\[ \int \frac {x^{3/2} (A+B x)}{a+b x+c x^2} \, dx \]
Optimal antiderivative \[ \frac {2 B \,x^{\frac {3}{2}}}{3 c}-\frac {2 \left (-A c +b B \right ) \sqrt {x}}{c^{2}}+\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {x}}{\sqrt {b -\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {2}\, \left (b^{2} B -A b c -a B c +\frac {-2 a A \,c^{2}+A \,b^{2} c +3 a b B c -B \,b^{3}}{\sqrt {-4 a c +b^{2}}}\right )}{c^{\frac {5}{2}} \sqrt {b -\sqrt {-4 a c +b^{2}}}}+\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {x}}{\sqrt {b +\sqrt {-4 a c +b^{2}}}}\right ) \sqrt {2}\, \left (b^{2} B -A b c -a B c +\frac {2 a A \,c^{2}-A \,b^{2} c -3 a b B c +B \,b^{3}}{\sqrt {-4 a c +b^{2}}}\right )}{c^{\frac {5}{2}} \sqrt {b +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(x**(3/2)*(B*x+A)/(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 \[ \text {Timed out} \]_____________________________________________________