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Integrate[(a + b*x + c*x^2)^(1/2 + n), x] ==
(2^(1/2 + n)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)* ((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^ (-1/2 - n)*(a + x*(b + c*x))^(1/2 + n)* Hypergeometric2F1[-1/2 - n, 3/2 + n, 5/2 + n, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/ (2*Sqrt[b^2 - 4*a*c])])/(c*(3 + 2*n))

                        2 1/2 + n
Integrate[(a + b x + c x )       , x] == 

  1/2 + n            2
(2        (b - Sqrt[b  - 4 a c] + 2 c x) 
 
               2
     b + Sqrt[b  - 4 a c] + 2 c x -(1/2) - n
    (----------------------------)           
                 2
           Sqrt[b  - 4 a c]
 
                     1/2 + n
    (a + x (b + c x))        
 
                        1       3      5
    Hypergeometric2F1[-(-) - n, - + n, - + n, 
                        2       2      2
 
                2
     -b + Sqrt[b  - 4 a c] - 2 c x
     -----------------------------]) / (c (3 + 2 n))
                  2
          2 Sqrt[b  - 4 a c]