Geodesic
On the integral of a polynomial with multiple roots
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXII, Tome 482 (2019), pp. 28-44

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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. 
@article{ZNSL_2019_482_a2,
     author = {A. E. Guterman and S. V. Danielyan},
     title = {On the integral of a polynomial with multiple roots},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {28--44},
     publisher = {mathdoc},
     volume = {482},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_482_a2/}
}
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A. E. Guterman; S. V. Danielyan. On the integral of a polynomial with multiple roots. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXII, Tome 482 (2019), pp. 28-44. http://geodesic.mathdoc.fr/item/ZNSL_2019_482_a2/