Integrating algebraic functions The function $y = f(x)$, restricted on the domain $ 0 < x < 1$ and satisfying
$$y^{5}+y^{4} + x = 0,$$
seems to be well-defined and smooth. So how does one integrate this thing? That is, what is $\int_{0}^{1} f(x) dx$? 
Of course, one can use Newton's method to approximate $f(x)$ for any given value of $x$ and numerically integrate the result. But this feels uninspiring. So I was wondering if there is a general "trick" or insight to integrating algebraic functions.

As MPW says, in general there is no reason to expect that the integral should be easy to compute. However, you can do the following. Differentiate the equation once to obtain $$0=5y'y^4+4y'y^3+1=y'y^3(y+4(y+1))+1. $$ Then use the equation to write $$ y+1 = -\frac{x}{y^4}$$ and plug this into the above to get $$-1= y'y^3(y+4(-\frac{x}{y^4}))=\frac{y'y^5-4xy'}{y},$$ or $$ y=4xy'-y'y^5.$$ Now you're in business: $$ \int_0^1 yd x=\int_0^1(4xy'-y'y^5)d x=4y(1)-4\int_0^1yd x -\frac{1}{6} y(1)^6.$$ Solving for the desired quantity you get $$ \int_0^1 yd x=-\frac{1}{30}y(1)^6+\frac{4}{5}y(1). $$
