Creation of a Reduction Formula using Trignometric Substitution?

Question

As the title states, I would greatly appreciate some guidance concerning the creation of a reduction formula for the following integral:

I have attempted the question myself. The following is the my attempt, which utilizes IBP. I am stuck at the simplification of the right-hand side integral, and am wondering if there is a way past this (or if I should use a different method altogether):

Thank you.

Answer 1

$$\int\frac{x^n+a^2x^{n-2}}{\sqrt{x^2+a^2}}\,dx=\int\frac{x^n}{\sqrt{x^2+a^2}}\,dx+a^2\int\frac{x^{n-2}}{\sqrt{x^2+a^2}}\,dx$$

Answer 2

You can bring the term $-(n-1)\int\frac{x^n}{\sqrt{x^2+a^2}}$ on the LHS and add it with the rest, so you get a recurrence.