A single insert statement is taking, occasionally, more than 2 seconds. The inserts are potentially concurrent, as it depends on our site traffic which can result in 200 inserts per minute.
The table has more than 150M rows, 4 indexes and is accessed using a simple select statement for reporting purposes.
How to speed up the inserts considering that all indexes are required?
You haven't provided many details but it seems like you need partitions.
An insertion operation in an database index has, in general, an O(logN) time complexity where N is the number of rows in the table. If your table is really huge even logN may become too much.
So, to address that scalability issue you can make use of index partitions to transparently split up your table indexes in smaller internal pieces and reduce that N without changing your application or SQL scripts.
https://dev.mysql.com/doc/refman/5.7/en/partitioning-overview.html
[EDIT]
Considering information initially added in the comments and now updated in the question itself.
There are a few not mutually exclusive improvements:
Check the output of EXPLAIN for that SELECT and remove indexes not being used, or, otherwise, combine them in a single index.
Make the inserts in batch
Partitioning still an option.
Alternatively, change your approach: save the data to a nosql database like redis and populate the mysql table asynchronously for reporting purpose.