jmc

We first simplify each smaller fractional expression, by multiplying by the conjugate of the denominator: $$\frac{1}{\sqrt{2}+1}=\frac{1}{\sqrt{2}+1}\cdot \frac{\sqrt{2}-1}{\sqrt{2}-1}=\sqrt{2}-1$$and $$\frac{2}{\sqrt{3}-1}=\frac{2}{\sqrt{3}-1}\cdot \frac{\sqrt{3}+1}{\sqrt{3}+1}=\sqrt{3}+1.$$Therefore, the given expression becomes $$\frac{1}{(\sqrt{2}-1)+(\sqrt{3}+1)}=\frac{1}{\sqrt{2}+\sqrt{3}}.$$Multiplying by the conjugate one more time, we have $$\frac{1}{\sqrt{2}+\sqrt{3}}=\frac{1}{\sqrt{2}+\sqrt{3}}\cdot \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}=\boxed{\sqrt{3}-\sqrt{2}}.$$