This question is predicated on two false beliefs: First, that open immigration would immediately result in mass immigration, a possible but not likely outcome. Second, that "effectively integrated" should be the goal of permitting more immigrants.
If the federal government were to open its borders tomorrow morning, there would likely be an initial influx of new immigrants, which would fairly quickly increase the costs of moving for those who weren't quite as quick to immigrate. Markets would adjust, and immigration would stabilize.
Whether or not those immigrants would effectively integrate into their new host country depends on a lot of variables. While immigrants themselves sometimes struggle to adapt to their new society beyond survival, their children often thrive, and third-generation immigrants are quite well-integrated into society. While the "how" is an important question, it seems that American culture has proven for centuries that it can handle the adjustment many immigrants bring.
This question is from Faith Seeking Freedom: Updated & Expanded, available in paperback, PDF, and Kindle.