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

Time to compute: 0.49 second

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