What substitution do I use to integrate this?

Use the so-called Weierstrass substitution $$\cos(x)=\frac{1-t^2}{1+t^2}$$ and $$dx=\frac{2}{1+t^2}dt$$

This function may not have an elementary antiderivative but we can use special functions to evaluate it $$\begin{align} I=&\int\frac{\mathrm dx}{\sqrt{3+2\cos x}}\\ =&\frac1{\sqrt5}\int\frac{\mathrm dx}{\sqrt{1-\frac45\sin^2(x/2)}}\\ =&\frac2{\sqrt5}\int\frac{\mathrm du}{\sqrt{1-\frac45\sin^2u}}\\ \end{align}$$ Where $u=x/2$. Recall the definition of the incomplete elliptic integral of the first kind $$\mathrm F(\phi,k)=\int_0^\phi\frac{\mathrm{d}x}{\sqrt{1-k\sin^2x}}$$ So we immediately have that $$I=\frac2{\sqrt5}\mathrm{F}\bigg(\frac{x}2,\frac45\bigg)$$