Optimal. Leaf size=35 \[ -\frac {1}{16} (3-8 x) \sqrt {3 x-4 x^2}-\frac {9}{64} \sin ^{-1}\left (1-\frac {8 x}{3}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {626, 633, 222}
\begin {gather*} -\frac {9}{64} \text {ArcSin}\left (1-\frac {8 x}{3}\right )-\frac {1}{16} \sqrt {3 x-4 x^2} (3-8 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 626
Rule 633
Rubi steps
\begin {align*} \int \sqrt {3 x-4 x^2} \, dx &=-\frac {1}{16} (3-8 x) \sqrt {3 x-4 x^2}+\frac {9}{32} \int \frac {1}{\sqrt {3 x-4 x^2}} \, dx\\ &=-\frac {1}{16} (3-8 x) \sqrt {3 x-4 x^2}-\frac {3}{64} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,3-8 x\right )\\ &=-\frac {1}{16} (3-8 x) \sqrt {3 x-4 x^2}-\frac {9}{64} \sin ^{-1}\left (1-\frac {8 x}{3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 55, normalized size = 1.57 \begin {gather*} \frac {1}{32} \sqrt {-x (-3+4 x)} \left (-6+16 x+\frac {9 \log \left (-2 \sqrt {x}+\sqrt {-3+4 x}\right )}{\sqrt {x} \sqrt {-3+4 x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 28, normalized size = 0.80
method | result | size |
default | \(\frac {9 \arcsin \left (-1+\frac {8 x}{3}\right )}{64}-\frac {\left (3-8 x \right ) \sqrt {-4 x^{2}+3 x}}{16}\) | \(28\) |
risch | \(-\frac {\left (-3+8 x \right ) x \left (-3+4 x \right )}{16 \sqrt {-x \left (-3+4 x \right )}}+\frac {9 \arcsin \left (-1+\frac {8 x}{3}\right )}{64}\) | \(33\) |
meijerg | \(-\frac {9 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {3}\, \left (3-8 x \right ) \sqrt {-\frac {4 x}{3}+1}}{9}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {2 \sqrt {3}\, \sqrt {x}}{3}\right )}{2}\right )}{16 \sqrt {\pi }}\) | \(47\) |
trager | \(\left (-\frac {3}{16}+\frac {x}{2}\right ) \sqrt {-4 x^{2}+3 x}+\frac {9 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-8 x \RootOf \left (\textit {\_Z}^{2}+1\right )+4 \sqrt {-4 x^{2}+3 x}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{64}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 36, normalized size = 1.03 \begin {gather*} \frac {1}{2} \, \sqrt {-4 \, x^{2} + 3 \, x} x - \frac {3}{16} \, \sqrt {-4 \, x^{2} + 3 \, x} - \frac {9}{64} \, \arcsin \left (-\frac {8}{3} \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.60, size = 38, normalized size = 1.09 \begin {gather*} \frac {1}{16} \, \sqrt {-4 \, x^{2} + 3 \, x} {\left (8 \, x - 3\right )} - \frac {9}{32} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 3 \, x}}{2 \, 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 \sqrt {- 4 x^{2} + 3 x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.93, size = 27, normalized size = 0.77 \begin {gather*} \frac {1}{16} \, \sqrt {-4 \, x^{2} + 3 \, x} {\left (8 \, x - 3\right )} + \frac {9}{64} \, \arcsin \left (\frac {8}{3} \, x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 26, normalized size = 0.74 \begin {gather*} \frac {9\,\mathrm {asin}\left (\frac {8\,x}{3}-1\right )}{64}+\left (\frac {x}{2}-\frac {3}{16}\right )\,\sqrt {3\,x-4\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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