7.3 Problem number 249

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

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

command

integrate(1/(c+d/x)/(a+b/x)^(1/2),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

\[ -b^{2} {\left (\frac {2 \, d^{2} \arctan \left (\frac {d \sqrt {\frac {a x + b}{x}}}{\sqrt {b c d - a d^{2}}}\right )}{\sqrt {b c d - a d^{2}} b^{2} c^{2}} + \frac {\sqrt {\frac {a x + b}{x}}}{{\left (a - \frac {a x + b}{x}\right )} a b c} - \frac {{\left (b c + 2 \, a d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a b^{2} c^{2}}\right )} \]