<s>[INST] Calculate:$\int \frac{1}{(x+1)^\frac{3}{4}(x+2)^{\frac{5}{4}}}\ dx$ Calculate following integration
$$\int \frac{1}{(x+1)^\frac{3}{4}(x+2)^{\frac{5}{4}}}\ dx$$ 
 [/INST] We have
$$I=\int \frac{1}{(x+1)^\frac{3}{4}(x+2)^{\frac{5}{4}}}\ dx=\int\frac{dx}{\left(\frac{x+1}{x+2}\right)^{3/4}(x+2)^2}$$ 
Now we pose $t=\frac{x+1}{x+2}$ then $x=\frac{1-2t}{t-1}$ and $dx=\frac{dt}{(t-1)^2}$, so
$$I=\int t^{-3/4}dt=4t^{1/4}+C,$$
hence
$$I=4\left(\frac{x+1}{x+2}\right)^{1/4}+C.$$
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