\[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{5 d \,e^{3} \left (e x +d \right )^{4}}+\frac {\left (-B e +2 C d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d \,e^{3} \left (e x +d \right )^{3}}-\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{15 d^{2} e^{3} \left (e x +d \right )^{3}}-\frac {C \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{3}}-\frac {2 C \sqrt {-e^{2} x^{2}+d^{2}}}{e^{3} \left (e x +d \right )} \]
command
integrate((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -C \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \frac {2 \, {\left (\frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} C d^{2} e^{\left (-2\right )}}{x} + \frac {165 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} C d^{2} e^{\left (-4\right )}}{x^{2}} + \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} C d^{2} e^{\left (-6\right )}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} C d^{2} e^{\left (-8\right )}}{x^{4}} + 24 \, C d^{2} + B d e + \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} B d e^{\left (-1\right )}}{x} - \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} B d e^{\left (-3\right )}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} B d e^{\left (-5\right )}}{x^{3}} + 4 \, A e^{2} + \frac {25 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} A e^{\left (-2\right )}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} A e^{\left (-4\right )}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} A e^{\left (-6\right )}}{x^{4}} + \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} A}{x}\right )} e^{\left (-3\right )}}{15 \, d^{2} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________