speeding up Solve(ing) the Diophantine equation

Trying to solve the Diophantine equation (or variation?) using Solve in order to generate some chemical formulas. I need to generate all possible solutions. The code is following:

mx1 = 1007825; mx2 = 12000000; mx3 = 14003074; mx4 = 15994915; mx5 = 18998403;
mx6 = 30973762; mx7 = 31972071; mx8 = 34968853; mx9 = 78918338; mx10 = 126904468;
up = 722126458;
dn = 722120458;
x1max = 70; x2max = 125; x3max = 25; x4max = 27; x5max = 10; x6max = \
0; x7max = 0; x8max = 0; x9max = 0; x10max = 0;

Solve[{dn <=
mx1*x1 + mx2*x2 + mx3*x3 + mx4*x4 + mx5*x5 + mx6*x6 + mx7*x7 +
mx8*x8 + mx9*x9 + mx10*x10 <= up, x1max >= x1 >= 0,
x2max >= x2 >= 10, x3max >= x3 >= 0, x4max >= x4 >= 0,
x5max >= x5 >= 0, x6max >= x6 >= 0, x7max >= x7 >= 0,
x8max >= x8 >= 0, x9max >= x9 >= 0, x10max >= x10 >= 0}, {x1, x2,
x3, x4, x5, x6, x7, x8, x9, x10}, Integers]; // AbsoluteTiming

{0.2670152, Null}

As I am increasing number of variables time increase. Addition of x6max=10 - time=2.15s; x7max=9 - time=32.34s

with addition of x8max=8, x9max=8 and x10max=2 timing is ??

I am looking for suggestions how to speed up the process. One to several seconds time frame would be great.

Reduce and FindInstance perform roughly similar. Brute forcing doesn’t work effectively either.

Tried to work out solution in different way; like generating 2 sets (e.g. for x1 to x5 and x6 to x6) with all possible combinations, sorting them (decreasing/increasing order) and than sum elements. It doesn’t perform fast either considering 1 million element tables.