Mixed cubic and quadratic Laplace endpoints ($I(\lambda )=A{\lambda }^{-1/3}+B{\lambda }^{-1/2}+O({\lambda }^{-1})$)

If a nonnegative smooth phase has an isolated cubic zero at one endpoint and an isolated quadratic zero at the other, their leading Laplace's method contributions scale respectively as ${\lambda }^{-1/3}$ and ${\lambda }^{-1/2}$. The first two global terms therefore come from different endpoints, rather than from the first two terms at the dominant endpoint. When the cubic endpoint has no quartic phase term and constant amplitude, its next local term starts at ${\lambda }^{-1}$; cubic corrections at the quadratic endpoint also start at that order.