Vector Calculus · Superchallenge problem

Consider all ellipses in the plane that are centred at the origin, enclose area $\pi$, and may point in any direction; each is traversed once counterclockwise. Find the smallest possible value of $\displaystyle\oint_E -y^3\,dx + \left(x^3 + x^2y\right)dy$ over all such ellipses $E$. Give your answer in terms of $\pi$.