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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[x^n*Sqrt[2*a*x - x^2], 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 

           n               2
Integrate[x  Sqrt[2 a x - x ], 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: 0.27 second
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