How can I find the minimum of $F(\vec{x}) = \sum_{i=1}^{n} (a_i x_i)^2 \left(\sum_{i \neq j}^{n} (a_i x_i) (a_j x_j)\right)^{-1}$?

Question

I am trying to find the critical points of $F$. Computing the $k$th partial derivative of $F$ I get $$\frac{\partial F}{\partial x_k} = 2a_kx_k \cdot \left(\sum_{i \neq j} (a_i x_i)(a_j x_j)\right)^{-1} - \sum_{i=1}^n (a_i x_i)^2 \left(\sum_{i \neq j}^n (a_i x_i) (a_j x_j)\right)^{-2}\sum_{i \neq k} 2 a_k a_i x_i.$$ Solving for $DF(\vec{x}) = 0$ seems very unwieldy. How should I find the minimizer of $F$?