Optimal. Leaf size=25 \[ -\frac{1}{6 (3 x+2) \sqrt{9 x^2+12 x+4}} \]
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Rubi [A] time = 0.0026011, antiderivative size = 25, 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 = {607} \[ -\frac{1}{6 (3 x+2) \sqrt{9 x^2+12 x+4}} \]
Antiderivative was successfully verified.
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Rule 607
Rubi steps
\begin{align*} \int \frac{1}{\left (4+12 x+9 x^2\right )^{3/2}} \, dx &=-\frac{1}{6 (2+3 x) \sqrt{4+12 x+9 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0056801, size = 20, normalized size = 0.8 \[ -\frac{3 x+2}{6 \left ((3 x+2)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 17, normalized size = 0.7 \begin{align*} -{\frac{2+3\,x}{6} \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 = 1.6986, size = 12, normalized size = 0.48 \begin{align*} -\frac{1}{6 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05615, size = 34, normalized size = 1.36 \begin{align*} -\frac{1}{6 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \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 \frac{1}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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