TIP122 or Relay + H-Bridge

I'm building a project that controls a DC motor, which needs to stay on for 30 - 60 minutes. I made it work with a TIP122 transistor and it's working fine. Now I'm asking you if it will be more "efficient" to use a on-off (set/reset) relay, so I don't have power consumption and voltage drop because of the transistor (I'll be running this on 4 AA baterries).
Since the relay I have needs polarity inversion to switch between states, I will also need an H-bridge to control it via Arduino: will this configuration save battery power or it will be negligile so I better stay with the TIP122?

Thank you

You're probably better to go with a logic-level mosfet.