Optimal. Leaf size=19 \[ -\frac{\sin ^{-1}\left (\frac{3-4 x}{\sqrt{41}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0137482, 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, 216} \[ -\frac{\sin ^{-1}\left (\frac{3-4 x}{\sqrt{41}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{4+3 x-2 x^2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{41}}} \, dx,x,3-4 x\right )}{\sqrt{82}}\\ &=-\frac{\sin ^{-1}\left (\frac{3-4 x}{\sqrt{41}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.00997, size = 19, normalized size = 1. \[ -\frac{\sin ^{-1}\left (\frac{3-4 x}{\sqrt{41}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{2}\arcsin \left ({\frac{4\,\sqrt{41}}{41} \left ( x-{\frac{3}{4}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44799, size = 22, normalized size = 1.16 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arcsin \left (-\frac{1}{41} \, \sqrt{41}{\left (4 \, x - 3\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75203, size = 93, normalized size = 4.89 \begin{align*} -\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-2 \, x^{2} + 3 \, x + 4} - 2 \, \sqrt{2}}{2 \, x}\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{- 2 x^{2} + 3 x + 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08688, size = 22, normalized size = 1.16 \begin{align*} \frac{1}{2} \, \sqrt{2} \arcsin \left (\frac{1}{41} \, \sqrt{41}{\left (4 \, x - 3\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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