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Integrate[Sqrt[x]/(-b + a*x^4), x] ==
(2*Sqrt[2]*ArcTan[1 - (Sqrt[2]*a^(1/8)*Sqrt[x])/ b^(1/8)] - 2*Sqrt[2]* ArcTan[1 + (Sqrt[2]*a^(1/8)*Sqrt[x])/b^(1/8)] + 4*ArcTan[(a^(1/8)*Sqrt[x])/b^(1/8)] + 2*Log[-b^(1/8) + a^(1/8)*Sqrt[x]] - 2*Log[b^(1/8) + a^(1/8)*Sqrt[x]] - Sqrt[2]*Log[b^(1/4) - Sqrt[2]*a^(1/8)*b^(1/8)* Sqrt[x] + a^(1/4)*x] + Sqrt[2]*Log[b^(1/4) + Sqrt[2]*a^(1/8)*b^(1/8)* Sqrt[x] + a^(1/4)*x])/(8*a^(3/8)*b^(5/8))



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

                               1/8
                      Sqrt[2] a    Sqrt[x]
(2 Sqrt[2] ArcTan[1 - --------------------] - 
                               1/8
                              b
 
                                  1/8
                         Sqrt[2] a    Sqrt[x]
    2 Sqrt[2] ArcTan[1 + --------------------] + 
                                  1/8
                                 b
 
              1/8
             a    Sqrt[x]
    4 ArcTan[------------] + 
                  1/8
                 b
 
            1/8    1/8
    2 Log[-b    + a    Sqrt[x]] - 
 
           1/8    1/8
    2 Log[b    + a    Sqrt[x]] - 
 
                 1/4            1/8  1/8
    Sqrt[2] Log[b    - Sqrt[2] a    b    Sqrt[x] + 
 
        1/4
       a    x] + Sqrt[2] 
 
          1/4            1/8  1/8            1/4
     Log[b    + Sqrt[2] a    b    Sqrt[x] + a    x])\
 
         3/8  5/8
   / (8 a    b   )

Time to compute: 0.15 second

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