Optimal. Leaf size=40 \[ -\frac {1}{3} \sqrt {-3 x^2+4 x+2}-\frac {2 \sin ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {640, 619, 216} \begin {gather*} -\frac {1}{3} \sqrt {-3 x^2+4 x+2}-\frac {2 \sin ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 619
Rule 640
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx &=-\frac {1}{3} \sqrt {2+4 x-3 x^2}+\frac {2}{3} \int \frac {1}{\sqrt {2+4 x-3 x^2}} \, dx\\ &=-\frac {1}{3} \sqrt {2+4 x-3 x^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{40}}} \, dx,x,4-6 x\right )}{3 \sqrt {30}}\\ &=-\frac {1}{3} \sqrt {2+4 x-3 x^2}-\frac {2 \sin ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 1.00 \begin {gather*} \frac {1}{9} \left (-3 \sqrt {-3 x^2+4 x+2}-2 \sqrt {3} \sin ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 60, normalized size = 1.50 \begin {gather*} -\frac {1}{3} \sqrt {-3 x^2+4 x+2}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {2}-\sqrt {-3 x^2+4 x+2}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 55, normalized size = 1.38 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (3 \, x - 2\right )}}{3 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 31, normalized size = 0.78 \begin {gather*} \frac {2}{9} \, \sqrt {3} \arcsin \left (\frac {1}{10} \, \sqrt {10} {\left (3 \, x - 2\right )}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 30, normalized size = 0.75 \begin {gather*} \frac {2 \sqrt {3}\, \arcsin \left (\frac {3 \sqrt {10}\, \left (x -\frac {2}{3}\right )}{10}\right )}{9}-\frac {\sqrt {-3 x^{2}+4 x +2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.89, size = 31, normalized size = 0.78 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \arcsin \left (-\frac {1}{10} \, \sqrt {10} {\left (3 \, x - 2\right )}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 46, normalized size = 1.15 \begin {gather*} -\frac {\sqrt {-3\,x^2+4\,x+2}}{3}-\frac {\sqrt {3}\,\ln \left (\sqrt {-3\,x^2+4\,x+2}+\frac {\sqrt {3}\,\left (3\,x-2\right )\,1{}\mathrm {i}}{3}\right )\,2{}\mathrm {i}}{9} \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 \frac {x}{\sqrt {- 3 x^{2} + 4 x + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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