\[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (-3 a c +b^{2}\right ) x^{2}}{c^{2} \left (-4 a c +b^{2}\right )}-\frac {b \,x^{4}}{2 c \left (-4 a c +b^{2}\right )}+\frac {x^{6} \left (b \,x^{2}+2 a \right )}{2 \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {\left (6 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right ) \arctanh \! \left (\frac {2 c \,x^{2}+b}{\sqrt {-4 a c +b^{2}}}\right )}{c^{3} \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {b \ln \! \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c^{3}} \]
command
integrate(x**11/(c*x**5+b*x**3+a*x)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \left (- \frac {b}{2 c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{2 c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x^{2} + \frac {- 5 a^{2} b c - 16 a^{2} c^{4} \left (- \frac {b}{2 c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{2 c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + a b^{3} + 8 a b^{2} c^{3} \left (- \frac {b}{2 c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{2 c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - b^{4} c^{2} \left (- \frac {b}{2 c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{2 c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right )}{6 a^{2} c^{2} - 6 a b^{2} c + b^{4}} \right )} + \left (- \frac {b}{2 c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{2 c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x^{2} + \frac {- 5 a^{2} b c - 16 a^{2} c^{4} \left (- \frac {b}{2 c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{2 c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + a b^{3} + 8 a b^{2} c^{3} \left (- \frac {b}{2 c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{2 c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - b^{4} c^{2} \left (- \frac {b}{2 c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{2 c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right )}{6 a^{2} c^{2} - 6 a b^{2} c + b^{4}} \right )} + \frac {- 3 a^{2} b c + a b^{3} + x^{2} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{8 a^{2} c^{4} - 2 a b^{2} c^{3} + x^{4} \left (8 a c^{5} - 2 b^{2} c^{4}\right ) + x^{2} \left (8 a b c^{4} - 2 b^{3} c^{3}\right )} + \frac {x^{2}}{2 c^{2}} \]