\[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {c}{7 a^{2} x^{7}}+\frac {-a d +2 b c}{5 a^{3} x^{5}}+\frac {-a^{2} e +2 a b d -3 b^{2} c}{3 a^{4} x^{3}}+\frac {-a^{3} f +2 a^{2} b e -3 a \,b^{2} d +4 b^{3} c}{a^{5} x}+\frac {b \left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) x}{2 a^{5} \left (b \,x^{2}+a \right )}+\frac {\left (-3 a^{3} f +5 a^{2} b e -7 a \,b^{2} d +9 b^{3} c \right ) \arctan \! \left (\frac {x \sqrt {b}}{\sqrt {a}}\right ) \sqrt {b}}{2 a^{\frac {11}{2}}} \]
command
integrate((f*x**6+e*x**4+d*x**2+c)/x**8/(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
\[ \frac {\sqrt {- \frac {b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right ) \log {\left (- \frac {a^{6} \sqrt {- \frac {b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right )}{3 a^{3} b f - 5 a^{2} b^{2} e + 7 a b^{3} d - 9 b^{4} c} + x \right )}}{4} - \frac {\sqrt {- \frac {b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right ) \log {\left (\frac {a^{6} \sqrt {- \frac {b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right )}{3 a^{3} b f - 5 a^{2} b^{2} e + 7 a b^{3} d - 9 b^{4} c} + x \right )}}{4} + \frac {- 30 a^{4} c + x^{8} \left (- 315 a^{3} b f + 525 a^{2} b^{2} e - 735 a b^{3} d + 945 b^{4} c\right ) + x^{6} \left (- 210 a^{4} f + 350 a^{3} b e - 490 a^{2} b^{2} d + 630 a b^{3} c\right ) + x^{4} \left (- 70 a^{4} e + 98 a^{3} b d - 126 a^{2} b^{2} c\right ) + x^{2} \left (- 42 a^{4} d + 54 a^{3} b c\right )}{210 a^{6} x^{7} + 210 a^{5} b x^{9}} \]