Begin with the equation above and the assumption that any leftover constants after integrating always equal c. We can simplify to the 7 simpler expressions below. Consider only up to before the statement regarding their sum, that the sum is positive. There are at this point four possible tuples (a, b, c, d). They are -2, -1, 0, and 2; -3, 0, 1, and 3; -4, 1, 2, and 4; and -5, 2, 3, and 5. Now, the first and last of these have sums that are out of bounds. The first of the remaining options is eliminated by the final clue. This leaves us with an answer of (a, b, c, d) = (-4, 1, 2, 4).
a,\ b,\ c,\ d\in\Z\\[8pt] a\leqslant b\leqslant c\leqslant d\\[8pt] a+b=-3\\[8pt] 2c-d-b\lt3\\[8pt] 2b-c-a\gt0\\[8pt] 0\lt a+b+c+d\lt5\\[8pt] \sqrt{bc}\notin\Z\\[16pt] \Rightarrow\therefore(a,b,c,d)=(-4,1,2,4)Solution to Seriocomic Set
a,\,b,\,c,\,d\in\mathbb Z\\[8pt]
a\leqslant b\leqslant c\leqslant d\\[8pt]
a+b+3=a+d=\cos'{\pi}\\[8pt]
\int \frac{d-c}a+1\;\;\delta a\bigg\lt\cos'{(a+b)\pi}\bigg\lt\int\frac{1-a}b-2\;\;\delta b\\[8pt]
0\lt|bc|+1\leqslant a+b+c+d\lt5\\[8pt]
bc\notin\{a^2|a\in\mathbb N\}