National Federation rules.
A1 is dribbling near the sideline when he loses his balance and steps on the line or actually runs a few steps out of bounds. His hand is never in contact with the ball while he is out of bounds. The ball continues to bounce along inbounds. A1 regains his balance come inbounds and continues to bounce the already bouncing ball.(He does not pick up the bouncing ball) Is this a good play or is out of bounce the correct call here? I need a case book or rule book reference if possible. Thanks.

Judgement call.

1)If you think that the player lost control of the ball and couldn't immediately dribble, it's an interrupted dribble as per rule 4-15-5, and it's also not a violation as per R4-15-6(d).
2) If you think that the player could have continued dribbling after going OOB, then it's a violation as per rule 9-3NOTE.