\[ \int \frac {\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}}{\sqrt {1-4 x+x^2}+\left (1-4 x+x^2\right )^{3/2}-\left (1-4 x+x^2\right )^{5/2}} \, dx \]
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, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {{\left ({\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{2} + 4 \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 6\right )} \log \left (x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}{2 \, {\left (2 \, {\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{3} + 12 \, {\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{2} + 17 \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 32\right )}} - \frac {{\left ({\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{2} - 4 \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 6\right )} \log \left (x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}{2 \, {\left (2 \, {\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{3} - 12 \, {\left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{2} + 17 \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 32\right )}} + \frac {{\left ({\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{2} + 4 \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 6\right )} \log \left (x + \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}{2 \, {\left (2 \, {\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{3} + 12 \, {\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}^{2} + 17 \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 32\right )}} - \frac {{\left ({\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{2} - 4 \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 6\right )} \log \left (x - \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} - 2\right )}{2 \, {\left (2 \, {\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{3} - 12 \, {\left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 2\right )}^{2} + 17 \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {7}{2}} + 32\right )}} \]