Optimal. Leaf size=23 \[ \frac {1}{6} (2+3 x) \sqrt {-4-12 x-9 x^2} \]
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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 = {623}
\begin {gather*} \frac {1}{6} (3 x+2) \sqrt {-9 x^2-12 x-4} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
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 = 0.00, size = 27, normalized size = 1.17 \begin {gather*} \frac {x \sqrt {-(2+3 x)^2} (4+3 x)}{4+6 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 27, normalized size = 1.17
method | result | size |
gosper | \(\frac {x \left (3 x +4\right ) \sqrt {-\left (2+3 x \right )^{2}}}{4+6 x}\) | \(27\) |
default | \(\frac {x \left (3 x +4\right ) \sqrt {-\left (2+3 x \right )^{2}}}{4+6 x}\) | \(27\) |
risch | \(\frac {2 \sqrt {-\left (2+3 x \right )^{2}}\, x}{2+3 x}+\frac {3 \sqrt {-\left (2+3 x \right )^{2}}\, x^{2}}{2 \left (2+3 x \right )}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 30, normalized size = 1.30 \begin {gather*} \frac {1}{2} \, \sqrt {-9 \, x^{2} - 12 \, x - 4} x + \frac {1}{3} \, \sqrt {-9 \, x^{2} - 12 \, x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.32, size = 9, normalized size = 0.39 \begin {gather*} \frac {3}{2} i \, x^{2} + 2 i \, x \end {gather*}
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 \begin {gather*} \int \sqrt {- \left (3 x + 2\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.83, size = 26, normalized size = 1.13 \begin {gather*} -\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 ) \end {gather*}
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 \begin {gather*} \frac {\left (3\,x+2\right )\,\sqrt {-{\left (3\,x+2\right )}^2}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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