\[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx \]
Optimal antiderivative \[ -\frac {\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {7}{2}}}{5 a c \,x^{5}}-\frac {\left (-a d +b c \right )^{4} \left (3 a d +7 b c \right ) \arctanh \left (\frac {\sqrt {c}\, \sqrt {b x +a}}{\sqrt {a}\, \sqrt {d x +c}}\right )}{128 a^{\frac {9}{2}} c^{\frac {5}{2}}}-\frac {\left (-a d +b c \right )^{2} \left (3 a d +7 b c \right ) \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}{192 a^{3} c^{2} x^{2}}+\frac {\left (-a d +b c \right ) \left (3 a d +7 b c \right ) \left (d x +c \right )^{\frac {5}{2}} \sqrt {b x +a}}{240 a^{2} c^{2} x^{3}}+\frac {\left (3 a d +7 b c \right ) \left (d x +c \right )^{\frac {7}{2}} \sqrt {b x +a}}{40 a \,c^{2} x^{4}}+\frac {\left (-a d +b c \right )^{3} \left (3 a d +7 b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}{128 a^{4} c^{2} x} \]
command
integrate((d*x+c)^(5/2)*(b*x+a)^(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 \[ \text {Timed out} \]_______________________________________________________