I am worried and want to be safe

I just had sex for the first time (it was my bfs first time too), the first time he came after he pulled out and the condom didn't break or leak or anything. The second time (with a new condom) he came but didn't pull out, he said the it slide a tiny bit cause he went limp. The condom again didn't break, but I'm worried that some sperm may have came out in me. I start birth control a week ago.