I have many huge algebraic expressions such as:
$$\frac{8Y}{1+x}-\frac{(1-Y)}{x}+\frac{Kx(1+5x)^{3/5}}{2}$$
where $\ Y=\dfrac{Kx(1+x)^{n+2}}{(n+4)(1+5x)^{2/5}}+\dfrac{7-10x-x^2}{7(1+x)^2}+\dfrac{Ax}{(1+5x)^{2/5}(1+x)^2}\ $ and $A,n$ are constants.
To simplify these expressions by hand is taking me a lot of time and there is also the danger of making a mistake. I am looking for a free software on the internet using which I can simplify these expressions. Does anyone have any recommendations?
On
Try WolframAlpha, http://www.wolframalpha.com/. It is free on the internet. But another one you can also try which is very easy to use is Maple. I cannot tell if it is free.
On
Note that if you set $\rm\ z = (5x+1)^{1/5}\ $ then your computations reduce to rational function arithmetic combined with the rewrite rule $\rm\: z^5\ \to\ 5x+1\ $ with the following expressions
$$\frac{8Y}{1+x}-\frac{(1-Y)}{x}+\frac{Kxz^3}{2}$$
where $\ Y\ =\ \dfrac{Kx(1+x)^{n+2}}{(n+4)z^2}+\dfrac{7-10x-x^2}{7(1+x)^2}+\dfrac{Ax}{(z(1+x))^2}\ $ and $A,n$ are constants.
This is so simple that it can be done by hand. When using computer algebra systems you need to be sure that they can effectively compute with algebraic functions, or that they can effectively handle said rewrite rule implementing this simple special case. For example, in Macsyma (or Maxima, e.g. in Sage) one may use $\rm\:radcan\:$ (RADical CANonicalize) or, alternatively, set $\rm\:algebraic:true\:$ and do $\rm\:tellrat(\:z^5 =\: 5*x+1)\ $ and then employ the $\rm\:rat\:$ function to normalize such "rational" expressions.
The most popular open-source symbolic math software is Maxima: http://maxima.sourceforge.net/ It's pretty sophisticated. You could also check out sage, yacas, and axiom.