Integrate: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$

Question

I am trying to solve the integral: $$ \int_{0}^{\infty}{x^{n-2} \over b\,\left(1 + ax^{\frac{n - 1}{n - 2}}\right)}\,\sin\left(bx\right)\,\mathrm{d}x \quad \mbox{where}\ x \in \mathbb{R}\ \mbox{and}\ a, b, n\ \mbox{are positive real constants.} $$

I have tried several things, including adding a complex phase and taking the vanishing limit and some substitutions, but I have not been successful so far. Mathematica is also confused by the integral.

Any ideas on how to go about doing this ?. Are there any obvious methods to solve this ( or even just write in terms of gamma functions or something workable ) that I have been overlooking ?.

Thanks !.