Optimal. Leaf size=23 \[ \frac {1}{6} (3 x+2) \sqrt {-9 x^2-12 x-4} \]
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Rubi [A] time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.17 \[ \frac {x \sqrt {-(3 x+2)^2} (3 x+4)}{6 x+4} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.85, size = 9, normalized size = 0.39 \[ \frac {3}{2} i \, x^{2} + 2 i \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.41, size = 26, normalized size = 1.13 \[ -\frac {1}{2} i \, {\left (3 \, x^{2} + 4 \, x\right )} \mathrm {sgn}\left (-3 \, x - 2\right ) - \frac {2}{3} i \, \mathrm {sgn}\left (-3 \, x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 27, normalized size = 1.17 \[ \frac {\left (3 x +4\right ) \sqrt {-\left (3 x +2\right )^{2}}\, x}{6 x +4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 30, normalized size = 1.30 \[ \frac {1}{2} \, \sqrt {-9 \, x^{2} - 12 \, x - 4} x + \frac {1}{3} \, \sqrt {-9 \, x^{2} - 12 \, x - 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 18, normalized size = 0.78 \[ \frac {\left (3\,x+2\right )\,\sqrt {-{\left (3\,x+2\right )}^2}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- \left (3 x + 2\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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