Total Energy of Rocket Increasing in Earth-Moon System

Matt wrote:
So to summarise, the increase in the total energy of the rocket as it
approaches the Moon is due to the Moon's kinetic and potential energy
not being considered. The thing I'm having a problem with is that the
*only* way the kinetic energy of a rocket in freefall can increase is
when there is a corresponding decrease in the gravitational potential
between the rocket and the body in question (i.e. the Moon).

It's the _total_ energy of the Moon that compensates. (There's a tiny
contribution from the total energy of the Earth as well.) The
conserved quantity is the total energy of the system.

Also, where does this extra kinetic energy come from? It can't be due
to a decrease in the Moon's kinetic energy, as the Moon is also being
gravitationally attracted to the rocket. The rocket's trajectory
takes
it just ahead of where the Moon will be in its orbital path, so the
Moon's kinetic energy will increase rather then decrease.

It sounds as though the Moon's potential energy should increase and
its kinetic energy should decrease. This would make the rocket's
total energy decrease. It's hard to say for sure, though, without
knowing more details. Alternatively, I may simply be confused.
Orbits are tricky!