11.23 Problem number 52

\[ \int \frac {(d+e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx \]

Optimal antiderivative \[ \frac {2 e^{2} \left (e x +d \right )}{5 d^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {e^{2} \left (6 e x +5 d \right )}{5 d^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {9 e^{2} \arctanh \! \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{2 d^{7}}+\frac {2 e^{2} \left (11 e x +10 d \right )}{5 d^{7} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{6} x^{2}}-\frac {2 e \sqrt {-e^{2} x^{2}+d^{2}}}{d^{7} x} \]

command

integrate((e*x+d)^2/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \text {could not integrate} \]

Giac 1.7.0 via sagemath 9.3 output

\[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left (2 \, {\left (x {\left (\frac {11 \, x e^{7}}{d^{7}} + \frac {10 \, e^{6}}{d^{6}}\right )} - \frac {25 \, e^{5}}{d^{5}}\right )} x - \frac {45 \, e^{4}}{d^{4}}\right )} x + \frac {30 \, e^{3}}{d^{3}}\right )} x + \frac {27 \, e^{2}}{d^{2}}\right )}}{5 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac {9 \, e^{2} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{2 \, d^{7}} + \frac {x^{2} {\left (\frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{4}}{x} + e^{6}\right )}}{8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{7}} - \frac {{\left (\frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{7} e^{8}}{x} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{7} e^{6}}{x^{2}}\right )} e^{\left (-8\right )}}{8 \, d^{14}} \]