Optimal. Leaf size=42 \[ \frac{1}{12} \left (4 x^2+12 x+9\right )^{3/2}-\frac{3}{8} (2 x+3) \sqrt{4 x^2+12 x+9} \]
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Rubi [A] time = 0.0079994, antiderivative size = 42, normalized size of antiderivative = 1., 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 = {640, 609} \[ \frac{1}{12} \left (4 x^2+12 x+9\right )^{3/2}-\frac{3}{8} (2 x+3) \sqrt{4 x^2+12 x+9} \]
Antiderivative was successfully verified.
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Rule 640
Rule 609
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*}
Mathematica [A] time = 0.0064007, size = 30, normalized size = 0.71 \[ \frac{x^2 \sqrt{(2 x+3)^2} (4 x+9)}{6 (2 x+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 27, normalized size = 0.6 \begin{align*}{\frac{{x}^{2} \left ( 4\,x+9 \right ) }{18+12\,x}\sqrt{ \left ( 3+2\,x \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70717, size = 59, normalized size = 1.4 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53914, size = 26, normalized size = 0.62 \begin{align*} \frac{2}{3} \, x^{3} + \frac{3}{2} \, x^{2} \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 x \sqrt{\left (2 x + 3\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31284, size = 42, normalized size = 1. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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