integrate $ \int \frac {x dx}{\sqrt {1+x^{10}} } $ This is a tough one. Thanks.
$$\int \frac {x dx}{\sqrt {1+x^{10}} } $$
This is not a homework problem. I spend 10 hours over the course of 3 days on this. I tried:
1) substituting u for x^5 to get a tangent-like quantity in the denominator
2)subsituting u for 1+x^10
3) substituting u for sqrt(1+x^10)
4) integration by parts. This leaves me with $$\int \frac { dx}{\sqrt {1+x^{10}} } $$ which I still find difficult.
5). Multiplying denominator and numerator by sqrt(1+x^10), and then integrating.

As mentioned in the comments, you can at least simplify this integral somewhat via the substitution $u=x^2$:
$$\int\frac{x\,\mathrm{d}x}{\sqrt{1+x^{10}}}=\frac12\int\frac{\mathrm{d}u}{\sqrt{1+u^5}}.$$
Next, substituting $t=-u^5$ puts the integral in a form recognizable as the definition of an incomplete beta function: for non-negative $t$,
$$\frac12\int\frac{\mathrm{d}u}{\sqrt{1+u^5}}=\frac{\sqrt[5]{-1}}{10}\int\frac{t^{-\frac45}}{\sqrt{1-t}}\mathrm{d}t=\frac{\sqrt[5]{-1}}{10}\operatorname{B}{\left(t;\frac15,\frac12\right)}+\text{const}.$$
