t^2 = 1 + x^{-\frac{1}{2}} => x^{-\frac{1}{2}} = t^2-1 => x^{-1} = (t^2-1)^2 => x = (t^2-1)^{-2} => dx = -2(t^2-1)^{-3}*2tdt = -4t(t^2-1)^{-3}dt\\" alt="\int \frac{\sqrt{1+\sqrt{x}}}{x\sqrt[4]{x^3} } dx = \int ((1+x^{\frac{1}{2}})^\frac{1}{2}*x^{-\frac{7}{4}})dx = \int (x^{-\frac{7}{4}}*(x^{\frac{1}{2}}+1)^\frac{1}{2})dx; \\|m=-\frac{7}{4}\\| a = 1\\|n = \frac{1}{2}\\ |b = 1\\|p=\frac{1}{2}\\\frac{m+1}{n} + p = -1 \in Z => t^2 = 1 + x^{-\frac{1}{2}} => x^{-\frac{1}{2}} = t^2-1 => x^{-1} = (t^2-1)^2 => x = (t^2-1)^{-2} => dx = -2(t^2-1)^{-3}*2tdt = -4t(t^2-1)^{-3}dt\\" align="absmiddle" class="latex-formula">
![\int \frac{\sqrt{1+\sqrt{x}}}{x\sqrt[4]{x^3} } dx = \int \frac{\sqrt{1+\sqrt{(t^2-1)^{-2}}}}{(t^2-1)^{-2*\frac{7}{4} }} * -4t(t^2-1)^{-3}dt = \int \frac{\sqrt{1 + \frac{1}{t^2-1}}}{(t^2-1)^{-\frac{7}{2} }} *-4t(t^2-1)^{-3}dt =](https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7B%5Csqrt%7B1%2B%5Csqrt%7Bx%7D%7D%7D%7Bx%5Csqrt%5B4%5D%7Bx%5E3%7D%20%7D%20dx%20%3D%20%5Cint%20%5Cfrac%7B%5Csqrt%7B1%2B%5Csqrt%7B%28t%5E2-1%29%5E%7B-2%7D%7D%7D%7D%7B%28t%5E2-1%29%5E%7B-2%2A%5Cfrac%7B7%7D%7B4%7D%20%7D%7D%20%2A%20-4t%28t%5E2-1%29%5E%7B-3%7Ddt%20%3D%20%5Cint%20%5Cfrac%7B%5Csqrt%7B1%20%2B%20%5Cfrac%7B1%7D%7Bt%5E2-1%7D%7D%7D%7B%28t%5E2-1%29%5E%7B-%5Cfrac%7B7%7D%7B2%7D%20%7D%7D%20%2A-4t%28t%5E2-1%29%5E%7B-3%7Ddt%20%3D "\int \frac{\sqrt{1+\sqrt{x}}}{x\sqrt[4]{x^3} } dx = \int \frac{\sqrt{1+\sqrt{(t^2-1)^{-2}}}}{(t^2-1)^{-2*\frac{7}{4} }} * -4t(t^2-1)^{-3}dt = \int \frac{\sqrt{1 + \frac{1}{t^2-1}}}{(t^2-1)^{-\frac{7}{2} }} *-4t(t^2-1)^{-3}dt =")
