Optimal. Leaf size=34 \[ -\frac{1}{3} \sqrt{-3 x^2+5 x-2}-\frac{5 \sin ^{-1}(5-6 x)}{6 \sqrt{3}} \]
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Rubi [A] time = 0.0091497, antiderivative size = 34, 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} \[ -\frac{1}{3} \sqrt{-3 x^2+5 x-2}-\frac{5 \sin ^{-1}(5-6 x)}{6 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{-2+5 x-3 x^2}} \, dx &=-\frac{1}{3} \sqrt{-2+5 x-3 x^2}+\frac{5}{6} \int \frac{1}{\sqrt{-2+5 x-3 x^2}} \, dx\\ &=-\frac{1}{3} \sqrt{-2+5 x-3 x^2}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,5-6 x\right )}{6 \sqrt{3}}\\ &=-\frac{1}{3} \sqrt{-2+5 x-3 x^2}-\frac{5 \sin ^{-1}(5-6 x)}{6 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0087704, size = 34, normalized size = 1. \[ -\frac{1}{3} \sqrt{-3 x^2+5 x-2}-\frac{5 \sin ^{-1}(5-6 x)}{6 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 27, normalized size = 0.8 \begin{align*}{\frac{5\,\arcsin \left ( -5+6\,x \right ) \sqrt{3}}{18}}-{\frac{1}{3}\sqrt{-3\,{x}^{2}+5\,x-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41195, size = 35, normalized size = 1.03 \begin{align*} \frac{5}{18} \, \sqrt{3} \arcsin \left (6 \, x - 5\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 5 \, x - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09634, size = 155, normalized size = 4.56 \begin{align*} -\frac{5}{18} \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{-3 \, x^{2} + 5 \, x - 2}{\left (6 \, x - 5\right )}}{6 \,{\left (3 \, x^{2} - 5 \, x + 2\right )}}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 5 \, 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{- \left (x - 1\right ) \left (3 x - 2\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12041, size = 35, normalized size = 1.03 \begin{align*} \frac{5}{18} \, \sqrt{3} \arcsin \left (6 \, x - 5\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 5 \, x - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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