longdog after a bitch
To proceed with something right away; in a hurry.
When Harry found out he won the lottery, he quit his job and took off like a longdog after a bitch to collect his winnings.

A longdog a crossbreed between two sightdogs (example: Saluki x Greyhound - Deerhound x Greyhound - Whippet x Greyhound)

Bitch is a female dog.