How can I integrate $\sqrt{ \frac{x}{ax^3 + b}}$ analytically? How can I calculate  $$\int^1_0\sqrt{\frac{x}{ax^3 + b}} \mathrm  dx$$ analytically?
I searched my integral table, but I haven't found the solution. 
But using WolframAlpha, I could find the analytic result like this,
$$\int \sqrt{\frac{x}{ax^3 + b}} \mathrm dx = \frac{
2 \sqrt{\frac{x}{ax^3+b}} \sqrt{ax^3+b} \log{(\sqrt{a}\sqrt{ax^3 +b} + ax^{3/2})}}{3\sqrt{a}\sqrt{x}} + C \\= \frac{
2\log{(\sqrt{a}\sqrt{ax^3 +b} + ax^{3/2})}}{3\sqrt{a}} + C$$
How can I obtain this result?

Hint...substitute $u^2=x^3$ and you have a standard $\operatorname{arsinh}$ integral
