This description is very plausible, but entirely wrong. It was almost completely the opposite of what you're saying. The Muslim mathematicians used fewer symbols than the Greek tradition they inherited for almost the entire timeline of medieval Arabic/Islamic mathematics. The "first textbook" you're referring too, Al-Khwarizmi's Al-jabr wa'l muqabalah, the one ultimately responsible for the word "algebra", did not use any symbols at all, and wrote everything out in words.

Greek mathematicians started to use something like symbols (abbreviated letters with fixed positional meaning) by the time of Diophantus around 3rd century CE. The Arab mathematicians did not adopt that when they translated the Greek texts, and for the first 500 years of their work, wrote everything out fully. Moreover, it is those texts devoid of any symbolic systems that were translated into Latin and used to help fuel the European tradition in 12th-13th centuries CE. Even though some Islamic mathematicians later did develop the beginnings of a symbolic notation, in 14th-15th centuries, this happened roughly in parallel with the Europeans inventing their own symbols, and did not influence the modern tradition that derives from those European symbols.

To be sure, Muslim mathematicians were much better algebraists than the Greeks. But that was because Greeks never quite reached the idea of decoupling numbers (and unknown quantities) from geometry and manipulating them as separate objects on their own. The Muslim mathematicians were able to do that (and as a result, much more), despite not having any symbolic system at their disposal.