# Nasty integral

Calculus Level 3

We have the following identity for positive values of $$a$$:

$\int_{0}^{\frac{a}{2}} \sqrt{x^{2}+ax} \text{ d}x=\frac{a^2}{4}\left(\sqrt{n_1}-\ln \left(\sqrt{n_2+\sqrt{n_3}}\right)\right)$

where $$n_1$$, $$n_2$$, and $$n_3$$ are positive integers. What is the value of $$n_1+n_2+n_3$$?

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