\[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {c}{3 a^{3} x^{3}}+\frac {-a d +3 b c}{a^{4} x}+\frac {\left (\frac {b^{2} c}{a^{2}}-\frac {b d}{a}+e -\frac {a f}{b}\right ) x}{4 a \left (b \,x^{2}+a \right )^{2}}+\frac {\left (a^{3} f +3 a^{2} b e -7 a \,b^{2} d +11 b^{3} c \right ) x}{8 a^{4} b \left (b \,x^{2}+a \right )}+\frac {\left (a^{3} f +3 a^{2} b e -15 a \,b^{2} d +35 b^{3} c \right ) \arctan \! \left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{8 a^{\frac {9}{2}} b^{\frac {3}{2}}} \]
command
integrate((f*x**6+e*x**4+d*x**2+c)/x**4/(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
\[ - \frac {\sqrt {- \frac {1}{a^{9} b^{3}}} \left (a^{3} f + 3 a^{2} b e - 15 a b^{2} d + 35 b^{3} c\right ) \log {\left (- a^{5} b \sqrt {- \frac {1}{a^{9} b^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{9} b^{3}}} \left (a^{3} f + 3 a^{2} b e - 15 a b^{2} d + 35 b^{3} c\right ) \log {\left (a^{5} b \sqrt {- \frac {1}{a^{9} b^{3}}} + x \right )}}{16} + \frac {- 8 a^{3} b c + x^{6} \left (3 a^{3} b f + 9 a^{2} b^{2} e - 45 a b^{3} d + 105 b^{4} c\right ) + x^{4} \left (- 3 a^{4} f + 15 a^{3} b e - 75 a^{2} b^{2} d + 175 a b^{3} c\right ) + x^{2} \left (- 24 a^{3} b d + 56 a^{2} b^{2} c\right )}{24 a^{6} b x^{3} + 48 a^{5} b^{2} x^{5} + 24 a^{4} b^{3} x^{7}} \]