Optimal. Leaf size=91 \[ \frac {3}{64} \sqrt {3-x} \sqrt {-2+x}+\frac {1}{32} (3-x)^{3/2} \sqrt {-2+x}-\frac {1}{8} (3-x)^{5/2} \sqrt {-2+x}-\frac {1}{4} (3-x)^{5/2} (-2+x)^{3/2}-\frac {3}{128} \sin ^{-1}(5-2 x) \]
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Rubi [A]
time = 0.01, 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 = {52, 55, 633,
222} \begin {gather*} -\frac {3}{128} \text {ArcSin}(5-2 x)-\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 55
Rule 222
Rule 633
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} \text {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 = 0.09, size = 82, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {-6+5 x-x^2} \left (\sqrt {-2+x} \left (675-1095 x+650 x^2-168 x^3+16 x^4\right )+3 \sqrt {-3+x} \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-3+x}{-2+x}}}\right )\right )}{64 (-3+x) \sqrt {-2+x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 89, normalized size = 0.98
method | result | size |
risch | \(\frac {\left (16 x^{3}-120 x^{2}+290 x -225\right ) \left (-3+x \right ) \sqrt {-2+x}\, \sqrt {\left (-2+x \right ) \left (3-x \right )}}{64 \sqrt {-\left (-3+x \right ) \left (-2+x \right )}\, \sqrt {3-x}}+\frac {3 \sqrt {\left (-2+x \right ) \left (3-x \right )}\, \arcsin \left (2 x -5\right )}{128 \sqrt {-2+x}\, \sqrt {3-x}}\) | \(86\) |
default | \(\frac {\left (3-x \right )^{\frac {3}{2}} \left (-2+x \right )^{\frac {5}{2}}}{4}+\frac {\sqrt {3-x}\, \left (-2+x \right )^{\frac {5}{2}}}{8}-\frac {\sqrt {3-x}\, \left (-2+x \right )^{\frac {3}{2}}}{32}-\frac {3 \sqrt {3-x}\, \sqrt {-2+x}}{64}+\frac {3 \sqrt {\left (-2+x \right ) \left (3-x \right )}\, \arcsin \left (2 x -5\right )}{128 \sqrt {-2+x}\, \sqrt {3-x}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, 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.
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Fricas [A]
time = 1.46, 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.
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Sympy [C] Result contains complex when optimal does not.
time = 11.55, 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.
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Giac [A]
time = 1.25, 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.
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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.
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