Optimal. Leaf size=18 \[ -\frac{6-x}{2 \sqrt{x^2+4}} \]
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Rubi [A] time = 0.0040183, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {637} \[ -\frac{6-x}{2 \sqrt{x^2+4}} \]
Antiderivative was successfully verified.
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Rule 637
Rubi steps
\begin{align*} \int \frac{2+3 x}{\left (4+x^2\right )^{3/2}} \, dx &=-\frac{6-x}{2 \sqrt{4+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0091034, size = 16, normalized size = 0.89 \[ \frac{x-6}{2 \sqrt{x^2+4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 13, normalized size = 0.7 \begin{align*}{\frac{x-6}{2}{\frac{1}{\sqrt{{x}^{2}+4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20106, size = 27, normalized size = 1.5 \begin{align*} \frac{x}{2 \, \sqrt{x^{2} + 4}} - \frac{3}{\sqrt{x^{2} + 4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.80335, size = 66, normalized size = 3.67 \begin{align*} \frac{x^{2} + \sqrt{x^{2} + 4}{\left (x - 6\right )} + 4}{2 \,{\left (x^{2} + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.66411, size = 20, normalized size = 1.11 \begin{align*} \frac{x}{2 \sqrt{x^{2} + 4}} - \frac{3}{\sqrt{x^{2} + 4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.20806, size = 16, normalized size = 0.89 \begin{align*} \frac{x - 6}{2 \, \sqrt{x^{2} + 4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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