\[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a d +b c \right )^{2} \left (a d +b c \right ) \left (7 a^{2} d^{2}+2 a b c d +7 b^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {c}\, \sqrt {b x +a}}{\sqrt {a}\, \sqrt {d x +c}}\right )}{128 a^{\frac {9}{2}} c^{\frac {9}{2}}}-\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{5 x^{5}}-\frac {\left (a d +b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}{40 a c \,x^{4}}+\frac {\left (7 a^{2} d^{2}-2 a b c d +7 b^{2} c^{2}\right ) \sqrt {b x +a}\, \sqrt {d x +c}}{240 a^{2} c^{2} x^{3}}-\frac {\left (a d +b c \right ) \left (35 a^{2} d^{2}-46 a b c d +35 b^{2} c^{2}\right ) \sqrt {b x +a}\, \sqrt {d x +c}}{960 a^{3} c^{3} x^{2}}+\frac {\left (105 a^{4} d^{4}-40 a^{3} b c \,d^{3}-34 a^{2} b^{2} c^{2} d^{2}-40 a \,b^{3} c^{3} d +105 b^{4} c^{4}\right ) \sqrt {b x +a}\, \sqrt {d x +c}}{1920 a^{4} c^{4} x} \]
command
integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^6,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________