7.9 Problem number 264

\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2} \, dx \]

Optimal antiderivative \[ \frac {b \left (6 a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right )}{3 a^{2} c^{2} \left (-a d +b c \right )^{2} \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}+\frac {d \left (-2 a d +b c \right )}{a \,c^{2} \left (-a d +b c \right ) \left (a +\frac {b}{x}\right )^{\frac {3}{2}} \left (c +\frac {d}{x}\right )}+\frac {x}{a c \left (a +\frac {b}{x}\right )^{\frac {3}{2}} \left (c +\frac {d}{x}\right )}-\frac {d^{\frac {7}{2}} \left (-4 a d +9 b c \right ) \arctan \! \left (\frac {\sqrt {d}\, \sqrt {a +\frac {b}{x}}}{\sqrt {-a d +b c}}\right )}{c^{3} \left (-a d +b c \right )^{\frac {7}{2}}}-\frac {\left (4 a d +5 b c \right ) \arctanh \! \left (\frac {\sqrt {a +\frac {b}{x}}}{\sqrt {a}}\right )}{a^{\frac {7}{2}} c^{3}}+\frac {b \left (-2 a d +b c \right ) \left (a^{2} d^{2}-a b c d +5 b^{2} c^{2}\right )}{a^{3} c^{2} \left (-a d +b c \right )^{3} \sqrt {a +\frac {b}{x}}} \]

command

integrate(1/(a+b/x)^(5/2)/(c+d/x)^2,x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \text {Exception raised: TypeError} \]

Giac 1.7.0 via sagemath 9.3 output

\[ -\frac {1}{3} \, b^{3} {\left (\frac {3 \, {\left (9 \, b c d^{4} - 4 \, a d^{5}\right )} \arctan \left (\frac {d \sqrt {\frac {a x + b}{x}}}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b^{6} c^{6} - 3 \, a b^{5} c^{5} d + 3 \, a^{2} b^{4} c^{4} d^{2} - a^{3} b^{3} c^{3} d^{3}\right )} \sqrt {b c d - a d^{2}}} - \frac {2 \, {\left (a b c - a^{2} d + \frac {6 \, {\left (a x + b\right )} b c}{x} - \frac {12 \, {\left (a x + b\right )} a d}{x}\right )} x}{{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} {\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}} + \frac {3 \, {\left (b^{4} c^{4} \sqrt {\frac {a x + b}{x}} - 4 \, a b^{3} c^{3} d \sqrt {\frac {a x + b}{x}} + 6 \, a^{2} b^{2} c^{2} d^{2} \sqrt {\frac {a x + b}{x}} - 4 \, a^{3} b c d^{3} \sqrt {\frac {a x + b}{x}} + 2 \, a^{4} d^{4} \sqrt {\frac {a x + b}{x}} + \frac {{\left (a x + b\right )} b^{3} c^{3} d \sqrt {\frac {a x + b}{x}}}{x} - \frac {3 \, {\left (a x + b\right )} a b^{2} c^{2} d^{2} \sqrt {\frac {a x + b}{x}}}{x} + \frac {3 \, {\left (a x + b\right )} a^{2} b c d^{3} \sqrt {\frac {a x + b}{x}}}{x} - \frac {2 \, {\left (a x + b\right )} a^{3} d^{4} \sqrt {\frac {a x + b}{x}}}{x}\right )}}{{\left (a^{3} b^{5} c^{5} - 3 \, a^{4} b^{4} c^{4} d + 3 \, a^{5} b^{3} c^{3} d^{2} - a^{6} b^{2} c^{2} d^{3}\right )} {\left (a b c - a^{2} d - \frac {{\left (a x + b\right )} b c}{x} + \frac {2 \, {\left (a x + b\right )} a d}{x} - \frac {{\left (a x + b\right )}^{2} d}{x^{2}}\right )}} - \frac {3 \, {\left (5 \, b c + 4 \, a d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3} b^{3} c^{3}}\right )} \]