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-(2^(1/2 + n)*x^(-1 + n)*(x/a)^(1/2 - n)*Sqrt[-(x*(-2*a + x))]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, 1 - x/(2*a)])


Integrate[x^n/Sqrt[2*a*x - x^2], x] ==
-(2^(1/2 + n)*x^(-1 + n)*(x/a)^(1/2 - n)* Sqrt[-(x*(-2*a + x))]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, 1 - x/(2*a)])



                  n
                 x
Integrate[----------------, x] == 
                        2
          Sqrt[2 a x - x ]

   1/2 + n  -1 + n  x 1/2 - n
-(2        x       (-)        Sqrt[-(x (-2 a + x))] 
                    a
 
                      1  1      3       x
    Hypergeometric2F1[-, - - n, -, 1 - ---])
                      2  2      2      2 a

Time to compute: 0.11 second

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