Let
#\frac{6x^2+1}{x^2(x-1)^3}=A/(x-1)+B/(x-1)^2+C/(x-1)^3+D/x+{E}/(x^2)#
#\frac{6x^2+1}{x^2(x-1)^3}=frac{Ax^2(x-1)^2+Bx^2(x-1)+Cx^2+(Dx+E)(x-1)^3}{x^2(x-1)^3}#
#6x^2+1=(A+D)x^4+(-2A+B-3D+E)x^3+(A-B+C+3D-3E)x^2+(-D+3E)x-E#
Comparing the corresponding coefficients on both the sides we get
#A+D=0\ .........(1)#
#-2A+B-3D+E=0\ .........(2)#
#A-B+C+3D-3E=6\ ............(3)#
#-D+3E=0\ ............(4)#
#E=-1\ ..........(5)#
Solving all above five linear equations, we get
#A=3, B=-2, C=7, D=-3, E=-1#
Now, setting above values , the partial fractions are given as follows
#\frac{6x^2+1}{x^2(x-1)^3}=3/(x-1)-2/(x-1)^2+7/(x-1)^3-3/x-1/x^2#
Now, integrating above equation w.r.t. #x#, we get
#\int \frac{6x^2+1}{x^2(x-1)^3}\ dx=\int 3/(x-1)\ dx-\int2/(x-1)^2\ dx+\int 7/(x-1)^3\ dx-\int 3/x\ dx-\int 1/x^2\ dx#
#=3\int \frac{dx}{x-1}-2\int (x-1)^{-2}\ dx+7\int (x-1)^{-3}\ dx-3\int dx/x-\int x^{-2}\ dx#
#=3\ln|x-1|+2/(x-1)-14/(x-1)^2-3\ln|x|+1/x+C#
#=1/x+2/(x-1)-14/(x-1)^2+3\ln|(x-1)/x|+C#
