Optimal. Leaf size=23 \[ \frac{1}{12} (3 x+2) \left (9 x^2+12 x+4\right )^{3/2} \]
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Rubi [A] time = 0.0028427, 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} \[ \frac{1}{12} (3 x+2) \left (9 x^2+12 x+4\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin{align*} \int \left (4+12 x+9 x^2\right )^{3/2} \, dx &=\frac{1}{12} (2+3 x) \left (4+12 x+9 x^2\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0087792, size = 20, normalized size = 0.87 \[ \frac{1}{12} (3 x+2) \left ((3 x+2)^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 35, normalized size = 1.5 \begin{align*}{\frac{x \left ( 27\,{x}^{3}+72\,{x}^{2}+72\,x+32 \right ) }{4\, \left ( 2+3\,x \right ) ^{3}} \left ( \left ( 2+3\,x \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70056, size = 41, normalized size = 1.78 \begin{align*} \frac{1}{4} \,{\left (9 \, x^{2} + 12 \, x + 4\right )}^{\frac{3}{2}} x + \frac{1}{6} \,{\left (9 \, x^{2} + 12 \, x + 4\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08727, size = 46, normalized size = 2. \begin{align*} \frac{27}{4} \, x^{4} + 18 \, x^{3} + 18 \, x^{2} + 8 \, 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 \left (9 x^{2} + 12 x + 4\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20392, size = 69, normalized size = 3. \begin{align*} \frac{27}{4} \, x^{4} \mathrm{sgn}\left (3 \, x + 2\right ) + 18 \, x^{3} \mathrm{sgn}\left (3 \, x + 2\right ) + 18 \, x^{2} \mathrm{sgn}\left (3 \, x + 2\right ) + 8 \, x \mathrm{sgn}\left (3 \, x + 2\right ) + \frac{4}{3} \, \mathrm{sgn}\left (3 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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