\[ \int \frac {c+d x^2+e x^4+f x^6}{x^5 \sqrt {a+b x^2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (8 a^{2} e -4 a b d +3 b^{2} c \right ) \arctanh \left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{8 a^{\frac {5}{2}}}+\frac {f \sqrt {b \,x^{2}+a}}{b}-\frac {c \sqrt {b \,x^{2}+a}}{4 a \,x^{4}}+\frac {\left (-4 a d +3 b c \right ) \sqrt {b \,x^{2}+a}}{8 a^{2} x^{2}} \]
command
integrate((f*x**6+e*x**4+d*x**2+c)/x**5/(b*x**2+a)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ f \left (\begin {cases} \frac {x^{2}}{2 \sqrt {a}} & \text {for}\: b = 0 \\\frac {\sqrt {a + b x^{2}}}{b} & \text {otherwise} \end {cases}\right ) - \frac {c}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} c}{8 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {\sqrt {b} d \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} + \frac {3 b^{\frac {3}{2}} c}{8 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} + \frac {b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} - \frac {3 b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {5}{2}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________