Help on finding the closed form of the integral

Question

Can anyone help me to find closed solution of the integral $$\int_0^{1-e^{-\lambda x}}\frac{u^{b-1}\,(1-u)^{a+c-1}}{[1-(1-e^{-\lambda_1 t_1})u]^{a+b+c}}\,{\rm d}u,$$

where $a,b,c,t_1,\lambda,\lambda_1>0\,.$

Answer 1

Hint. You have an Euler-type integral, you may then rewrite your initial integral in terms of the Appel hypergeometric function $$ F_1(a,b_1,b_2,c;x,y)=\dfrac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_0^1\frac{v^{a-1}(1-v)^{c-a-1}}{(1-xv)^{-b_1}(1-yv)^{-b_2}}~dv $$ where $\Re c>\Re a>0$ with the change of variable $$ v=\frac{u}{1-e^{-\lambda x}}. $$