Closed-form Integral Solution without Hypergeometric2F1Regularized ! ! !

*To*: mathgroup at smc.vnet.net*Subject*: [mg70219] Closed-form Integral Solution without Hypergeometric2F1Regularized ! ! !*From*: "Jeffrey Tan" <M.L.Tan at open.ac.uk>*Date*: Sat, 7 Oct 2006 07:09:06 -0400 (EDT)

Dear All, I am trying to evaluate the definite integral of the following function, but encountered the problem of Hypergeometric2F1Regularized. Input : Integrate[x^n*Sqrt[(C + x)/(L - x)], {x, 0, L}, Assumptions -> C ≥ 0 && L ≥ 0 && n ≥ 0] Output : \!\(If[C > 0 && L > 0, L\^n\ \@\(C\ L\)\ \@π\ Gamma[1 + n]\ Hypergeometric2F1Regularized[\(-\(1\/2\)\), 1 + n, 3\/2 + n, \(-\(L\/C\)\)], Integrate[x\^n\ \@\(\(C + x\)\/\(L - \x\)\), {x, 0, L}, Assumptions -> C ≤ 0 || L ≤ 0]]\) Here, for n = 0, 1, 2.... two conditions apply 1. L>= C >= 0, OR 2. C>= L >= 0 However, suppose I put n = 0, 1, 2,...10 individually in the integration, I'll get a closed-form solution without the complexity of Hypergeometric2F1Regularized. Could anyone suggest any possibility of avoiding the presence of "Hypergeometric2F1Regularized", in order to make the integral more approachable in calculation? Many thanks in advance. Cheers, Jeffrey M.L.Tan