ASM1166 is a 6 port controller, the one you linked has 8 pots, suggesting it has a port multiplier and so not recommended.
LSI is a safer bet since they are made by LSI (or Avago/Broadcom now), unless you get a fake one, these Asmedia and JMB controllers are different, the chips themselves are good but a no brand controller based on one might be good or not, when possible suggest buying from known brands, like Startech or Syba for example.