\[ \int \frac {c+d x^2+e x^4+f x^6}{x^9 \sqrt {a+b x^2}} \, dx \]
Optimal antiderivative \[ -\frac {b \left (-64 a^{3} f +48 a^{2} b e -40 a \,b^{2} d +35 b^{3} c \right ) \arctanh \left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{128 a^{\frac {9}{2}}}-\frac {c \sqrt {b \,x^{2}+a}}{8 a \,x^{8}}+\frac {\left (-8 a d +7 b c \right ) \sqrt {b \,x^{2}+a}}{48 a^{2} x^{6}}-\frac {\left (48 a^{2} e -40 a b d +35 b^{2} c \right ) \sqrt {b \,x^{2}+a}}{192 a^{3} x^{4}}+\frac {\left (-64 a^{3} f +48 a^{2} b e -40 a \,b^{2} d +35 b^{3} c \right ) \sqrt {b \,x^{2}+a}}{128 a^{4} x^{2}} \]
command
integrate((f*x**6+e*x**4+d*x**2+c)/x**9/(b*x**2+a)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {c}{8 \sqrt {b} x^{9} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {d}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {e}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} c}{48 a x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} d}{24 a x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} e}{8 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {\sqrt {b} f \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} - \frac {7 b^{\frac {3}{2}} c}{192 a^{2} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 b^{\frac {3}{2}} d}{48 a^{2} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b^{\frac {3}{2}} e}{8 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {35 b^{\frac {5}{2}} c}{384 a^{3} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 b^{\frac {5}{2}} d}{16 a^{3} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {35 b^{\frac {7}{2}} c}{128 a^{4} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} - \frac {3 b^{2} e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {5}{2}}} + \frac {5 b^{3} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {7}{2}}} - \frac {35 b^{4} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{128 a^{\frac {9}{2}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________