14.35 Problem number 812

\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx \]

Optimal antiderivative \[ -\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{11 d e \left (e x +d \right )^{10}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{99 d^{2} e \left (e x +d \right )^{9}} \]

command

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \frac {2 \, {\left (\frac {11 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {451 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {396 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {2376 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {1386 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-10\right )}}{x^{5}} + \frac {3234 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-12\right )}}{x^{6}} + \frac {924 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-14\right )}}{x^{7}} + \frac {1254 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-16\right )}}{x^{8}} + \frac {99 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} e^{\left (-18\right )}}{x^{9}} + \frac {99 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} e^{\left (-20\right )}}{x^{10}} + 10\right )} e^{\left (-1\right )}}{99 \, d^{2} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{11}} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________