If you count 8ths, you are right that it is divided in 3/8 plus 5/8, but this is not the time signature, it's only the rhythm within figures that adds up to a repetitive pattern. If you add those two together you will get 8/8, and as a time signature, it's more natural to use 4/4 instead, since it's not very common to use 8ths for the time signature.
The fact that the pattern is built up around 3/8 plus 5/8, just makes the rhythm more interesting — it has little to do with the time signature. This way of braking with the basic beat where notes "hang over" and the next weighted note starts in the middle of next beat is called syncopation. We say that the notes are syncopated. This is a quite common method to make the rhythm more interesting.
There are are some special cases, where you have a phrase with odd numbered eighths, like a pattern that matches 7/8 and cannot be converted to x/4 (the pattern in your case is really 8/8).
Otherwise, the answer and comments by @MatthewRead are very good with details on this.
I listened some more, and I see that after the first "versey part", there are triplets to add to the confusion. Triplets are basically 3 notes distributed evenly over the same amount of time that normally 2 notes would fit. So eights triplets would be 3 notes in the space of 2 eights, which is in our case 3 notes evenly distributed in one beat in a 4/4 time signature. You also have later quarter triplets, which is 3 notes over 2 quarters. This is harder to get right because it feels like it doesn't quite follow the ground beat. You have to identify these patterns comparing to the ground beat, which is the steady 4/4 beat you hear throughout the whole thing in the bass. You need to notice, though, that during the quarter triplets, the bass beat is silenced a bit (still there, though), and even some times the bass goes out and follows the triplets instead ( that does not mean that the time signature changes, though...)