ToHypergeometric[t^b (1 - t)^c (1+tz)^a,t] returns
u^a Gamma[b+1] Gamma[c+1]/Gamma[b+c+2] Hypergeometric2F1[-a,b+1,b+c+2,-z/u].
Remember that \textrm{Re}(b) >0 and
\textrm{Re} (c-b) > 0 should hold
(need not be set in Mathematica).
Overview, HypergeometricAC, HypergeometricIR, HypergeometricSE.
ToHypergeometric[t^b (1 - t)^c (1 + t z)^a, t]\frac{\Gamma (b+1) \Gamma (c+1) \, _2F_1(-a,b+1;b+c+2;-z)}{\Gamma (b+c+2)}
ToHypergeometric[w t^(b - 1) (1 - t)^(c - b - 1) (1 - t z)^-a, t]\frac{w \Gamma (b) \Gamma (c-b) \, _2F_1(a,b;c;z)}{\Gamma (c)}
ToHypergeometric[t^b (1 - t)^c (u + t z)^a, t]\frac{u^a \Gamma (b+1) \Gamma (c+1) \, _2F_1\left(-a,b+1;b+c+2;-\frac{z}{u}\right)}{\Gamma (b+c+2)}
ToHypergeometric[w t^(b - 1) (1 - t)^(c - b - 1) (u - t z)^-a, t]\frac{w u^{-a} \Gamma (b) \Gamma (c-b) \, _2F_1\left(a,b;c;\frac{z}{u}\right)}{\Gamma (c)}