In regards to posts 60 and posts 63 above, I did a calculation assuming the total energy of the rocket stays constant using the gravitational potentials of the earth and moon, and that calculation showed the rocket should speed up considerably, up to about 2600 m.p.h. I determined that calculation to be in error though.
The moon, the source of the second gravitational potential was moving at about 2200 m.p.h. Thereby, the rocket didn't move across the gravitational potential enough (in the reference frame of the earth) to get the complete ## dW=\vec{F} \cdot d \vec{s}=\Delta(mv^2/2) ##. Instead, much of the closure in distance was because of the motion of the moon. The way I initially treated it as a static/fixed gravitational potential was incorrect. To apply the method of gravitational potential to get the change in ## v^2 ##, it seems it is necessary to go to the reference frame of the moon. Thereby I have concluded that the nasa data showing no increase in speed measured w.r.t. the earth, (the speed stayed nearly steady at about one thousand m.p.h.), even in its closest approach to the moon, was apparently completely valid.
I thought I was somewhat clever in applying the use of two gravitational potentials to compute the ## v ##, but it seems that if one or both of the two sources are moving, the method can not be used to compute the kinetic energy of the 3rd body in the simple straightforward manner that I tried to employ.
Note: This was the first time I ever tried to work a problem involving two gravitational potentials at the same time. It would have worked if the moon were stationary, but it isn't.