Sorry for answering this so late ... I have only just seen the question (by chance).
The Pearson family is conventionally defined in terms of 4 parameters (not 5). Solving the Pearson coefficients in terms of moments involves assuming the density tends to zero at the extremum of the tails, solving a recurrence relation, and is a little bit of work, but Mathematica does it very nicely indeed. See section 5.2D of our book, "Mathematical Statistics with Mathematica" which provides the derivation; if interested, a free download is available here:
Section 5.2E of Chapter 5 extends to a 5-parameter model based on a third-order (cubic) polynomial, instead of the standard second-order (quadratic) model ... and the same method works just as nicely.