I got as far as:
$$\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right) $$ $$=\frac{2}{\sqrt{\pi}} \int dx \int^{c(x-b)}_0 dy {\sqrt{x^2+a}} e^{-A x^2 - y^2}$$ $$=\frac{-2 c}{\sqrt{\pi}} \int db \int dx {\sqrt{x^2+a}} e^{-A x^2 - c^2 (x-b)^2} $$ $$=\frac{-2 c}{\sqrt{\pi}} \int db e^{-b^2 \left\{ c^2 + \frac{c^4}{A+c^2} \right\} } \int dx {\sqrt{x^2+a}} e^{ - \left(A+c^2 \right) \left( x-\frac{b c^2}{A+c^2} \right)^2} $$
Hint:
$\int\sqrt{x^2+a}e^{-Ax^2}\text{erf}(c(x-b))~dx$
$=\dfrac{2}{\sqrt\pi}\int\sum\limits_{n=0}^\infty\dfrac{(-1)^nc^{2n+1}(x-b)^{2n+1}\sqrt{x^2+a}e^{-Ax^2}}{n!(2n+1)}~dx$
$=\dfrac{2}{\sqrt\pi}\int\sum\limits_{n=0}^\infty\sum\limits_{k=0}^{2n+1}\dfrac{(-1)^{n-k+1}(2n)!b^{2n-k+1}c^{2n+1}x^k\sqrt{x^2+a}e^{-Ax^2}}{n!k!(2n-k+1)!}~dx$