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

Integrate[(a + b*x^2)^(-1 - m), x] ==
(x*(1 + (b*x^2)/a)^m*Hypergeometric2F1[1/2, 1 + m, 3/2, -((b*x^2)/a)])/(a*(a + b*x^2)^m) 

                  2 -1 - m
Integrate[(a + b x )      , x] == 

           2
        b x  m                   1         3
(x (1 + ----)  Hypergeometric2F1[-, 1 + m, -, 
         a                       2         2
 
          2
       b x                  2 m
     -(----)]) / (a (a + b x ) )
        a
Time to compute: 0.03 second
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