After some algebra, Wolfram online integrator gave me the following: $\tag{1} \int (1-a-t)^{N-2}\ \sqrt{2t-t^2}\ \text{d} t = c\ \cdot t^{3/2}\ \operatorname{F}_1 \left( \frac{3}{2}; -\frac{1}{2}, 2-N; \frac{5}{2}; \frac{t}{2}, \frac{t}{1-a}\right)+C$ where:
- $\operatorname{F}_1$ is the Appell hypergeometric function of two variables defined by the expansion: $\operatorname{F}_1(\alpha; \beta, \beta^\prime ; \gamma; x,y):= \sum_{m,n=0}^\infty \frac{(\alpha)_{m+n}\ (\beta)_m\ (\beta^\prime)_n}{m!\ n!\ (\gamma)_{m+n}}\ x^m\ y^n$ ($(a_k):=\frac{\Gamma (a+k)}{\Gamma(a)}$ is the Pochhammer symbol), which converges in the region $|x|,|y|<1$;
- $N\geq 3$ an integer, $a\in ]0,1[$ and $t\in [0,1-a]$ (so the RHside makes sense);
- $c=c(a,N)$ is a known constant and $C$ is an arbitrary constant (coming from indefinite integration).
My question is:
How can I get (1) without using any software?
I suppose a series expansion of the integrand and term by term integration should be used, but I could not figure out how to do explicit computations. Neither I succeeded in simplifying the derivative of the LHside to get the integrand...
Any hint will be appreciated.