\[ \int \frac {A+B x+C x^2}{(d+e x)^4 \sqrt {d^2-e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{7 d \,e^{3} \left (e x +d \right )^{4}}+\frac {C \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{3} \left (e x +d \right )^{3}}-\frac {\left (6 A \,e^{2}+8 B d e +13 C \,d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{70 d^{2} e^{3} \left (e x +d \right )^{3}}-\frac {\left (6 A \,e^{2}+8 B d e +13 C \,d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{105 d^{3} e^{3} \left (e x +d \right )^{2}}-\frac {\left (6 A \,e^{2}+8 B d e +13 C \,d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{105 d^{4} e^{3} \left (e x +d \right )} \]
command
integrate((C*x^2+B*x+A)/(e*x+d)^4/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (\frac {56 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} C d^{2} e^{\left (-2\right )}}{x} + \frac {168 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} C d^{2} e^{\left (-4\right )}}{x^{2}} + \frac {140 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} C d^{2} e^{\left (-6\right )}}{x^{3}} + \frac {140 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} C d^{2} e^{\left (-8\right )}}{x^{4}} + 8 \, C d^{2} + 13 \, B d e + \frac {91 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} B d e^{\left (-1\right )}}{x} + \frac {168 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} B d e^{\left (-3\right )}}{x^{2}} + \frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} B d e^{\left (-5\right )}}{x^{3}} + \frac {175 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} B d e^{\left (-7\right )}}{x^{4}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} B d e^{\left (-9\right )}}{x^{5}} + 36 \, A e^{2} + \frac {441 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} A e^{\left (-2\right )}}{x^{2}} + \frac {630 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} A e^{\left (-4\right )}}{x^{3}} + \frac {630 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} A e^{\left (-6\right )}}{x^{4}} + \frac {315 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} A e^{\left (-8\right )}}{x^{5}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} A e^{\left (-10\right )}}{x^{6}} + \frac {147 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} A}{x}\right )} e^{\left (-3\right )}}{105 \, d^{4} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{7}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________