\[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right )^{3} \left (7 a^{2} d^{2}+6 a b c d +3 b^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {c}\, \sqrt {b x +a}}{\sqrt {a}\, \sqrt {d x +c}}\right )}{128 a^{\frac {7}{2}} c^{\frac {9}{2}}}-\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}}{5 x^{5}}-\frac {\left (a d +3 b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}{40 c \,x^{4}}-\frac {\left (\frac {3 b^{2} c}{a}+12 b d -\frac {7 a \,d^{2}}{c}\right ) \sqrt {b x +a}\, \sqrt {d x +c}}{240 c \,x^{3}}+\frac {\left (-35 a^{3} d^{3}+61 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +15 b^{3} c^{3}\right ) \sqrt {b x +a}\, \sqrt {d x +c}}{960 a^{2} c^{3} x^{2}}-\frac {\left (-105 a^{4} d^{4}+190 a^{3} b c \,d^{3}-36 a^{2} b^{2} c^{2} d^{2}-30 a \,b^{3} c^{3} d +45 b^{4} c^{4}\right ) \sqrt {b x +a}\, \sqrt {d x +c}}{1920 a^{3} c^{4} x} \]
command
integrate((b*x+a)^(3/2)*(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 \[ \text {Timed out} \]_______________________________________________________