Why is the moon's motion more complicated than that of those planets? I am of course not an expert on astronomy. But from the literature, I have the impression that the lunar motion is very complicated. It seems that Kepler can make sense of the motion of the planets but cannot do that for the moon. 

So, why is the moon's motion more complicated than that of planets like Mars?
Keplerian motion would be a correct way to explain the Moon's orbit about the Earth if the Sun, Venus, Jupiter, and all the other planets weren't present, if the Moon wasn't slowly receding from the Earth due to tidal acceleration, and if Newtonian gravity properly described the underlying physics. None of these is the case.

In the case of modeling the behaviors of the major objects that comprise the solar system (i.e., the Sun and the eight planets), perturbations involving planets interacting with one another, particularly with Jupiter, make the orbits of the planets slightly non-Keplerian. That the universe is relativistic rather than Newtonian famously makes Mercury's orbit precess in a way that even accounting for planetary perturbations cannot explain. That said, Keplerian motion is a fairly good approximation with regard to the Sun and the eight planets.

The same is not true for the Moon. It's orbit is notably non-Keplerian, even over the course of just a month. Ptolemy (after the fact) and Brahe (also after the fact) both noticed irregularities in the Moon's motion that cannot be explained in terms of Keplerian elliptical orbits in which area swept out per unit time is constant.

Newton explained some of these effects by noticing that the Sun is a huge perturber of the Moon's orbit about the Earth. Newton was not quite successful in this endeavor; the mathematical tools needed to model the Moon's motion hadn't been developed. The full development of classical lunar theory took two hundred years after Newton's time.

Another complicating factor is that the Moon is gradually receding from the Earth due to tidal acceleration. That the Moon is slowly receding from the Earth was first hypothesized by Halley, one of Newton's contemporaries. A century after the first publication of Newton's Principia, Laplace erroneously attributed this lunar recession solely to perturbations from the Sun. In the next hundred years, others corrected Laplace's calculations and found that solar perturbations are unable to explain this recession. George Darwin (Charles' son) explained the recession in terms of the tides. The full explanation? People are still writing papers on this subject.

Relativity makes for yet another complication. Relativistic effects on the Moon are very hard to see given the many orders of magnitude larger complications due to perturbations by the Sun, Venus, Jupiter, ..., and tidal acceleration.