Vector Calculus · Challenge problem

For a point $(a, 0)$ let $\mathbf V_a = \left(\dfrac{-y}{(x - a)^2 + y^2},\; \dfrac{x - a}{(x - a)^2 + y^2}\right)$, and let $\mathbf F = \mathbf V_0 + 2\,\mathbf V_3 + (0,\; x)$. Evaluate $\displaystyle\oint_C\mathbf F\cdot d\mathbf r$, where $C$ is the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{4} = 1$ traversed once counterclockwise. Give your answer in terms of $\pi$.