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-(2^(3/2 - n)*x^(-1 - n)*(x/a)^(-1/2 + n)*(-(x*(-2*a + x)))^(3/2)*Hypergeometric2F1[3/2, -1/2 + n, 5/2, 1 - x/(2*a)])/3


Integrate[Sqrt[2*a*x - x^2]/x^n, x] ==
-(2^(3/2 - n)*x^(-1 - n)*(x/a)^(-1/2 + n)* (-(x*(-2*a + x)))^(3/2)*Hypergeometric2F1[3/2, -1/2 + n, 5/2, 1 - x/(2*a)])/3



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

   3/2 - n  -1 - n  x -(1/2) + n                  3/2
-(2        x       (-)           (-(x (-2 a + x)))    
                    a
 
                       3    1       5       x
     Hypergeometric2F1[-, -(-) + n, -, 1 - ---]) / 3
                       2    2       2      2 a

Time to compute: 0.17 second

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