\[ \int \frac {1}{(d+e x) \sqrt {\frac {-c d^2+b d e}{e^2}+b x+c x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 e \sqrt {-\frac {d \left (-b e +c d \right )}{e^{2}}+b x +c \,x^{2}}}{\left (-b e +2 c d \right ) \left (e x +d \right )} \]
command
integrate(1/(e*x+d)/((b*d*e-c*d^2)/e^2+b*x+c*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{\sqrt {c} x e + \sqrt {c} d - \sqrt {c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________