It sounds more mainly like ADHD however you can have both ADHD and ASD (aspergers, they now use the term ASD under the new diagnosis)
I will warn you, you will struggle to get a diagnosis for a child who is functioning as well as yours, the average age in the UK is still around 9, it can be a very long road of years of appointments and tests. That however doesnt mean you shouldnt start or keep pushing if you feel there is something different from his peer group