Summation?

The limit $$\lim\limits_{n \to \infty} \left( \dfrac{n}{n^2 + 3} + \dfrac{n}{n^2 + 12} + \dfrac{n}{n^2 + 27} + \ldots + \dfrac{1}{4n} \right) = \dfrac{a\sqrt{b}\pi}{c}$$ for some positive integers $a, b, c$ such that $\gcd(a, c) = 1$ and $b$ is squarefree. Find $a + b + c$.