Optimal. Leaf size=12 \[ \frac{\sin ^{-1}(6 x+5)}{\sqrt{3}} \]
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Rubi [A] time = 0.0044271, antiderivative size = 12, 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} \[ \frac{\sin ^{-1}(6 x+5)}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2-5 x-3 x^2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,-5-6 x\right )}{\sqrt{3}}\\ &=\frac{\sin ^{-1}(5+6 x)}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0068594, size = 12, normalized size = 1. \[ \frac{\sin ^{-1}(6 x+5)}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 12, normalized size = 1. \begin{align*}{\frac{\arcsin \left ( 6\,x+5 \right ) \sqrt{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42477, size = 15, normalized size = 1.25 \begin{align*} \frac{1}{3} \, \sqrt{3} \arcsin \left (6 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03931, size = 115, normalized size = 9.58 \begin{align*} -\frac{1}{3} \, \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 ) \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} - 5 x - 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06858, size = 15, normalized size = 1.25 \begin{align*} \frac{1}{3} \, \sqrt{3} \arcsin \left (6 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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