Optimal. Leaf size=36 \[ -\frac{\tan ^{-1}\left (\frac{4-5 x}{2 \sqrt{2} \sqrt{-3 x^2+5 x-2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0108753, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {724, 204} \[ -\frac{\tan ^{-1}\left (\frac{4-5 x}{2 \sqrt{2} \sqrt{-3 x^2+5 x-2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 724
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{-2+5 x-3 x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,\frac{-4+5 x}{\sqrt{-2+5 x-3 x^2}}\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{4-5 x}{2 \sqrt{2} \sqrt{-2+5 x-3 x^2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0097152, size = 30, normalized size = 0.83 \[ \frac{\tan ^{-1}\left (\frac{5 x-4}{2 \sqrt{-6 x^2+10 x-4}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 29, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 5\,x-4 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{-3\,{x}^{2}+5\,x-2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49993, size = 27, normalized size = 0.75 \begin{align*} \frac{1}{2} \, \sqrt{2} \arcsin \left (\frac{5 \, x}{{\left | x \right |}} - \frac{4}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03531, size = 115, normalized size = 3.19 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-3 \, x^{2} + 5 \, x - 2}{\left (5 \, x - 4\right )}}{4 \,{\left (3 \, x^{2} - 5 \, 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 \frac{1}{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 = 1.13379, size = 59, normalized size = 1.64 \begin{align*} -\frac{1}{3} \, \sqrt{6} \sqrt{3} \arctan \left (\frac{1}{12} \, \sqrt{6}{\left (\frac{5 \,{\left (2 \, \sqrt{3} \sqrt{-3 \, x^{2} + 5 \, x - 2} - 1\right )}}{6 \, x - 5} - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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