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Closed-form Integral Solution without Hypergeometric2F1Regularized ! ! !


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


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