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).
HypInt[exp, t]
Hypergeometric2F1[a,b,c,z]
exp
Gamma[c]/(Gamma[b] Gamma[c-b]) Integratedx[t,0,1] t^(b-1) (1-t)^(c-b-1) (1-t z)^(-a)
Overview, Series2.
Hypergeometric2F1[a, b, c, z] HypInt[%, t]
\, _2F_1(a,b;c;z)
tb−1Γ(c)(1−tz)−a(1−t)−b+c−1∫01dt Γ(b)Γ(c−b)\frac{t^{b-1} \Gamma (c) (1-t z)^{-a} (1-t)^{-b+c-1} \int _0^1dt\, }{\Gamma (b) \Gamma (c-b)}Γ(b)Γ(c−b)tb−1Γ(c)(1−tz)−a(1−t)−b+c−1∫01dt