Material from this lecture will be fair game for the next assignment and exam. You may look in other sources or seek my assistance. For astro majors and minors only. Marion and Thornton do not ask you to show that the resulting integral really is an elliptic integral, so you do not need to manipulate your expression into one of the standard forms for an elliptic integral. You do need to do the expansion.
Section II.B and requested screen capture and program list.