HypInt[exp, t] substitutes all Hypergeometric2F1[a,b,c,z] in exp with Gamma[c]/(Gamma[b] Gamma[c-b]) Integratedx[t,0,1] t^(b-1) (1-t)^(c-b-1) (1-t z)^(-a).
Hypergeometric2F1[a, b, c, z]
HypInt[%, t]\, _2F_1(a,b;c;z)
\frac{t^{b-1} \Gamma (c) (1-t z)^{-a} (1-t)^{-b+c-1} \int _0^1dt\, }{\Gamma (b) \Gamma (c-b)}