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



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

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

Time to compute: 0.08 second

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