The proposition
(x^4+1)/(x^5+6 x^3)=(x^4+1)/(x^3(x^2+6))=c_1/x+c_2/x^2+c_3/x^3+c_4/(x+i sqrt(6))+c_5/(x-i sqrt(6))x4+1x5+6x3=x4+1x3(x2+6)=c1x+c2x2+c3x3+c4x+i√6+c5x−i√6
The c_kck can be determined according to some techniques. The most elementar is
1) Performing the fractions addition
(x^4+1)/(x^5+6 x^3) =(1 - 6 c_3 -6 c_2x -(6 c_1 + c_3)x^2+ (i sqrt[6] c_4 - i sqrt[6] c_5-c_2)x^3+(1 - c_1 - c_4 - c_5)x^4)/(x^3 (6 + x^2))x4+1x5+6x3=1−6c3−6c2x−(6c1+c3)x2+(i√6c4−i√6c5−c2)x3+(1−c1−c4−c5)x4x3(6+x2)
2) Choosing c_kck such that
x^4+1= (1 - 6 c_3 -6 c_2x -(6 c_1 + c_3)x^2+ (i sqrt[6] c_4 - i sqrt[6] c_5-c_2)x^3+(1 - c_1 - c_4 - c_5)x^4), forall x in RR
3) Solving the conditions
{(1 - 6 c_3=0), (-6 c_2=0), (6 c_1 + c_3=0), (-c_2 + i sqrt[6] c_4 -
i sqrt[6] c_5=0), (1 - c_1 - c_4 - c_5=0):}
obtaining
c_1 = -1/36, c_2 = 0, c_3 = 1/6, c_4 = 37/72, c_5 = 37/72
(x^4+1)/(x^5+6 x^3)=1/(6 x^3) - 1/(36 x) + 37/(72 ( x-i sqrt[6] )) + 37/(
72 (x+i sqrt[6]))
Note. We can avoid the complex expansion doing
(c'_3x+c'_4)/(x^2+6) instead of c_4/(x+i sqrt(6))+c_5/(x-i sqrt(6))
