Is there a fast and accurate way to evaluate integrals of the form: $$\int_0^r dz\, e^{i z} j_n(z)$$ where $j_n(z)$ is the spherical Bessel function of order $n$, and $r>0$? Mathematica yields a hypergeometric function. This evaluates slowly. It also exhibits some numerical imperfections for large $n$ and $r$: for example, for $n=21$, in the range $120<r<140$.
FWIW, although the above integral increases without bound as $r\rightarrow\infty$, I wish to evaluate: $$\int_0^r dz\, e^{i z} \left(a j_{n-2}(z) + b j_{n}(z) + c j_{n+2}(z)\right)$$ where $b=a+c$. This approaches a constant as $r\rightarrow \infty$, because $j_{n}$ on the one hand, and $j_{n-2}$ and $j_{n+2}$ on the other, have opposite phase at large $z$.