Satisfied after raising sleuth's rates (5)
I believe the answer is:
tempi
'rates' is the definition.
(tempo is a kind of rate)
'satisfied after raising sleuth's' is the wordplay.
'satisfied after raising' becomes 'tem' (I can't explain this - if you can you should believe this answer much more).
'sleuth' becomes 'pi' (private investigator).
'tem'+'pi'='TEMPI'
Can you help me to learn more?
(Other definitions for tempi that I've seen before include "Times" , "Speeds, rhythms of musical pieces" , "Rates" , "Beats" , "Speeds for musicians" .)