Question #41776

Consider the curve $C: y = 2 + \int_{1}^{x} \frac{6t^2}{t^3+1} dt$. Let $L_1$ and $L_2$ be the tangent and normal lines to $C$ at the point $P(1, 2)$. Let $S$ be the feasible region in the first quadrant defined by the half-planes containing the origin bounded by $L_1$ and $L_2$. An objective function $Z = c x + d y$, where $c$ and $d$ are coprime positive integers, is maximized over $S$. If the maximum value $M$ of $Z$ is attained at infinitely many points on the boundary of $S$, what is the value of $c^2 + d^2 + M^2$?