\[ \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
integrate(((x**2-4*x+1)**(1/2)+(x**2-4*x+1)**(3/2))/((x**2-4*x+1)**(1/2)+(x**2-4*x+1)**(3/2)-(x**2-4*x+1)**(5/2)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - 2 \operatorname {RootSum} {\left (4400 t^{4} - 140 t^{2} + 1, \left ( t \mapsto t \log {\left (3080 t^{3} - 54 t + x - 2 \right )} \right )\right )} - \operatorname {RootSum} {\left (4400 t^{4} - 14340 t^{2} + 400 t + 1, \left ( t \mapsto t \log {\left (\frac {102640 t^{3}}{18179} - \frac {19200 t^{2}}{18179} - \frac {75898 t}{4081} + x + \frac {21202}{199969} \right )} \right )\right )} + 4 \operatorname {RootSum} {\left (4400 t^{4} - 1000 t^{2} + 80 t - 1, \left ( t \mapsto t \log {\left (\frac {275 t^{3}}{7} - \frac {110 t^{2}}{7} - \frac {305 t}{28} + x + \frac {9}{28} \right )} \right )\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Exception raised: SympifyError} \]________________________________________________________________________________________