On the integral of a polynomial with multiple roots

A full integral of a polynomial is defined as its integral with the property that any multiple root of the polynomial is a root of this integral. The paper investigates relationships between the existence of a full integral and the form of a polynomial. In particular, it is proved that the full integral exists if the polynomial has no more than one multiple root. The full integral does not exist if the number of multiple roots strictly exceeds the number of simple roots increased by one.