1. Sustitución trigonométrica:
La expresión $(2 + 3x^2)$ sugiere el uso de la identidad $1 + \tan^2 \theta = \sec^2 \theta$.
Sea $\sqrt{3}x = \sqrt{2} \tan \theta \implies x = \frac{\sqrt{2}}{\sqrt{3}} \tan \theta$.
Diferencial: $dx = \frac{\sqrt{2}}{\sqrt{3}} \sec^2 \theta \, d\theta$.

2. Transformación del denominador:
$$ (2 + 3x^2)^{3/2} = (2 + 2\tan^2 \theta)^{3/2} = (2\sec^2 \theta)^{3/2} = 2\sqrt{2} \sec^3 \theta $$

3. Integración:
$$ \int \frac{\frac{\sqrt{2}}{\sqrt{3}} \sec^2 \theta}{2\sqrt{2} \sec^3 \theta} d\theta = \frac{1}{2\sqrt{3}} \int \frac{1}{\sec \theta} d\theta = \frac{1}{2\sqrt{3}} \int \cos \theta \, d\theta $$
$$ = \frac{1}{2\sqrt{3}} \sin \theta + C $$

4. Retorno a $x$:
Si $\tan \theta = \frac{\sqrt{3}x}{\sqrt{2}}$, entonces $\sin \theta = \frac{\sqrt{3}x}{\sqrt{2 + 3x^2}}$.
$$ \frac{1}{2\sqrt{3}} \cdot \frac{\sqrt{3}x}{\sqrt{2 + 3x^2}} + C = \frac{x}{2\sqrt{2 + 3x^2}} + C $$
$$ \boxed{\frac{x}{2\sqrt{2 + 3x^2}} + C} $$