$$ \text R(x)= \text A x^{\left(\frac{\lambda+1}{2}\right)} e^{-\eta x / 2} \text F_{1}\left(-n, \lambda+\frac{3}{2}, x\right) $$
where $\text A$ is normalization constant. Using the normalization condition and the relation between the Laguerre polynomials and confluent hypergeometric function, the normalization constant $\text A$ is given by
$$ \text A=\eta^{\left(\frac{\lambda+3 / 2}{2}\right)} \sqrt{\frac{2 \Gamma\left(n+\lambda+\frac{3}{2}\right)}{n !}}\left[\Gamma\left(\lambda+\frac{3}{2}\right)\right]^{-1} $$
where we have applied the following relation
$$ \int_{0}^{\infty} t^{q} e^{-t} \text L_{n}^{q}(t) \text L_{n^{\prime}}^{q}(t) d t=\frac{\Gamma(n+q+1)}{n !} \delta n n^{\prime} $$