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