\[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {b \,x^{2}}{2 c \left (-4 a c +b^{2}\right )}+\frac {x^{4} \left (b \,x^{2}+2 a \right )}{2 \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {b \left (-6 a c +b^{2}\right ) \arctanh \! \left (\frac {2 c \,x^{2}+b}{\sqrt {-4 a c +b^{2}}}\right )}{2 c^{2} \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {\ln \! \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{2}} \]
command
integrate(x**7/(c*x**4+b*x**2+a)**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 \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) \log {\left (x^{2} + \frac {- 32 a^{2} c^{3} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) + 8 a^{2} c + 16 a b^{2} c^{2} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) - a b^{2} - 2 b^{4} c \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) \log {\left (x^{2} + \frac {- 32 a^{2} c^{3} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) + 8 a^{2} c + 16 a b^{2} c^{2} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right ) - a b^{2} - 2 b^{4} c \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{4 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \frac {2 a^{2} c - a b^{2} + x^{2} \left (3 a b c - b^{3}\right )}{8 a^{2} c^{3} - 2 a b^{2} c^{2} + x^{4} \left (8 a c^{4} - 2 b^{2} c^{3}\right ) + x^{2} \left (8 a b c^{3} - 2 b^{3} c^{2}\right )} \]