# Integrate an expression and simplify the result

I have to find the indefinite integral of the following:

$$\int\sqrt{ax^4 + bx^3 + cx^2 + dx + e} dx$$

where a, b, c, d, e are constant of the order 0.0001 or less but always greater than 0.

I have used the following code:

int = Integrate[Sqrt[(a*x^4 + b*x^3 + c*x^2 + d*x + e)], x]


which gives out a lengthy solution. I have tried to simplify the solution using

Simplify[int, Assumptions -> 0 < a < 0.0001 && 0 < b < 0.0001 && 0 < c < 0.0001 && 0 < d < 0.0001 && 0 < e < 0.0001]


I have tried simplifying the solution with FullSimplify, but it is taking a lot of time and the kernel crashed meanwhile.

Is there a way I can solve the integral and simplify its answer so that I can adopt the simplified form of general solution outside Mathematica?

• You do realize that this is a non-elementary elliptic integral, yes? – J. M.'s technical difficulties Mar 18 '16 at 12:48
• @J.M. My apologies, I am not familiar with non-elementary elliptic integrals. Since I have a very basic calculus background, I will look for it. – mrkbtr Mar 18 '16 at 12:52
• "a very basic calculus background" - you're in for a wild ride down the rabbit hole, then. You can first look at this. Then, if you can, try to look for the book by Byrd and Friedman; this is a book entirely devoted to handling integrals of this sort. – J. M.'s technical difficulties Mar 18 '16 at 13:03
• As is almost always our favorite tool and its keen helpers have some vitamins on board, i.e. see Elliptic Integral of the First Kind – user9660 Mar 18 '16 at 13:25
• Use this as csub[Hold[Evaluate[yourIntegralResult]], {}, n] varying n to extract common subexpressions – Dr. belisarius Mar 18 '16 at 15:13