Apply partial fraction decomposition:
x^2/((x-3)^2(x+4)) = A/(x-3)+B/(x-3)^2 +C/(x+4)x2(x−3)2(x+4)=Ax−3+B(x−3)2+Cx+4
x^2/((x-3)^2(x+4)) = (A(x-3)(x+4)+B(x+4)+C(x-3)^2)/((x-3)^2(x+4))x2(x−3)2(x+4)=A(x−3)(x+4)+B(x+4)+C(x−3)2(x−3)2(x+4)
x^2 = A(x^2+x-12)+Bx+4B+C(x^2-6x+9))x2=A(x2+x−12)+Bx+4B+C(x2−6x+9))
x^2 = Ax^2+ Ax-12A +Bx+4B +Cx^2 -6Cx+9Cx2=Ax2+Ax−12A+Bx+4B+Cx2−6Cx+9C
x^2 = (A+C)x^2 + (A+B-6C)x - (12A-4B-9C)x2=(A+C)x2+(A+B−6C)x−(12A−4B−9C)
{(A+C=1),(A+B-6C=0),(12A-4B-9C=0):}
{(A=1-C),(B-7C=-1),(4B+21C=12):}
{(A=1-C),(-4B+28C=4),(4B+21C=12):}
{(A=33/49),(B=9/7),(C=16/49):}
Then:
int (x^2dx)/((x-3)^2(x+4)) =33/49int dx /(x-3)+9/7int dx/(x-3)^2 +16/49int dx/(x+4)
int (x^2dx)/((x-3)^2(x+4)) =33/49ln abs(x-3)-9/(7(x-3)) +16/49ln abs(x+4)+C