Question 356207
factor the denomiator {{{x+2x^2+x^3)=x*(x+1)^2}}}
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now set 
{{{ int ( 1/ (x+2x^2+x^3), dx ) }}} = {{{int ((A/x+B/(x+1)+c/(x+1)^2))}}}
this integrates into
{{{A*ln(abs(x))+B*ln(abs(x+1))-2C*(X+1)^(-1)}}}+ constant
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find A, B, and C by multiplying {{{A/x+B/(x+1)+c/(x+1)^2)}}} by the common denominator {{{x*(x+1)^2}}} and setting the numerator equal to its original numerator of 1.

{{{A*(x+1)^2+B*x*(x+1)+c*x=1}}}
select 3 different values of x and solve for A, B and C
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choose 3 convenient X values ie -1,0,1
x=0: {{{ A*(0+1)^2 + b*0*(0+1)+c*0=1}}}   => A=1
x=-1:  {{{ A*(-1+1)^2 + b*-1*(-1+1)+c*(-1)=1}}}   => -C=1 or C=-1

you get the idea?