\[ \int \frac {x \left (1+a+a^2+x^2+a x^2+b x^2+2 a b x^2+b x^4+b^2 x^4\right )}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {1}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {\arctanh \left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a -b}}\right )}{\sqrt {a -b}}-\frac {1}{b \sqrt {b \,x^{2}+a}} \]
command
integrate(x*(b**2*x**4+b*x**4+2*a*b*x**2+a*x**2+b*x**2+a**2+x**2+a+1)/(x**2+1)/(b*x**2+a)**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {\operatorname {atan}{\left (\frac {\sqrt {a + b x^{2}}}{\sqrt {- a + b}} \right )}}{\sqrt {- a + b}} - \frac {1}{b \sqrt {a + b x^{2}}} - \frac {1}{3 b \left (a + b x^{2}\right )^{\frac {3}{2}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________