Child is laughing and playing around.
The child is having a good time and cracking up.
Laughing and playing around implies having a good time.
Cracking up means laughing, and playing around is a way of having a good time.
Cracking up is another way of saying laughing; playing implies one is having a good time; so, child having a good time and cracking up is a rephrasing of child is laughing and playing around.