It is easy to prove $abcd|abc+abd+acd+cda+1$ So we get $f(a,b,c,d)=1/a+1/b+1/c+1/d+1/abcd$ is at least $1$. So from here we get that $min(a,b,c,d)=1,2,3,4$ and using that $f(a,b,c,d)$ is cecreasing on each variable, we may plug in $a=1,2,3,4$ and find possible values for the remaining numbers. This gives not too much hand work to check solutions.