Solve the integral $\int \frac{dx}{\:\sqrt[4]{\left(x+2\right)^5\cdot \left(x-1\right)^3}}$

Question

Here is a indefinite integral must be solved. Help, who knows. Although it would be like casual. $$\int \frac{dx}{\:\sqrt[4]{\left(x+2\right)^5\cdot \left(x-1\right)^3}}$$

Answer 1

I think I may see something.

$$u=x+2,du=dx$$

$$\int\frac{dx}{\sqrt[4]{(x+2)^5(x-1)^3}}=\int\frac{du}{\sqrt[4]{u^5(u-3)^3}}=$$

$$\int\frac{du}{u^2\sqrt[4]{(\frac{u-3}{u}})^3}=\int\frac{(1-\frac3u)^{-3/4}du}{u^2}$$

$$t=1-\frac3u,dt=\frac{3du}{u^2}$$

$$\int\frac{(1-\frac3u)^{-3/4}du}{u^2}=\frac13\int t^{-3/4}dt=\frac43\sqrt[4]t=\frac43\sqrt[4]{1-\frac3{x+2}}+C$$

Answer 2

Setting $\frac{x-1}{x+2}=u$ gives you $$dx=\frac{(x+2)^2}{3}du.$$ So, $$\int(x+2)^{-\frac 54}(x-1)^{-\frac 34}dx=\int\frac 13u^{-\frac 34}du=\frac 43\sqrt[4]{\frac{x-1}{x+2}}+C.$$