How to integrate:
$\displaystyle _{2}F_{1}(B;C;D;Ex^{2})\,Ax$
where $\displaystyle _{2}F_{1}(...)$ is the hypergeometric function, x is the independent variable and A, B, C, D, and E are constants.
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