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