\[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx \]
Optimal antiderivative \[ \frac {\left (a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {5}{2}}}{8 a \,c^{2} x^{4}}-\frac {\left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {5}{2}}}{5 a c \,x^{5}}+\frac {3 \left (-a d +b c \right )^{4} \left (a d +b c \right ) \arctanh \left (\frac {\sqrt {c}\, \sqrt {b x +a}}{\sqrt {a}\, \sqrt {d x +c}}\right )}{128 a^{\frac {7}{2}} c^{\frac {7}{2}}}+\frac {\left (-a d +b c \right )^{2} \left (a d +b c \right ) \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}{64 a^{2} c^{3} x^{2}}+\frac {\left (-a d +b c \right ) \left (a d +b c \right ) \left (d x +c \right )^{\frac {5}{2}} \sqrt {b x +a}}{16 a \,c^{3} x^{3}}-\frac {3 \left (-a d +b c \right )^{3} \left (a d +b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}{128 a^{3} c^{3} x} \]
command
integrate((b*x+a)^(3/2)*(d*x+c)^(3/2)/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 \[ \text {Timed out} \]_______________________________________________________