I have to solve some integrals of the form: $$\int_0^{x_0} dx \, j_n( R ) \cdot \frac{p(x)}{R^n}$$ where $R=\sqrt{x^2 + 2 a c x + c^2}$ , $j_n$ is the spherical Bessel function of order n, p(x) is a polynomial and $|a|<1$.
I have tried changing variable from x to R but this doesn't work: e.g. for the simple case $n=0$ and $p(x)=1$, I obtain something like: $$\int dR \, j_0( R ) \cdot \frac{R}{\sqrt{(R-bc )(R+bc)}} = \int dR \, \frac{\sin(R)}{\sqrt{(R-bc )(R+bc)}}$$ with $b=(1-a^2)$.
Similar integrals of the type $$\int_0^{x_0} dx \, \frac{ j_n( x ) }{(x)^m}$$ with $m \le n$ are easily solvable expressing the Bessel functions in terms of sine and cosine and then integrating by parts, but, with the radical in the argument of the Bessel function, things get harder. Do you have any idea to manage that?