\[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^6} \, dx \]
Optimal antiderivative \[ -\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{9 d \,e^{3} \left (e x +d \right )^{6}}+\frac {C \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 e^{3} \left (e x +d \right )^{5}}-\frac {\left (11 C \,d^{2}+2 e \left (A e +2 B d \right )\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{42 d^{2} e^{3} \left (e x +d \right )^{5}}-\frac {\left (11 C \,d^{2}+2 e \left (A e +2 B d \right )\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{105 d^{3} e^{3} \left (e x +d \right )^{4}}-\frac {\left (11 C \,d^{2}+2 e \left (A e +2 B d \right )\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{315 d^{4} e^{3} \left (e x +d \right )^{3}} \]
command
integrate((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^6,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (\frac {36 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} C d^{2} e^{\left (-2\right )}}{x} + \frac {144 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} C d^{2} e^{\left (-4\right )}}{x^{2}} - \frac {84 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} C d^{2} e^{\left (-6\right )}}{x^{3}} + \frac {504 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} C d^{2} e^{\left (-8\right )}}{x^{4}} + \frac {420 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} C d^{2} e^{\left (-12\right )}}{x^{6}} + 4 \, C d^{2} + 11 \, B d e + \frac {99 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} B d e^{\left (-1\right )}}{x} + \frac {81 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} B d e^{\left (-3\right )}}{x^{2}} + \frac {609 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} B d e^{\left (-5\right )}}{x^{3}} + \frac {441 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} B d e^{\left (-7\right )}}{x^{4}} + \frac {945 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} B d e^{\left (-9\right )}}{x^{5}} + \frac {315 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} B d e^{\left (-11\right )}}{x^{6}} + \frac {315 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} B d e^{\left (-13\right )}}{x^{7}} + 58 \, A e^{2} + \frac {1143 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} A e^{\left (-2\right )}}{x^{2}} + \frac {2247 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} A e^{\left (-4\right )}}{x^{3}} + \frac {3843 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} A e^{\left (-6\right )}}{x^{4}} + \frac {3465 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} A e^{\left (-8\right )}}{x^{5}} + \frac {2625 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} A e^{\left (-10\right )}}{x^{6}} + \frac {945 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} A e^{\left (-12\right )}}{x^{7}} + \frac {315 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} A e^{\left (-14\right )}}{x^{8}} + \frac {207 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} A}{x}\right )} e^{\left (-3\right )}}{315 \, d^{4} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{9}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________