Rational functions

 

In Maple

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

• integration

  int( (3*a*b^2 - 5*a^2* b) / (a^4 - 2), a );

                                 2  1/2
                2  1/2    - 2 + a  2              3/4               3/4
           3/8 b  2    ln(-------------) - 5/4 b 2    arctan(1/2 a 2   )
                                 2  1/2
                          - 2 - a  2
  
                                     1/4
                         3/4    a + 2
                + 5/8 b 2    ln(--------)
                                     1/4
                                a - 2

• verification by derivation

  diff( ", a );

                /         1/2                2  1/2     1/2\
         2  1/2 |      a 2           (- 2 + a  2   ) a 2   |         2  1/2
        b  2    |2 ------------- + 2 ----------------------| (- 2 - a  2   )
                |         2  1/2                2  1/2 2   |
                \  - 2 - a  2           (- 2 - a  2   )    /
                                           2  1/2
                                    - 2 + a  2
  
                       1/2
                    b 2
         - 5/4 ---------------
                        2  1/2
               1 + 1/2 a  2
  
                      /                  1/4 \
                  3/4 |    1        a + 2    |       1/4
               b 2    |-------- - -----------| (a - 2   )
                      |     1/4         1/4 2|
                      \a - 2      (a - 2   ) /
         + 5/8 ------------------------------------------
                                     1/4
                                a + 2

  simplify( " );

                                                1/2
                                 (5 a - 3 b) b 2    a
                       - -----------------------------------
                               2  1/2        1/4        1/4
                         (2 + a  2   ) (a - 2   ) (a + 2   )

  simplify(numer(")/expand(denom(")));

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

• partial fraction decomposition

  convert((10*x^2- 11*x- 6) / (x^3- x^2- 2*x),parfrac,x);

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