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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[1/(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)])



                   1
Integrate[-------------------, x] == 
           n               2
          x  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: 0.15 second

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