My car sputters at idle in gear, and when I accelerate, especially around 45-55 miles per hour. When I accelerate hard, it will cough and sputter, then will accelerate normally, and run smoothly until I stop putting pressure on the gas peddle. Why?

The problem is intermittent. At idle if the engine is missing, I put the car in neutral and it idles perfectly until i put it in gear. Yes, it is an automatic. The fuel pump and filter are new. Spark plugs have 10K on them. When around 45 to 55 miles per hour and I am not accelerating hard, then do accelerate, it often sputters yet will work its way to smooth acceleration. The check engine light is not on, yet it seems to linger longer than usual after I start the car.