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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