Rational functions

 

In Mathematica

For comparison with other CAS choose from: Axiom Derive Macsyma Maple Reduce

• integration

  Integrate[ (3a b^2 - 5a^2 b)/(a^4 - 2), a ]

                   a
      -5 b ArcTan[----]
                   1/4
                  2
      ----------------- -
              1/4
           2 2
  
                   3/4  2            3/4
        (10 b - 3 2    b ) Log[-2 + 2    a]
        ----------------------------------- -
                         1/4
                      8 2
  
                   3/4  2           3/4
       (-10 b - 3 2    b ) Log[2 + 2    a]
       ----------------------------------- -
                        1/4
                     8 2
  
          2                  2
       3 b  Log[2 + Sqrt[2] a ]
       ------------------------
              4 Sqrt[2]

• verification by derivation

  D[%,a]

                                           2
               -5 b                   3 a b
      ----------------------- - ------------------ -
                        2                       2
                       a        2 (2 + Sqrt[2] a )
      2 Sqrt[2] (1 + -------)
                     Sqrt[2]
  
                    3/4  2                   3/4  2
         -10 b - 3 2    b          10 b - 3 2    b
       ---------------------- - -----------------------
                       3/4                       3/4
       4 Sqrt[2] (2 + 2    a)   4 Sqrt[2] (-2 + 2    a)

  Simplify[%]

      a b (-5 a + 3 b)
      ----------------
                4
          -2 + a

• partial fraction decomposition

  Apart[ (10x^2- 11x- 6)/(x^3- x^2- 2x)]

        2      3     5
      ------ + - + -----
      -2 + x   x   1 + x