Optimal. Leaf size=42 \[ -\frac {3}{8} (3+2 x) \sqrt {9+12 x+4 x^2}+\frac {1}{12} \left (9+12 x+4 x^2\right )^{3/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {654, 623}
\begin {gather*} \frac {1}{12} \left (4 x^2+12 x+9\right )^{3/2}-\frac {3}{8} (2 x+3) \sqrt {4 x^2+12 x+9} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
Rule 654
Rubi steps
\begin {align*} \int x \sqrt {9+12 x+4 x^2} \, dx &=\frac {1}{12} \left (9+12 x+4 x^2\right )^{3/2}-\frac {3}{2} \int \sqrt {9+12 x+4 x^2} \, dx\\ &=-\frac {3}{8} (3+2 x) \sqrt {9+12 x+4 x^2}+\frac {1}{12} \left (9+12 x+4 x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 30, normalized size = 0.71 \begin {gather*} \frac {x^2 \sqrt {(3+2 x)^2} (9+4 x)}{6 (3+2 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
2.
time = 0.43, size = 21, normalized size = 0.50
method | result | size |
default | \(\frac {\mathrm {csgn}\left (2 x +3\right ) \left (2 x +3\right )^{2} \left (-3+4 x \right )}{24}\) | \(21\) |
gosper | \(\frac {x^{2} \left (4 x +9\right ) \sqrt {\left (2 x +3\right )^{2}}}{12 x +18}\) | \(27\) |
risch | \(\frac {2 \sqrt {\left (2 x +3\right )^{2}}\, x^{3}}{3 \left (2 x +3\right )}+\frac {3 \sqrt {\left (2 x +3\right )^{2}}\, x^{2}}{2 \left (2 x +3\right )}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 44, normalized size = 1.05 \begin {gather*} \frac {1}{12} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} - \frac {3}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} x - \frac {9}{8} \, \sqrt {4 \, x^{2} + 12 \, x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.42, size = 11, normalized size = 0.26 \begin {gather*} \frac {2}{3} \, x^{3} + \frac {3}{2} \, x^{2} \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 x \sqrt {\left (2 x + 3\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.74, size = 31, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, x^{3} \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {3}{2} \, x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) - \frac {9}{8} \, \mathrm {sgn}\left (2 \, x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 23, normalized size = 0.55 \begin {gather*} \left (\frac {x^2}{3}+\frac {x}{4}-\frac {3}{8}\right )\,\sqrt {4\,x^2+12\,x+9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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