### Related

##### partfrac-plot2d

partfrac((9*x^2-1)/(2...

plot2d((9*x^2-1)/(2-8...

Calculate

##### partfrac

partfrac((C*s+K)/(M*s...

Calculate

##### partfrac

f(x):=(-8*x^6-8*x^5+2...

partfrac(f(x),x);

Calculate

##### partfrac

(2*x-1)/(x+1)^3=partf...

Calculate

##### partfrac

f(x):=((3*x)+7)/((x^2...

'partfrac(f(x),x)=par...

Calculate

##### partfrac

partfrac ( -x/(x^2+x ...

Calculate

##### partfrac

partfrac ((7*x+2)/((x...

Calculate

##### partfrac

fUp: (g*x^5 + h*x^4 ...

fDown: x^3 - x*b^2;

f: fUp/fDown;

Calculate

##### partfrac

partfrac(-6271.43239-...

Calculate

##### partfrac

partfrac((0.32*(z+0.9...

Calculate

### partfrac

Run Example
```(%i1)simp:false;
(%o1)                                false
(%o2)                                false
(%i3) 1/a, simp:false;
1
(%o3)                                  -
a
(%i4) 1/b, simp:false;
1
(%o4)                                  -
b
(%i5) 1/(a*b), simp:false;
1
(%o5)                                 ---
a b
(%i6) tellsimp(1/(a*b), ((1/a)*(1/b)));
(%o6)                         [/rule1, simpquot]
(%i7) 1/(a*b);
1
(%o7)                                 ---
a b
(%i8) disprule(all);
1     1  1
(%t8)                        /rule1 : --- -> - (-)
a b    a  b

(%o8)                                [%t8]
(%i9) 1/(a*b);
1
(%o9)                                 ---
a b
(%i10) properties("*");
(%o10)              [mirror symmetry, nary, rule, operator]
(%i11) rat(1/(a*b));
1
(%o11)/R/                             ---
a b
(%i12) map('ratsimp, 1/(a*b));
1
(%o12)                                ---
a b
(%i13) grind(1/(a*b));

1/(a*b)\$
(%o13)                               done
(%i14) partfrac(1/(a*b), a);
1  - 1
(%o14)                            + (- a   )
b
(%i15) a^-1;
- 1
(%o15)                               a
(%i16) ```
Run Example
```eo : (r1*r3-r2*r4)*ei/(r1*r2*(r3+r4)/rg+r3*r4*(r1+r2)/rg+(r1+r2)*(r3+r4));
ei (r1 r3 - r2 r4)
(%o1)       -------------------------------------------------------
r1 r2 (r4 + r3)   (r2 + r1) r3 r4
--------------- + --------------- + (r2 + r1) (r4 + r3)
rg                rg
(%i2) deo : ((r1+dr1)*(r3+dr3)-(r2+dr2)*(r4+dr4))*ei/((r1+dr1)*(r2+dr2)*((r3+dr3)+(r4+dr4))/rg+(r3+dr3)*(r4+dr4)*((r1+dr1)+(r2+dr2))/rg+((r1+dr1)+(r2+dr2))*((r3+dr3)+(r4+dr4)));
(%o2) (ei ((r1 + dr1) (r3 + dr3) - (r2 + dr2) (r4 + dr4)))
(r1 + dr1) (r2 + dr2) (r4 + r3 + dr4 + dr3)
/(-------------------------------------------
rg
(r2 + r1 + dr2 + dr1) (r3 + dr3) (r4 + dr4)
+ -------------------------------------------
rg
+ (r2 + r1 + dr2 + dr1) (r4 + r3 + dr4 + dr3))
(%i3) factor(deo/eo);
(%o3) ((r2 r4 + dr2 r4 - r1 r3 - dr1 r3 + dr4 r2 - dr3 r1 + dr2 dr4 - dr1 dr3)
(r2 r4 rg + r1 r4 rg + r2 r3 rg + r1 r3 rg + r2 r3 r4 + r1 r3 r4 + r1 r2 r4
+ r1 r2 r3))/((r2 r4 - r1 r3) (r2 r4 rg + r1 r4 rg + dr2 r4 rg + dr1 r4 rg + r2 r3 rg + r1 r3 rg + dr2 r3 rg + dr1 r3 rg + dr4 r2 rg + dr3 r2 rg + dr4 r1 rg + dr3 r1 rg + dr2 dr4 rg + dr1 dr4 rg + dr2 dr3 rg + dr1 dr3 rg + r2 r3 r4
+ r1 r3 r4 + dr2 r3 r4 + dr1 r3 r4 + r1 r2 r4 + dr3 r2 r4 + dr1 r2 r4
+ dr3 r1 r4 + dr2 r1 r4 + dr2 dr3 r4 + dr1 dr3 r4 + dr1 dr2 r4 + r1 r2 r3
+ dr4 r2 r3 + dr1 r2 r3 + dr4 r1 r3 + dr2 r1 r3 + dr2 dr4 r3 + dr1 dr4 r3
+ dr1 dr2 r3 + dr4 r1 r2 + dr3 r1 r2 + dr3 dr4 r2 + dr1 dr4 r2 + dr1 dr3 r2
+ dr3 dr4 r1 + dr2 dr4 r1 + dr2 dr3 r1 + dr2 dr3 dr4 + dr1 dr3 dr4
+ dr1 dr2 dr4 + dr1 dr2
dr3))
(%i4) partfrac(deo/eo,rg);
2                             2
(%o4) ((r2  + (r1 + dr2) r2 + dr2 r1) r4
2                      2                              2
+ ((r2  + (dr2 - dr1) r2 - r1  + (dr2 - dr1) r1) r3 + dr4 r2
2
+ ((dr4 - dr3) r1 + dr2 dr4 - dr1 dr3) r2 - dr3 r1  + (dr2 dr4 - dr1 dr3) r1)
2             2
r4 + ((- r1 - dr1) r2 - r1  - dr1 r1) r3
2                                                   2
+ (dr4 r2  + ((dr4 - dr3) r1 + dr2 dr4 - dr1 dr3) r2 - dr3 r1
2                          2
+ (dr2 dr4 - dr1 dr3) r1) r3)/((r2  + (r1 + dr2 + dr1) r2) r4
2                      2                                        2
+ ((r2  + (dr2 + dr1) r2 - r1  + (- dr2 - dr1) r1) r3 + (dr4 + dr3) r2
+ ((dr4 + dr3) r1 + (dr2 + dr1) dr4 + (dr2 + dr1) dr3) r2) r4
2                       2
+ (- r1 r2 - r1  + (- dr2 - dr1) r1) r3
2
+ ((- dr4 - dr3) r1 r2 + (- dr4 - dr3) r1
+ ((- dr2 - dr1) dr4 + (- dr2 - dr1) dr3) r1) r3)
3                                                2
- (((dr3 + dr1) r2  + (2 dr3 r1 + (2 dr2 + dr1) dr3 + 2 dr1 dr2) r2
2                                          2
+ ((dr3 + dr2) r1  + ((3 dr2 + dr1) dr3 + dr1 dr2) r1 + (dr2  + dr1 dr2) dr3
2                     2    2        2                         2
+ dr1 dr2 ) r2 + (dr2 dr3 + dr2 ) r1  + ((dr2  + dr1 dr2) dr3 + dr1 dr2 ) r1)
3             3                                                2    2
r4  + ((2 dr1 r2  + ((- dr3 - dr1) r1 - dr1 dr3 + 4 dr1 dr2 - dr1 ) r2
2
+ ((2 dr2 - 2 dr3) r1  + ((- dr2 - 3 dr1) dr3 + dr1 dr2) r1
2                 2      2                           3
+ (- dr1 dr2 - dr1 ) dr3 + 2 dr1 dr2  - dr1  dr2) r2 + (- dr3 - dr2) r1
2                2
+ ((- dr2 - 2 dr1) dr3 + 2 dr2  - 2 dr1 dr2) r1
2                 2      2
+ ((- dr1 dr2 - dr1 ) dr3 + 2 dr1 dr2  - dr1  dr2) r1) r3
3                    2
+ ((2 dr3 + 2 dr1) dr4 + dr1 dr3) r2  + ((4 dr3 dr4 - dr3  - dr1 dr3) r1
2                   2
+ ((4 dr2 + 2 dr1) dr3 + 4 dr1 dr2) dr4 - dr1 dr3  + (2 dr1 dr2 - dr1 ) dr3)
2                                2              2
r2  + (((2 dr3 + 2 dr2) dr4 - 2 dr3  + dr2 dr3) r1
2
+ (((6 dr2 + 2 dr1) dr3 + 2 dr1 dr2) dr4 + (- dr2 - 3 dr1) dr3 ) r1
2                             2                        2     2
+ ((2 dr2  + 2 dr1 dr2) dr3 + 2 dr1 dr2 ) dr4 + (- dr1 dr2 - dr1 ) dr3
2      2                      2              3
+ (dr1 dr2  - dr1  dr2) dr3) r2 + (- dr3  - dr2 dr3) r1
2                           2       2
+ ((2 dr2 dr3 + 2 dr2 ) dr4 + (- dr2 - 2 dr1) dr3  + (dr2  - 2 dr1 dr2) dr3)
2           2                             2                        2     2
r1  + (((2 dr2  + 2 dr1 dr2) dr3 + 2 dr1 dr2 ) dr4 + (- dr1 dr2 - dr1 ) dr3
2      2                 2                   3
+ (dr1 dr2  - dr1  dr2) dr3) r1) r4  + (((dr4 + dr1) r2
2    2
+ ((2 dr4 - 2 dr1) r1 + (2 dr2 + dr1) dr4 + 2 dr1 dr2 - 2 dr1 ) r2
2                                          2
+ ((dr4 + dr2) r1  + ((3 dr2 + dr1) dr4 - dr1 dr2) r1 + (dr2  + dr1 dr2) dr4
2        2                   3                 2                2
+ dr1 dr2  - 2 dr1  dr2) r2 - 2 dr2 r1  + (dr2 dr4 + dr2  - 4 dr1 dr2) r1
2                         2        2            2
+ ((dr2  + dr1 dr2) dr4 + dr1 dr2  - 2 dr1  dr2) r1) r3
3
+ (((dr3 + 3 dr1) dr4 + dr1 dr3) r2  + (((dr3 - dr1) dr4 - 3 dr1 dr3) r1
2                          2         2
+ (2 dr2 dr3 + 6 dr1 dr2 - dr1 ) dr4 + (2 dr1 dr2 - 3 dr1 ) dr3) r2
2
+ (((3 dr2 - dr3) dr4 + dr2 dr3) r1  + (((2 dr2 - 2 dr1) dr3 + 2 dr1 dr2) dr4
2      2                 2      2
- 2 dr1 dr2 dr3) r1 + ((dr2  - dr1 ) dr3 + 3 dr1 dr2  - dr1  dr2) dr4
2        2                                                  3
+ (dr1 dr2  - 3 dr1  dr2) dr3) r2 + ((- dr3 - dr2) dr4 - 3 dr2 dr3) r1
2                        2                     2
+ ((- 2 dr1 dr3 + 3 dr2  - 2 dr1 dr2) dr4 + (dr2  - 6 dr1 dr2) dr3) r1
2      2                 2      2
+ (((dr2  - dr1 ) dr3 + 3 dr1 dr2  - dr1  dr2) dr4
2        2                                    2                  3
+ (dr1 dr2  - 3 dr1  dr2) dr3) r1) r3 + ((dr3 + dr1) dr4  + dr1 dr3 dr4) r2
2         2                         2
+ ((2 dr3 dr4  + (- dr3  - dr1 dr3) dr4 - dr1 dr3 ) r1
2
+ ((2 dr2 + dr1) dr3 + 2 dr1 dr2) dr4
2               2           2    2    2
+ ((2 dr1 dr2 - dr1 ) dr3 - dr1 dr3 ) dr4 - dr1  dr3 ) r2
2                   2         2
+ (((dr3 + dr2) dr4  + (dr2 dr3 - 2 dr3 ) dr4) r1
2                      2
+ (((3 dr2 + dr1) dr3 + dr1 dr2) dr4  + (- dr2 - 3 dr1) dr3  dr4
2            2                         2     2
- dr1 dr2 dr3 ) r1 + ((dr2  + dr1 dr2) dr3 + dr1 dr2 ) dr4
2     2           2      2                    2        2
+ ((- dr1 dr2 - dr1 ) dr3  + (dr1 dr2  - dr1  dr2) dr3) dr4 - dr1  dr2 dr3 )
2                         2    3
r2 + ((- dr3  - dr2 dr3) dr4 - dr2 dr3 ) r1
2     2                       2       2
+ ((dr2 dr3 + dr2 ) dr4  + ((- dr2 - 2 dr1) dr3  + (dr2  - 2 dr1 dr2) dr3) dr4
2    2         2                         2     2
- 2 dr1 dr2 dr3 ) r1  + (((dr2  + dr1 dr2) dr3 + dr1 dr2 ) dr4
2     2           2      2                    2        2
+ ((- dr1 dr2 - dr1 ) dr3  + (dr1 dr2  - dr1  dr2) dr3) dr4 - dr1  dr2 dr3 )
2    2
r1) r4 + (((- dr4 - dr1) r1 - dr1 dr4 - dr1 ) r2
2                                                        2
+ (- 2 dr4 r1  + ((- dr2 - 3 dr1) dr4 - dr1 dr2) r1 + (- dr1 dr2 - dr1 ) dr4
2                           3                                       2
- dr1  dr2) r2 + (- dr4 - dr2) r1  + ((- dr2 - 2 dr1) dr4 - 2 dr1 dr2) r1
2           2            3
+ ((- dr1 dr2 - dr1 ) dr4 - dr1  dr2) r1) r3
2              3          2
+ ((dr4  + dr1 dr4) r2  + ((2 dr4  + (- 2 dr3 - dr1) dr4 - 2 dr1 dr3) r1
2                                 2             2        2
+ (2 dr2 + dr1) dr4  + (- 2 dr1 dr3 + 2 dr1 dr2 - dr1 ) dr4 - 2 dr1  dr3) r2
2                        2                     2
+ ((dr4  + (dr2 - 4 dr3) dr4) r1  + ((3 dr2 + dr1) dr4
2               2
+ (- 2 dr2 - 6 dr1) dr3 dr4 - 2 dr1 dr2 dr3) r1 + (dr2  + dr1 dr2) dr4
2               2      2                 2
+ ((- 2 dr1 dr2 - 2 dr1 ) dr3 + dr1 dr2  - dr1  dr2) dr4 - 2 dr1  dr2 dr3) r2
3
+ ((- 2 dr3 - dr2) dr4 - 2 dr2 dr3) r1
2                               2
+ (dr2 dr4  + ((- 2 dr2 - 4 dr1) dr3 + dr2  - 2 dr1 dr2) dr4 - 4 dr1 dr2 dr3)
2        2               2                        2               2
r1  + ((dr2  + dr1 dr2) dr4  + ((- 2 dr1 dr2 - 2 dr1 ) dr3 + dr1 dr2
2                 2                2
- dr1  dr2) dr4 - 2 dr1  dr2 dr3) r1) r3
2                  3
+ (((dr3 + dr1) dr4  + dr1 dr3 dr4) r2
2         2                         2
+ ((2 dr3 dr4  + (- dr3  - dr1 dr3) dr4 - dr1 dr3 ) r1
2
+ ((2 dr2 + dr1) dr3 + 2 dr1 dr2) dr4
2               2           2    2    2
+ ((2 dr1 dr2 - dr1 ) dr3 - dr1 dr3 ) dr4 - dr1  dr3 ) r2
2                   2         2
+ (((dr3 + dr2) dr4  + (dr2 dr3 - 2 dr3 ) dr4) r1
2                      2
+ (((3 dr2 + dr1) dr3 + dr1 dr2) dr4  + (- dr2 - 3 dr1) dr3  dr4
2            2                         2     2
- dr1 dr2 dr3 ) r1 + ((dr2  + dr1 dr2) dr3 + dr1 dr2 ) dr4
2     2           2      2                    2        2
+ ((- dr1 dr2 - dr1 ) dr3  + (dr1 dr2  - dr1  dr2) dr3) dr4 - dr1  dr2 dr3 )
2                         2    3
r2 + ((- dr3  - dr2 dr3) dr4 - dr2 dr3 ) r1
2     2                       2       2
+ ((dr2 dr3 + dr2 ) dr4  + ((- dr2 - 2 dr1) dr3  + (dr2  - 2 dr1 dr2) dr3) dr4
2    2         2                         2     2
- 2 dr1 dr2 dr3 ) r1  + (((dr2  + dr1 dr2) dr3 + dr1 dr2 ) dr4
2     2           2      2                    2        2
+ ((- dr1 dr2 - dr1 ) dr3  + (dr1 dr2  - dr1  dr2) dr3) dr4 - dr1  dr2 dr3 )
2                          2       2                      2                                        2                                                                              2                       2                                          2
r1) r3)/(((r2  + (r1 + dr2 + dr1) r2) r4  + ((r2  + (dr2 + dr1) r2 - r1  + (- dr2 - dr1) r1) r3 + (dr4 + dr3) r2  + ((dr4 + dr3) r1 + (dr2 + dr1) dr4 + (dr2 + dr1) dr3) r2) r4 + (- r1 r2 - r1  + (- dr2 - dr1) r1) r3  + ((- dr4 - dr3) r1 r2 + (- dr4 - dr3) r1  + ((- dr2 - dr1) dr4 + (- dr2 - dr1) dr3) r1) r3) (((r2 + r1 + dr2 + dr1) r4 + (r2 + r1 + dr2 + dr1) r3 + (dr4 + dr3) r2 + (dr4 + dr3) r1 + (dr2 + dr1) dr4 + (dr2 + dr1) dr3) rg + ((r2 + r1 + dr2 + dr1) r3 + (r1 + dr3 + dr1) r2 + (dr3 + dr2) r1 + (dr2 + dr1) dr3 + dr1 dr2) r4 + ((r1 + dr4 + dr1) r2 + (dr4 + dr2) r1 + (dr2 + dr1) dr4 + dr1 dr2) r3 + ((dr4 + dr3) r1 + (dr3 + dr1) dr4 + dr1 dr3) r2 + ((dr3 + dr2) dr4 + dr2 dr3) r1 + ((dr2 + dr1) dr3 + dr1 dr2) dr4 + dr1 dr2
dr3))
(%i5) partfrac(eo,rg);
2                          2    2
(%o5) (((ei r2  + ei r1 r2) r3 + ei r1 r2 ) r4
2    2            2        2                   2      2
+ ((- ei r1 r2 - ei r1 ) r3  + (ei r1 r2  - ei r1  r2) r3) r4 - ei r1  r2 r3 )
/(((r2 + r1) r4 + (r2 + r1) r3) (((r2 + r1) r4 + (r2 + r1) r3) rg + ((r2 + r1) r3 + r1 r2) r4 + r1 r2 r3))
ei r2 r4 - ei r1 r3
- ---------------------------
(r2 + r1) r4 + (r2 + r1) r3
(%i6) partfrac((r1+dr1)*(r2+dr2)*((r3+dr3)+(r4+dr4))/rg+(r3+dr3)*(r4+dr4)*((r1+dr1)+(r2+dr2))/rg,rg);
(%o6) (((r2 + r1 + dr2 + dr1) r3 + (r1 + dr3 + dr1) r2 + (dr3 + dr2) r1
+ (dr2 + dr1) dr3 + dr1 dr2) r4 + ((r1 + dr4 + dr1) r2 + (dr4 + dr2) r1
+ (dr2 + dr1) dr4 + dr1 dr2) r3 + ((dr4 + dr3) r1 + (dr3 + dr1) dr4 + dr1 dr3)
r2 + ((dr3 + dr2) dr4 + dr2 dr3) r1 + ((dr2 + dr1) dr3 + dr1 dr2) dr4
+ dr1 dr2 dr3)/rg
(%i7) expand((((r2 + r1 )* r3 + (r1)* r2 )* r4 + ((r1)* r2 )* r3) /rg);
r2 r3 r4   r1 r3 r4   r1 r2 r4   r1 r2 r3
(%o7)              -------- + -------- + -------- + --------
rg         rg         rg         rg
(%i8) ((((r2 + r1 )* r3 + (r1)* r2 )* r4 + ((r1)* r2 )* r3) /rg)/((r1+r2)*(r3+r4));
((r2 + r1) r3 + r1 r2) r4 + r1 r2 r3
(%o8)                ------------------------------------
(r2 + r1) (r4 + r3) rg
(%i9) expand(r1*r2*(r3+r4)/rg+r3*r4*(r1+r2)/rg);
r2 r3 r4   r1 r3 r4   r1 r2 r4   r1 r2 r3
(%o9)              -------- + -------- + -------- + --------
rg         rg         rg         rg
(%i10) ```
Run Example
```f(z):=(5*z^2 -8*z +5)/((z-1)^2 *(z-i)^2 *(z+i)^2);
2
5 z  - 8 z + 5
(%o1)                 f(z) := --------------------------
2        2        2
(z - 1)  (z - i)  (z + i)
(%i2) partfrac(f(z),z);
3       2                               2
5 i  + 11 i  + 15 i + 5                 5 i  + 8 i + 5
(%o2) ------------------------------------- + -----------------------------
6       5       4      3                4      3      2         2
(4 i  + 12 i  + 12 i  + 4 i ) (z + i)   (4 i  + 8 i  + 4 i ) (z + i)
3       2                               2
5 i  - 11 i  + 15 i - 5                 5 i  - 8 i + 5
- ------------------------------------- + -----------------------------
6       5       4      3                4      3      2         2
(4 i  - 12 i  + 12 i  - 4 i ) (z - i)   (4 i  - 8 i  + 4 i ) (z - i)
2
2 i  + 6                         2
+ ------------------------------ + ------------------------
6      4      2                  4      2             2
(i  - 3 i  + 3 i  - 1) (z - 1)   (i  - 2 i  + 1) (z - 1)
(%i3) ratsimp(%);
(%o3)             (%i4) ```

### Related Help

Help for Partfrac