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 = 0.0225357, antiderivative size = 40, normalized size of antiderivative = 1., 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} \[ -\frac{1}{3} \sqrt{-3 x^2+4 x+2}-\frac{2 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 619
Rule 216
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*}
Mathematica [A] time = 0.0162402, size = 40, normalized size = 1. \[ \frac{1}{9} \left (-3 \sqrt{-3 x^2+4 x+2}-2 \sqrt{3} \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 30, normalized size = 0.8 \begin{align*} -{\frac{1}{3}\sqrt{-3\,{x}^{2}+4\,x+2}}+{\frac{2\,\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 = 1.51413, size = 42, normalized size = 1.05 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03475, size = 154, normalized size = 3.85 \begin{align*} -\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{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 \frac{x}{\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 = 1.15499, size = 42, normalized size = 1.05 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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