No, social integration in the EU is not successful. Several European countries have fallen into debt because of immigration. The free healthcare systems in Europe cannot handle all of the immigrants plus their own citizens. There are not enough jobs to go around, which has led to poverty and a rise in the crime rate.
Although the EU is supposed to be a union in which people are free to travel from country to country, people still frown upon immigrants from eastern European countries in western Europe. Also, while many Europeans are multilingual, there is not a language or culture of the EU, common among all people.