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Integrate[x^2/(-b + a*x^4)^2, x] ==
((4*b^(1/4)*x^3)/(b - a*x^4) - (2*ArcTan[(a^(1/4)*x)/b^(1/4)])/a^(3/4) - Log[-b^(1/4) + a^(1/4)*x]/a^(3/4) + Log[b^(1/4) + a^(1/4)*x]/a^(3/4))/(16*b^(5/4))

                2
               x
Integrate[------------, x] == 
                   4 2
          (-b + a x )

                       1/4
                      a    x
             2 ArcTan[------]
    1/4  3              1/4           1/4    1/4
 4 b    x              b        Log[-b    + a    x]
(--------- - ---------------- - ------------------- + 
        4           3/4                 3/4
 b - a x           a                   a
 
         1/4    1/4
    Log[b    + a    x]         5/4
    ------------------) / (16 b   )
            3/4
           a