make the sub

$u=e x^2$

$du=2ex~dx$

and find

$\dfrac a {2e} \displaystyle{\int} _2F_1(b,c,d,u) ~du=\dfrac a {2e} \frac{\Gamma (d) \left(\Gamma (d-1) \, _2\tilde{F}_1(b-1,c-1;d-1;u)-1\right)}{(b-1) (c-1) \Gamma (d-1)}$

$_2\tilde{F}_1(b,c,d,u)$ is the http://reference.wolfram.com/mathema...gularized.html