Optimal. Leaf size=23 \[ \frac{1}{6} (3 x+2) \sqrt{9 x^2+12 x+4} \]
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Rubi [A] time = 0.0031282, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {609} \[ \frac{1}{6} (3 x+2) \sqrt{9 x^2+12 x+4} \]
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin{align*} \int \sqrt{4+12 x+9 x^2} \, dx &=\frac{1}{6} (2+3 x) \sqrt{4+12 x+9 x^2}\\ \end{align*}
Mathematica [A] time = 0.0054391, size = 25, normalized size = 1.09 \[ \frac{x \sqrt{(3 x+2)^2} (3 x+4)}{6 x+4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 25, normalized size = 1.1 \begin{align*}{\frac{x \left ( 3\,x+4 \right ) }{4+6\,x}\sqrt{ \left ( 2+3\,x \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.76622, size = 41, normalized size = 1.78 \begin{align*} \frac{1}{2} \, \sqrt{9 \, x^{2} + 12 \, x + 4} x + \frac{1}{3} \, \sqrt{9 \, x^{2} + 12 \, x + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13546, size = 20, normalized size = 0.87 \begin{align*} \frac{3}{2} \, x^{2} + 2 \, x \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 \sqrt{9 x^{2} + 12 x + 4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24368, size = 35, normalized size = 1.52 \begin{align*} \frac{1}{2} \,{\left (3 \, x^{2} + 4 \, x\right )} \mathrm{sgn}\left (3 \, x + 2\right ) + \frac{2}{3} \, \mathrm{sgn}\left (3 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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