Integrate $\frac{5x^3 +2}{\sqrt{x^3+1}}$
$$\int\frac{5x^3+2}{\sqrt{x^3+1}}\,\mathrm{d}x$$
Sad to say this has really stumped me and nothing I have tried has worked.
I used Wolfram Alpha to find that the answer is simply $2x\sqrt{x^3+1}$ but it says the method is unavailable and I have no idea which method to use. Hints would be appreciated.
Solution 1:
Observe that $$\int \frac{5x^3+2}{\sqrt{x^3+1}}dx=\int \frac{x(5x^3+2)}{x\sqrt{x^3+1}}dx=\int \frac{5x^4+2x}{\sqrt{x^5+x^2}}dx$$ and make the $u$-substitution $u=x^5+x^2$.