HypergeometricIR[exp, t] substitutes for all Hypergeometric2F1[a,b,c,z] in exp by its Euler integral representation. The factor Integratedx[t, 0, 1] can be omitted by setting the option Integratedx -> False.
Overview, HypergeometricAC, HypergeometricSE, ToHypergeometric.
HypergeometricIR[Hypergeometric2F1[a, b, c, z], t]\frac{t^{b-1} \Gamma (c) (1-t z)^{-a} (1-t)^{-b+c-1}}{\Gamma (b) \Gamma (c-b)}
ToHypergeometric[t^b (1 - t)^c (1 + t z)^a, t]
HypergeometricIR[%, t]
\frac{\Gamma (b+1) \Gamma (c+1) \, _2F_1(-a,b+1;b+c+2;-z)}{\Gamma (b+c+2)}
t^b (1-t)^c (t z+1)^a