\[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{(e+f x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (4 e f \left (-2 a e f -b d f +3 b \,e^{2}\right )-c \left (-d^{2} f^{2}-6 d \,e^{2} f +15 e^{4}\right )\right ) \arctanh \! \left (\frac {\sqrt {f}\, \sqrt {e x +d}}{\sqrt {e}\, \sqrt {f x +e}}\right )}{4 e^{\frac {3}{2}} f^{\frac {7}{2}}}+\frac {2 \left (a +\frac {e \left (-b f +c e \right )}{f^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{\left (-d f +e^{2}\right ) \sqrt {f x +e}}+\frac {c \left (e x +d \right )^{\frac {3}{2}} \sqrt {f x +e}}{2 e \,f^{2}}+\frac {\left (4 e f \left (-2 a e f -b d f +3 b \,e^{2}\right )-c \left (-d^{2} f^{2}-6 d \,e^{2} f +15 e^{4}\right )\right ) \sqrt {e x +d}\, \sqrt {f x +e}}{4 e \,f^{3} \left (-d f +e^{2}\right )} \]
command
integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)/(f*x+e)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {{\left ({\left (x e + d\right )} {\left (\frac {2 \, {\left (x e + d\right )} c e^{\left (-1\right )}}{f} - \frac {{\left (3 \, c d f^{4} e^{2} - 4 \, b f^{4} e^{3} + 5 \, c f^{3} e^{4}\right )} e^{\left (-3\right )}}{f^{5}}\right )} + \frac {{\left (c d^{2} f^{4} e^{2} - 4 \, b d f^{4} e^{3} + 6 \, c d f^{3} e^{4} - 8 \, a f^{4} e^{4} + 12 \, b f^{3} e^{5} - 15 \, c f^{2} e^{6}\right )} e^{\left (-3\right )}}{f^{5}}\right )} \sqrt {x e + d}}{4 \, \sqrt {{\left (x e + d\right )} f e - d f e + e^{3}}} + \frac {{\left (c d^{2} f^{2} - 4 \, b d f^{2} e + 6 \, c d f e^{2} - 8 \, a f^{2} e^{2} + 12 \, b f e^{3} - 15 \, c e^{4}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {x e + d} \sqrt {f} e^{\frac {1}{2}} + \sqrt {{\left (x e + d\right )} f e - d f e + e^{3}} \right |}\right )}{4 \, f^{\frac {7}{2}}} \]