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"Integral From a x squared Plus b Plus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Plus One RightParenthesis Differential x IsGreater Integral From a x squared Plus b Minus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Minus One RightParenthesis Differential x Implies Integral From a x squared Minus b Plus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Plus One RightParenthesis Differential x IsGreater Integral From a x squared Plus b Minus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Minus One RightParenthesis Differential x" ?

Quiz:

Do you recognize this one ?

Summation On j From 1 Upto N Of y Index k Times the Product On k From 1 And k NotEqual to j Upto N Of x Minus x Index k Over x Index j Minus x Index k.

Others have already answered on why one should use symbols. I want to add that one shouldn't overuse symbols, as people sometimes do.

With too many symbols, statements get clustered and confusing. Out of lazyness, many like to write $exists$ quantors in the middle of a sentence. Others overuse e.g. $land$, etc. ($land$ is not a synonym for "and"!)

So basically, one shouldn't overuse symbols.

Consider this problem, taken from The Evolution of Algebra in Science, vol. 18, no. 452 (Oct 2, 1891) pp. 183-187 (taken from JSTOR, itself translated from work of Nesselman on a problem by Mohammed ibn Musa):

A square and ten of its roots are equal to nine and thirty units, that is, if you add ten roots to one square, the sum is equal to nine and thirty. The solution is as follows: halve the number of roots, that is, in this case, five; then multiply this by itself, and the result is five and twenty. Add this to the nine and thirty, which gives four and sixty; take the square root, or eight, and subtract from it half the number of roots, namely five, and there remains three: this is the root of the square which was required and square itself is nine.

This is how algebra used to be done; you have similar descriptions in Babylonian scribe tablets, Egyptian papyrii, Middle Age textbooks, etc.

Using symbols, the problem becomes, first, to solve $x^2+10x = 39$.
The process is to complete the square:
$$beginalign*
x^2 + 10x &= 39\
x^2 + 10x + 25 &= 64\
(x+5)^2 &= 64\
x+5 &= 8\
x &= 3
endalign*$$
Something that is much easier to do without too much thought, and certainly much less effort, than the decription. Also, the idea of completing the square is much simpler to explain in symbols than it is to do so rhetorically.

There are lots of reasons.

A major one is of course brevity. Using mathematical notation is much shorter than writing things out in full, which will make a huge difference to a 100+ page proof.

What's more, in mathematical notation, synonyms don't exist. We can say the same thing in English in two different ways, but there is only one true way to express things in notation

The third, and most significant, problem with this is: not everyone speaks English! While we read this notation in our language, other countries will read it in theirs and interpret the same meaning.

"One must be able to say at all times—instead of points, straight lines, and planes— tables, chairs, and beer mugs.” Hilbert

Because not only do we have to learn the symbols, in order to understand it we have to say the real meaning in our heads.

That doesn't match my personal experience. For me, if someone speaks a formula aloud, I have to reconstruct in my head how it looks before I can begin to understand what it means. (Sometimes "in my head" doesn't work, and I need to use paper instead).

As for advantages, here is a bit I once wrote for another answer:

Also: cheating is allowed. Very often, large parts of a formula are identical to large parts of a previous formula -- and the only thing that really matters is how it differs from the previous formula. In those cases it is expected that you'll just think to yourself, "oh, this thing is the same as that thing over there", without bothering to understand or remember exactly what "that thing" was in detail. You can just compare the relevant parts symbol for symbol without thinking.

In fact, this last point is part of the reason why formulas are designed to be compact and dense with information. It increases the chance that you can keep the entire formula in your visual short-term memory as you move your eyes from one formula to the next, and thereby make it easy to spot that some parts of them look alike, without even being conscious of the individual symbols. (Who knew all those inane spot-the-differences problems you find in kids' magazines actually train a highly relevant mathematical skill? They do!)