Optimal. Leaf size=45 \[ -\frac{1}{6} \sqrt{-3 x^2+4 x+2} (2-3 x)-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0159289, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {612, 619, 216} \[ -\frac{1}{6} \sqrt{-3 x^2+4 x+2} (2-3 x)-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 612
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \sqrt{2+4 x-3 x^2} \, dx &=-\frac{1}{6} (2-3 x) \sqrt{2+4 x-3 x^2}+\frac{5}{3} \int \frac{1}{\sqrt{2+4 x-3 x^2}} \, dx\\ &=-\frac{1}{6} (2-3 x) \sqrt{2+4 x-3 x^2}-\frac{1}{6} \sqrt{\frac{5}{6}} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{40}}} \, dx,x,4-6 x\right )\\ &=-\frac{1}{6} (2-3 x) \sqrt{2+4 x-3 x^2}-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0192693, size = 46, normalized size = 1.02 \[ \left (\frac{x}{2}-\frac{1}{3}\right ) \sqrt{-3 x^2+4 x+2}-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 35, normalized size = 0.8 \begin{align*} -{\frac{-6\,x+4}{12}\sqrt{-3\,{x}^{2}+4\,x+2}}+{\frac{5\,\sqrt{3}}{9}\arcsin \left ({\frac{3\,\sqrt{10}}{10} \left ( x-{\frac{2}{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.71788, size = 62, normalized size = 1.38 \begin{align*} \frac{1}{2} \, \sqrt{-3 \, x^{2} + 4 \, x + 2} x - \frac{5}{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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94372, size = 166, normalized size = 3.69 \begin{align*} \frac{1}{6} \, \sqrt{-3 \, x^{2} + 4 \, x + 2}{\left (3 \, x - 2\right )} - \frac{5}{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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- 3 x^{2} + 4 x + 2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26562, size = 49, normalized size = 1.09 \begin{align*} \frac{1}{6} \, \sqrt{-3 \, x^{2} + 4 \, x + 2}{\left (3 \, x - 2\right )} + \frac{5}{9} \, \sqrt{3} \arcsin \left (\frac{1}{10} \, \sqrt{10}{\left (3 \, x - 2\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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