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 = 0.00, 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} \[ \frac {1}{12} (3 x+2) \left (9 x^2+12 x+4\right )^{3/2} \]
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 1.04, size = 19, normalized size = 0.83 \[ \frac {27}{4} \, x^{4} + 18 \, x^{3} + 18 \, x^{2} + 8 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 45, normalized size = 1.96 \[ \frac {3}{4} \, {\left (3 \, x^{2} + 4 \, x\right )}^{2} \mathrm {sgn}\left (3 \, x + 2\right ) + 2 \, {\left (3 \, x^{2} + 4 \, x\right )} \mathrm {sgn}\left (3 \, x + 2\right ) + \frac {4}{3} \, \mathrm {sgn}\left (3 \, x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 1.52 \[ \frac {\left (27 x^{3}+72 x^{2}+72 x +32\right ) \left (\left (3 x +2\right )^{2}\right )^{\frac {3}{2}} x}{4 \left (3 x +2\right )^{3}} \]
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 \[ \frac {1}{4} \, {\left (9 \, x^{2} + 12 \, x + 4\right )}^{\frac {3}{2}} x + \frac {1}{6} \, {\left (9 \, x^{2} + 12 \, x + 4\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 19, normalized size = 0.83 \[ \frac {\left (9\,x+6\right )\,{\left (9\,x^2+12\,x+4\right )}^{3/2}}{36} \]
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 \[ \int \left (9 x^{2} + 12 x + 4\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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