Adams-Basforth m-Step Method

m=4;
llist=Table[{a+j h,f[t[j],w[j]]},{j,i-m+1,i}];
p[tt_]=Expand[InterpolatingPolynomial[llist,tt]];
w[i]+Factor[Expand[(∫_(a+i h)^(a+(i+1) h))p[tt]\[DifferentialD]tt]]

error=((Derivative[m+1][y][mu] (∏_(j=i-m+1)^i)(tt-(a+j h)))/(m!));
localtruncerror=((Expand[(∫_(a+i h)^(a+(i+1) h))error\[DifferentialD]tt])/h)

Adams-Moulton m-Step Method

m=4;
llist=Table[{a+j h,f[t[j],w[j]]},{j,i-m+1,i+1}];
p[tt_]=Expand[InterpolatingPolynomial[llist,tt]];
w[i]+Factor[Expand[(∫_(a+i h)^(a+(i+1) h))p[tt]\[DifferentialD]tt]]

error=((Derivative[m+2][y][mu] (∏_(j=i-m+1)^(i+1))(tt-(a+j h)))/((m+1)!));
localtruncerror=((Expand[(∫_(a+i h)^(a+(i+1) h))error\[DifferentialD]tt])/h)