How can I integrate $\sqrt{ \frac{x}{ax^3 + b}}$ analytically?

Question

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?

Answer 1

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