ 3 Sqrt[1 - 1 / (-2 + x)] / (-6 + 3 x - 6 (-2 + x) ^ 2 + 3 (-2 + x)

^ 3) - 12 I (-2 + x) Sqrt[1 - 1 / (-2 + x)] / (-6 + 3 x - 6 (-2 + x) ^ 2 + 3 (-2 + x) ^ 3) - 2 I Sqrt[1 - 1 /

(-2 + x)] / (-6 + 3 x - 6 (-2 + x) ^ 2 + 3 (-2 + x) ^ 3) + I 48 (-2 + x) ^ 2 Sqrt[1 - 1 / (-2 + x)] / (-6 + 3

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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	\(\frac {2}{3 \left (3-x \right )^{\frac {3}{2}} \left (-2+x \right )^{\frac {3}{2}}}+\frac {4}{\left (-2+x \right )^{\frac {3}{2}} \sqrt {3-x}}-\frac {16 \sqrt {3-x}}{3 \left (-2+x \right )^{\frac {3}{2}}}-\frac {32 \sqrt {3-x}}{3 \sqrt {-2+x}}\)	\(58\)

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)^

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Maxima [A]
time = 0.27, 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*}

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)

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Fricas [A]
time = 0.31, 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*}

-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 [C] Result contains complex when optimal does not.
time = 7.05, 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*}

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 -

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Giac [A]
time = 0.01, size = 173, normalized size = 2.19 \begin {gather*} -2 \left (\frac {-\frac {64}{3} \left (-\frac {-2 \sqrt {x-2}+2}{2 \sqrt {-x+3}}\right )^{3}+\frac {352 \left (-2 \sqrt {x-2}+2\right )}{\sqrt {-x+3}}}{512}+\frac {33 \left (-\frac {-2 \sqrt {x-2}+2}{2 \sqrt {-x+3}}\right )^{2}+1}{24 \left (-\frac {-2 \sqrt {x-2}+2}{2 \sqrt {-x+3}}\right )^{3}}+\frac {2 \left (-\frac {4}{3} \sqrt {-x+3} \sqrt {-x+3}+\frac {3}{2}\right ) \sqrt {-x+3} \sqrt {x-2}}{\left (x-2\right )^{2}}\right ) \end {gather*}

1/12*(sqrt(x - 2) - 1)^3/(-x + 3)^(3/2) + 11/4*(sqrt(x - 2) - 1)/sqrt(-x + 3) - 2/3*(8*x - 15)*sqrt(-x + 3)/(x

- 2)^(3/2) + 1/12*(-x + 3)^(3/2)*(33*(sqrt(x - 2) - 1)^2/(x - 3) - 1)/(sqrt(x - 2) - 1)^3

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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*}

-(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

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