software for algebraic simplifying expressions

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?

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I am not sure if wolfram alpha works like mathematica where one can define what $Y$ is, and then type in "Full Simplify[$\frac{8Y}{1+x} - \ldots + \frac{Kx(1+5x)^{3/2}}{2}$]." –  fpqc Jan 2 '12 at 15:05
sagemath.org –  David Mitra Jan 2 '12 at 15:23
just another doubt i just got, cant matlab be used? ( i know author is asking for a free software) –  Bhargav Jan 2 '12 at 18:30

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.

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Interesting. Are there statistics showing this is "most popular"? –  GEdgar Jan 2 '12 at 16:02
You can use Maxima inside of Sage, as well as many other features that are not in Maxima. –  Graphth Jan 2 '12 at 18:30

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.

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One can argue whether it still can be treated as rational function because of that nasty $(1+x)^{n+2}$. The difference between $x^5$ and $x^n$ however negligible it seems to be can turn to a stopper. Well, at least for math software OP is interested in. –  Yrogirg May 25 '12 at 16:44
@Yrogirg Yes, I am aware of that (I was a lead developer of Macsyma). –  Bill Dubuque May 25 '12 at 16:47