\[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \! \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}{\sqrt {c}\, \sqrt {-4 a c +b^{2}}}+\frac {\arctanh \! \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}{\sqrt {c}\, \sqrt {-4 a c +b^{2}}} \]
command
integrate((e*x+d)**(1/2)/(c*x**2+b*x+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ 2 e \operatorname {RootSum} {\left (t^{4} \left (256 a^{2} c^{3} e^{4} - 128 a b^{2} c^{2} e^{4} + 16 b^{4} c e^{4}\right ) + t^{2} \left (- 16 a b c e^{3} + 32 a c^{2} d e^{2} + 4 b^{3} e^{3} - 8 b^{2} c d e^{2}\right ) + a e^{2} - b d e + c d^{2}, \left ( t \mapsto t \log {\left (64 t^{3} a c^{2} e^{2} - 16 t^{3} b^{2} c e^{2} - 2 t b e + 4 t c d + \sqrt {d + e x} \right )} \right )\right )} \]