jmc

The common denominator desired is $\sqrt{5}\cdot \sqrt{7}=\sqrt{35}$. So, this expression becomes $$\frac{\sqrt{5}\cdot (\sqrt{5}\cdot \sqrt{7})+1\cdot \sqrt{7}+\sqrt{7}\cdot (\sqrt{5}\cdot \sqrt{7})+1\cdot \sqrt{5}}{\sqrt{35}}.$$Simplifying this gives $$\frac{5\sqrt{7}+\sqrt{7}+7\sqrt{5}+\sqrt{5}}{\sqrt{35}}=\frac{6\sqrt{7}+8\sqrt{5}}{\sqrt{35}}.$$To rationalize, multiply numerator and denominator by $\sqrt{35}$ to get $$\frac{6\sqrt{7}\sqrt{35}+8\sqrt{5}\sqrt{35}}{35}.$$Simplifying yields $\frac{42\sqrt{5}+40\sqrt{7}}{35}$, so the desired sum is $42+40+35=\boxed{117}$.