\[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {2 A}{a \sqrt {x}}-\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {x}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {2}\, \sqrt {c}\, \left (A +\frac {A b -2 B a}{\sqrt {-4 a c +b^{2}}}\right )}{a \sqrt {b -\sqrt {-4 a c +b^{2}}}}-\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {x}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {2}\, \sqrt {c}\, \left (A +\frac {-A b +2 B a}{\sqrt {-4 a c +b^{2}}}\right )}{a \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate((B*x+A)/x**(3/2)/(c*x**2+b*x+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________