Question about a function that is a ratio of gamma functions and appears to be strictly increasing for $x\ge 2$

Stirling's approximation says that $\log \Gamma(z) \sim z \log z + O(z)$. So $$\log \Gamma(\alpha x + 1)\sim (\alpha x + 1)\log(\alpha x + 1)+O(x)\sim \alpha x \log x + O(x),$$ and so certainly $\log f(x)\sim -\frac{1}{30}x\log x + O(x)$: your function must, eventually, start decreasing asymptotically to zero. But eventually is important. Let's try to estimate where this happens.

Consider the ratio $f(30y+30) / f(30y)$, where $y$ is an integer. You have $$ \frac{f(30y+30)}{f(30y)}=\frac{\Gamma(30y+31)\Gamma(15y+1)\Gamma(10y+1)\Gamma(6y+1)}{\Gamma(30y+1)\Gamma(15y+16)\Gamma(10y+11)\Gamma(6y+7)}=\frac{(30y+30)!(15y)!(10y)!(6y)!}{(30y)!(15y+15)!(10y+10)!(6y+6)!}=\frac{(30y+30)(30y+29)\cdots(30y+1)}{(15y+15)(15y+14)\cdots(15y+1)\cdot(10y+10)\cdots(10y+1)\cdot(6y+6)\cdots(6y+1)}\approx\frac{30^{30}}{15^{15}10^{10}6^{6}}\frac{1}{y}=\frac{1.008\times10^{12}}{y}, $$ where the $\approx$ is arrived at by pulling out the prefactors, counting powers of $y$, and ignoring the additive terms (which is justified when $y$ is large). This is going to be less than $1$ eventually... when $y$ exceeds a trillion. Your function will start decreasing when $x$ is greater than about $3\times 10^{13}.$ By that point, $f(x)$ will have many trillions of decimal digits, which I dare say is a bit more than Excel can handle.