Limit + Summation = Integration

Calculus Level 5

$\large \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{n^2}{n^3+k^3} = \frac{\sqrt{a}\pi+\ln{b}}{c}$

If $$a$$, $$b$$ and $$c$$ are positive integers and $$a$$ is square-free, find $$a+b+c$$.

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