According to OSM I would need 8 days and 23 h !!
↗23153m ↘23263m

Countries crossed : Switzerland, Germany, Austria, Slovakia, Hungary, Croatia, Serbia, Bulgaria, Romania

The stages, basically ... as they were used to determine this route.

Toulouse, France
Lyon, France
Mulhouse, France
Basel, Switzerland
Passau, Germany
Linz, Austria
Vienna, Austria
Budapest, Hungary
Mohács, Hungary
Belgrade, Serbia
Negotin, Negotin, Serbia
Брегово, Bulgaria
Svichtov, Bulgaria
Podul Prieteniei Giurgiu-Ruse, Romania
Constanţa, Romania

The contour lines, very indicative (there is normally not the second very short peak at 500m)