How would a child that has adhd and asperger syndrome behave

Observable : Behaviors depend on whether he has hyperactivity/impulsivity with his inattention. Inattention, distractibility, being "off-task" often, forgetfulness, impact his academics. Hyperactivity/impulsivity contribute to disruptive behavior.Since the as already causes non-compliance & seemingly oppositional behaviors, proper diagnosis & treatment of adhd ( in 40-70% of kids with asd's) is vital.

Not very differently: Adhd would present similarly in all children however there would some difference in children with asperger's as impulsivity is likely to present with increase stereotypes which are part of asperger's. Other aspects of asperger's might also become more pronounced. Frustration tolerance which is an issue might become more pronounced. Treatment of adhd is more tricky as well as asperger's children.

A technical point: adhd symptoms really should be understood to be part and parcel of aspergers. In DSM, it is incorrect to make a second diagnosis of adhd if the child is already on the autism spectrum. Unless the child is so severely adhd that limits function or puts them at risk e.g. climbing so much or running that they injure themselves multiple times per week.