Question Number 140339 by mohammad17 last updated on 06/May/21
$$\int\frac{{dx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }} \\ $$
Answered by Dwaipayan Shikari last updated on 06/May/21
$$\int\frac{{dx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }}=\underset{{n}\geqslant\mathrm{0}} {\sum}\int\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{n}} }{{n}!}\left(−{x}^{\mathrm{4}} \right)^{{n}} \\ $$$$={x}\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)_{{n}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)_{{n}} }{{n}!\left(\frac{\mathrm{5}}{\mathrm{4}}\right)_{{n}} }\left(−{x}^{\mathrm{4}} \right)^{{n}} ={x}\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{4}},\frac{\mathrm{5}}{\mathrm{4}};−{x}^{\mathrm{4}} \right)+{C} \\ $$