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.
On
I spend $10$ hours over the course of $3$ days on this.
You shouldn't have. Antiderivatives of the form $~\displaystyle\int\frac{dx}{\sqrt[n]{1+x^m}}~$ and $~\displaystyle\int\sqrt[n]{1+x^m}~dx~$, with both
m and n natural numbers, cannot be expressed in terms of elementary functions, unless either $m=1,~$ or $~m,n\le2$. This can be proven using either Liouville's theorem or the Risch algorithm, though I think that goes beyond the scope of this question. So unless you didn't have the definite
integral $~\displaystyle\int_0^\infty\frac{dx}{\sqrt[n]{1+x^m}}~$ in mind, which returns an expression in terms of the beta function, as
can be shown by letting $t=\dfrac1{1+x^m}$ , I am afraid that the only closed forms available are those in
terms of hypergeometric functions or elliptic integrals, where the former can be obtained by simply expanding the integrand into its binomial series, and then switching the order of summation and integration.
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 $t$,
$$\frac12\int\frac{\mathrm{d}u}{\sqrt{1+u^5}}=\frac{\sqrt[5]{-1}}{10}\int\frac{t^{-\frac45}}{\sqrt{1-t}}\mathrm{d}t=\frac{\sqrt[5]{-1}}{10}\operatorname{B}{\left(t;\frac15,\frac12\right)}+\text{const}.$$