Star formation rates (SFRs) larger than 1000 M&sun; yr-1 are observed in extreme starbursts. This leads to the formation of star clusters with masses >106 M&sun; in which crowding of the pre-stellar cores may lead to a change of the stellar initial mass function (IMF). Indeed, the large mass-to-light ratios of ultracompact dwarf galaxies and recent results on globular clusters suggest the IMF to become top-heavy with increasing star-forming density. We explore the implications of top-heavy IMFs in these very massive and compact systems for the integrated galactic initial mass function (IGIMF), which is the galaxy-wide IMF, in dependence of the SFR of galaxies. The resulting IGIMFs can have slopes, α3, for stars more massive than about 1 M&sun; between 1.5 and the Salpeter slope of 2.3 for an embedded cluster mass function (ECMF) slope (β) of 2.0, but only if the ECMF has no low-mass clusters in galaxies with major starbursts. Alternatively, β would have to decrease with increasing SFR > 10 M&sun; yr-1 such that galaxies with major starbursts have a top-heavy ECMF. The resulting IGIMFs are within the range of observationally deduced IMF variations with redshift.