Optimal. Leaf size=51 \[ -\frac {3}{64} (1-2 x) \sqrt {x-x^2}-\frac {1}{8} (1-2 x) \left (x-x^2\right )^{3/2}-\frac {3}{128} \sin ^{-1}(1-2 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {626, 633, 222}
\begin {gather*} -\frac {3}{128} \text {ArcSin}(1-2 x)-\frac {1}{8} (1-2 x) \left (x-x^2\right )^{3/2}-\frac {3}{64} (1-2 x) \sqrt {x-x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 626
Rule 633
Rubi steps
\begin {align*} \int \left (x-x^2\right )^{3/2} \, dx &=-\frac {1}{8} (1-2 x) \left (x-x^2\right )^{3/2}+\frac {3}{16} \int \sqrt {x-x^2} \, dx\\ &=-\frac {3}{64} (1-2 x) \sqrt {x-x^2}-\frac {1}{8} (1-2 x) \left (x-x^2\right )^{3/2}+\frac {3}{128} \int \frac {1}{\sqrt {x-x^2}} \, dx\\ &=-\frac {3}{64} (1-2 x) \sqrt {x-x^2}-\frac {1}{8} (1-2 x) \left (x-x^2\right )^{3/2}-\frac {3}{128} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1-2 x\right )\\ &=-\frac {3}{64} (1-2 x) \sqrt {x-x^2}-\frac {1}{8} (1-2 x) \left (x-x^2\right )^{3/2}-\frac {3}{128} \sin ^{-1}(1-2 x)\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 67, normalized size = 1.31 \begin {gather*} \frac {x \left (-3+x+26 x^2-40 x^3+16 x^4\right )+6 \sqrt {-1+x} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {-1+x}}{-1+\sqrt {x}}\right )}{64 \sqrt {-((-1+x) x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 42, normalized size = 0.82
method | result | size |
risch | \(\frac {\left (16 x^{3}-24 x^{2}+2 x +3\right ) x \left (x -1\right )}{64 \sqrt {-x \left (x -1\right )}}+\frac {3 \arcsin \left (2 x -1\right )}{128}\) | \(39\) |
default | \(-\frac {\left (1-2 x \right ) \left (-x^{2}+x \right )^{\frac {3}{2}}}{8}+\frac {3 \arcsin \left (2 x -1\right )}{128}-\frac {3 \left (1-2 x \right ) \sqrt {-x^{2}+x}}{64}\) | \(42\) |
meijerg | \(-\frac {3 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \left (80 x^{3}-120 x^{2}+10 x +15\right ) \sqrt {1-x}}{240}+\frac {i \sqrt {\pi }\, \arcsin \left (\sqrt {x}\right )}{16}\right )}{4 \sqrt {\pi }}\) | \(49\) |
trager | \(\left (-\frac {1}{4} x^{3}+\frac {3}{8} x^{2}-\frac {1}{32} x -\frac {3}{64}\right ) \sqrt {-x^{2}+x}-\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (2 x \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {-x^{2}+x}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{128}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 55, normalized size = 1.08 \begin {gather*} \frac {1}{4} \, {\left (-x^{2} + x\right )}^{\frac {3}{2}} x - \frac {1}{8} \, {\left (-x^{2} + x\right )}^{\frac {3}{2}} + \frac {3}{32} \, \sqrt {-x^{2} + x} x - \frac {3}{64} \, \sqrt {-x^{2} + x} + \frac {3}{128} \, \arcsin \left (2 \, x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.42, size = 43, normalized size = 0.84 \begin {gather*} -\frac {1}{64} \, {\left (16 \, x^{3} - 24 \, x^{2} + 2 \, x + 3\right )} \sqrt {-x^{2} + x} - \frac {3}{64} \, \arctan \left (\frac {\sqrt {-x^{2} + x}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- x^{2} + x\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.91, size = 35, normalized size = 0.69 \begin {gather*} -\frac {1}{64} \, {\left (2 \, {\left (4 \, {\left (2 \, x - 3\right )} x + 1\right )} x + 3\right )} \sqrt {-x^{2} + x} + \frac {3}{128} \, \arcsin \left (2 \, x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 39, normalized size = 0.76 \begin {gather*} \frac {3\,\mathrm {asin}\left (2\,x-1\right )}{128}+\frac {3\,\sqrt {x-x^2}\,\left (\frac {x}{2}-\frac {1}{4}\right )}{16}+\frac {{\left (x-x^2\right )}^{3/2}\,\left (x-\frac {1}{2}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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