Difficulty with evaluating an integral

[throneoo]

Homework Statement

How to evaluate the anti-derivative of [x+(7x^2)]^(1/3) ?

2. The attempt at a solution

I've attempted to do so via standard integration techniques , i.e. substitution and integration by parts , but the problem simply becomes increasingly complicated . I've even tried to manipulate it into a perfect square in hopes of getting an easier expression but that failed too.

My substitution : u=(x+7x^2) ; du=(1+14x)dx and x=(+/-(1+28u)^(0.5)-1)/14
making the intergrand u^(1/3)(1+28u)^-0.5

with by parts things get so complicated it terrified me.

Frustrated , I tried wolfram alpha ,only to get sth like this :
http://www4c.wolframalpha.com/Calculate/MSP/MSP4182022fe0gi4018be9000021e9375hf3bi1a91?MSPStoreType=image/gif&s=36&w=569.&h=56.

I don't even know what a hypergeometric function is

Can anyone help me ?

 

[HallsofIvy]

If Wolfram alpha is giving the answer in terms of a "hypergeometric" function, I would take that as an indication that the anti-derivative cannot be written in terms of "elementary" functions. Where did you get that integral and what reason do you have to believe that it can be written in terms of elementary functions?

 

[throneoo]

It was from the book Feynman's Tips on Physics , where the author mentions integration as a prerequisite for doing physics.

the 5 examples of integrand he gives are (1+6t) , (4t^2+2t^3) , (1+2t)^3 , (1+5t)^0.5 and [t+(7t^2)]^(1/3).
the first 4 examples' anti-derivatives are given in terms of "elementary functions" , while the last one is left with some question marks . so I thought the last one was meant as an exercise and decided to give it a try , assuming it's as simple as the other examples without any particular reasons.

Edit: upon a closer look into the text , Feynman says it isn't necessary to be able to integrate simple expressions and the last one is not possible to integrate in an easy fashion , implying it's not really as simple as it looks . lol

 

[exo]

This is a differential binomial.It's an expression of the type x^{m}(a+bx^{n})^{p}dx
In your case that becomes x^{1/3}(1+7x)^{1/3}dx
If \frac{m+1}{n} is an integer you can make the substitution u = ax^{n}+b
If \frac{m+1}{n}+p is an integer you can make the substitution u = a+bx^{-n}
In all other cases this integral cannot be expressed by elementary functions

 

[throneoo]

This is a differential binomial.It's an expression of the type x^{m}(a+bx^{n})^{p}
In your case that becomes x^{1/3}(1+7x)^{1/3}
If \frac{m+1}{n} is an integer you can make the substitution u = ax^{n}+b
If \frac{m+1}{n}+p is an integer you can make the substitution u = a+bx^{-n}
In all other cases this integral cannot be expressed by elementary functions

wow..differential binomials must be quite some significant objects to be studied so precisely...
Thanks for the help . I guess I can't go anywhere further in this problem with my current knowledge