Evaluating an integral using Gegenbauer polynomials

Question

I want to evaluate the following integral $$\int \frac{(r-r'\cos\theta')^2r'^2\,dr'\sin\theta'\,d\theta'\,d\phi'}{(r^2+r'^2-2rr'\cos\theta')^{3/2}}$$ Working that a little bit i end up with this expression. $$\frac{2\pi}{r^3} \int_0^\pi d\theta'\frac{r^2 (1-\frac{r'}{r}\cos\theta')^2\sin\theta'}{(1+(\frac{r'}{r})^2-2\frac{r'}{r}\cos\theta' )^{3/2}}\int_0^R dr'\,r'^2$$ Using Gegenbauer polynomials to express the denominator i get $ \frac{2\pi}{r} \int_0^\pi d\theta'(1-\frac{r'}{r}\cos\theta')^2\sin\theta'\sum C_n(\cos\theta')(\frac{r'}{r})^n \int_0^R dr'r'^2$ Doing some more calculations i get the following integrals. $\int_{-1}^1dx \sum C_n(x)$ $\int_{-1}^1dx \sum C_n(x)x^2$ $\int_{-1}^1dx \sum C_n(x)x$
Which i dont know how to evaluate since I'm missing the weight function $(1-x^2)$ in the integrals