For 5 mths old, we did a lot of crawling, singing songs, walking at parks, reading, outings with other babies, balls catching, rolling on the bed activities, listening to nursery songs, watching children playing at playground. My 5 mths old was v active too so I am always on the go. V tiring but that's the stage where they want to explore. We went library to borrow those baby books with mirror and those felt materials and that helped to occupy her for a while. Children are curious creatures. I gave her a mirror (with supervision) and she was surprised to see herself in it and it bought me some time.
Good thing that he is very active. But when he is very active, he shouldn't be demanding more attention from you. It seems that he is not occupied more by himself. Train him those wooden puzzles or stacking cups. It will keep him occupied for longer times. Give him those musical guitar or piano toys - sounds and lights attract him for longer time. And forget not, when he is such active in such games, he will get tired soon and will need feedings frequently and sleeps more. And finally you get some time for yourself.
bring the baby shopping