\[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {c}{5 a^{3} x^{5}}+\frac {-a d +3 b c}{3 a^{4} x^{3}}+\frac {-a^{2} e +3 a b d -6 b^{2} c}{a^{5} x}-\frac {\left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) x}{4 a^{4} \left (b \,x^{2}+a \right )^{2}}-\frac {\left (-3 a^{3} f +7 a^{2} b e -11 a \,b^{2} d +15 b^{3} c \right ) x}{8 a^{5} \left (b \,x^{2}+a \right )}-\frac {\left (-3 a^{3} f +15 a^{2} b e -35 a \,b^{2} d +63 b^{3} c \right ) \arctan \! \left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{8 a^{\frac {11}{2}} \sqrt {b}} \]
command
integrate((f*x**6+e*x**4+d*x**2+c)/x**6/(b*x**2+a)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {\sqrt {- \frac {1}{a^{11} b}} \left (3 a^{3} f - 15 a^{2} b e + 35 a b^{2} d - 63 b^{3} c\right ) \log {\left (- a^{6} \sqrt {- \frac {1}{a^{11} b}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{11} b}} \left (3 a^{3} f - 15 a^{2} b e + 35 a b^{2} d - 63 b^{3} c\right ) \log {\left (a^{6} \sqrt {- \frac {1}{a^{11} b}} + x \right )}}{16} + \frac {- 24 a^{4} c + x^{8} \left (45 a^{3} b f - 225 a^{2} b^{2} e + 525 a b^{3} d - 945 b^{4} c\right ) + x^{6} \left (75 a^{4} f - 375 a^{3} b e + 875 a^{2} b^{2} d - 1575 a b^{3} c\right ) + x^{4} \left (- 120 a^{4} e + 280 a^{3} b d - 504 a^{2} b^{2} c\right ) + x^{2} \left (- 40 a^{4} d + 72 a^{3} b c\right )}{120 a^{7} x^{5} + 240 a^{6} b x^{7} + 120 a^{5} b^{2} x^{9}} \]