\[ \int \frac {\sqrt {1+x} \left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
integrate((1+x)**(1/2)*(x**4-1)*(1+(1+x)**(1/2))**(1/2)/(x**4+1),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {4 \left (\sqrt {x + 1} + 1\right )^{\frac {7}{2}}}{7} - \frac {8 \left (\sqrt {x + 1} + 1\right )^{\frac {5}{2}}}{5} + \frac {4 \left (\sqrt {x + 1} + 1\right )^{\frac {3}{2}}}{3} - 8 \operatorname {RootSum} {\left (73786976294838206464 t^{16} + 1152921504606846976 t^{14} + 9007199254740992 t^{12} + 35184372088832 t^{10} + 223338299392 t^{8} + 671088640 t^{6} + 1048576 t^{4} + 1, \left ( t \mapsto t \log {\left (\frac {41505174165846491136 t^{15}}{247} + \frac {157625986957967360 t^{13}}{247} - \frac {1987917023019008 t^{11}}{247} - \frac {2508260900864 t^{9}}{19} - \frac {5368709120 t^{7}}{19} - \frac {1142423552 t^{5}}{247} - \frac {2942976 t^{3}}{247} - \frac {4548 t}{247} + \sqrt {\sqrt {x + 1} + 1} \right )} \right )\right )} - 8 \operatorname {RootSum} {\left (73786976294838206464 t^{16} + 1152921504606846976 t^{14} + 2206763817411543040 t^{12} - 26511424368934912 t^{10} + 15161114295795712 t^{8} + 1441095745536 t^{6} + 1137704960 t^{4} - 10240 t^{2} + 1, \left ( t \mapsto t \log {\left (\frac {25799730214666355850404468800693993472 t^{15}}{108743714938181051945} + \frac {403861901874869101489646848100532224 t^{13}}{108743714938181051945} + \frac {771609972177521912726952404102152192 t^{11}}{108743714938181051945} - \frac {1849519098743151228176137260630016 t^{9}}{21748742987636210389} + \frac {5300841208392062031314032384802816 t^{7}}{108743714938181051945} + \frac {656153520626827815900553936896 t^{5}}{108743714938181051945} + \frac {415462360730314920286783488 t^{3}}{108743714938181051945} + \frac {3116941937477144828564 t}{108743714938181051945} + \sqrt {\sqrt {x + 1} + 1} \right )} \right )\right )} + 16 \operatorname {RootSum} {\left (73786976294838206464 t^{16} + 1152921504606846976 t^{14} + 112589990684262400 t^{12} + 721279627821056 t^{10} + 46368466927616 t^{8} - 94086627328 t^{6} + 64487424 t^{4} - 14336 t^{2} + 1, \left ( t \mapsto t \log {\left (- \frac {220664868483694780793608994816 t^{15}}{14117716493} - \frac {3481032216261090566771572736 t^{13}}{14117716493} - \frac {337229516699789748290453504 t^{11}}{14117716493} - \frac {2207666506599210485284864 t^{9}}{14117716493} - \frac {138997739819058925142016 t^{7}}{14117716493} + \frac {260507999712158679040 t^{5}}{14117716493} - \frac {152968699413618688 t^{3}}{14117716493} + \frac {18852877983396 t}{14117716493} + \sqrt {\sqrt {x + 1} + 1} \right )} \right )\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Exception raised: SympifyError} \]________________________________________________________________________________________