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

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

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