The following functional $$ J[F]:=\int_0^\infty dF(x_1)\int_0^\infty dF(x_2) \sqrt{1+\frac{x_1}{a+x_2}} $$ must be maximized with respect to $F$, which is supposed to be the probability distribution of a random variable with mean $\mu$. Assume $a>0$.
My approach is to compute the first variation of the Lagrangian $$ J[F]-\lambda_1 \left( \int_0^\infty dF(x_1)-1 \right)-\lambda_2 \left( \int_0^\infty dF(x_1) x_1 - \mu \right) $$ and the problem reduces to finding the solution of the integral equation $$ C_0+C_1 z = \int_0^\infty dF(x) \left(\sqrt{1+\frac{z}{a+x}} -\sqrt{1+\frac{x}{a+z}} \right) $$ for some $C_0$ and $C_1$.
Problem is: How to find the solution of this integral equation? Any hint is welcome on this or related problem.