Perform the decomposition into partial fractions
#(2x^3)/((1-x^2)(1+x)^2)=(2x^3)/((1-x)(1+x)(1+x)^2)=(2x^3)/((1-x)(1+x)^3)#
#=A/(1-x)+B/(1+x)+C/(1+x)^2+D/(1+x)^3#
#=(A(1+x)^3+B(1-x)(1+x)^2+C(1-x)(1+x)+D(1-x))/(((1-x)(1+x)^3))#
The denominators are the same, compare the numerators
#2x^3=A(1+x)^3+B(1-x)(1+x)^2+C(1-x)(1+x)+D(1-x)#
Let #x=1#, #=>#, #2=8A#, #=>#, #A=1/4#
Let #x=-1#, #=>#, #-2=2D#, #=>#, #D=-1#
Coefficient of #x^3#
#2=A-B#, #=>#, #B=A-2=1/4-2=-7/4#
Coefficients of #x^2#
#0=3A-B-C#, #=>#, #C=3A-B=3/4+7/4=10/4=5/2#
Finally,
#(2x^3)/((1-x^2)(1+x)^2)=(1/4)/(1-x)+(-7/4)/(1+x)+(5/2)/(1+x)^2+(-1)/(1+x)^3#
