\[ \int \frac {1}{(d+e x)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {32 x}{715 d^{7} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {1}{15 d e \left (e x +d \right )^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2}{39 d^{2} e \left (e x +d \right )^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {6}{143 d^{3} e \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {16}{429 d^{4} e \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {16}{429 d^{5} e \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {128 x}{2145 d^{9} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {256 x}{2145 d^{11} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(1/(e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{2196480} \, {\left ({\left (\frac {143 \, {\left (675 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{2} + \frac {100 \, d}{x e + d} - 47\right )} e^{\left (-10\right )}}{d^{11} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (143 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {15}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 1650 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {13}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 8775 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {11}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 28600 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 64350 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 108108 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 150150 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 257400 \, d^{154} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14}\right )} e^{\left (-150\right )}}{d^{165} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{15}}\right )} e^{10} + \frac {262144 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{11}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________