Optimal. Leaf size=19 \[ -\frac{\sinh ^{-1}\left (\frac{2-3 x}{\sqrt{11}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0119958, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {619, 215} \[ -\frac{\sinh ^{-1}\left (\frac{2-3 x}{\sqrt{11}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{5-4 x+3 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{44}}} \, dx,x,-4+6 x\right )}{2 \sqrt{33}}\\ &=-\frac{\sinh ^{-1}\left (\frac{2-3 x}{\sqrt{11}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0082043, size = 18, normalized size = 0.95 \[ \frac{\sinh ^{-1}\left (\frac{3 x-2}{\sqrt{11}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 15, normalized size = 0.8 \begin{align*}{\frac{\sqrt{3}}{3}{\it Arcsinh} \left ({\frac{3\,\sqrt{11}}{11} \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 = 1.55264, size = 22, normalized size = 1.16 \begin{align*} \frac{1}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{11} \, \sqrt{11}{\left (3 \, x - 2\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03638, size = 111, normalized size = 5.84 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-2 \, \sqrt{3} \sqrt{3 \, x^{2} - 4 \, x + 5}{\left (3 \, x - 2\right )} - 18 \, x^{2} + 24 \, x - 19\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}{\sqrt{3 x^{2} - 4 x + 5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07872, size = 45, normalized size = 2.37 \begin{align*} -\frac{1}{3} \, \sqrt{3} \log \left (-\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - 4 \, x + 5}\right )} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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