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



             1
Integrate[--------, x] == 
                 4
          a + b x

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

Time to compute: 0.09 second

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