\[ \int \frac {(c+d x)^{5/2}}{x^6 \sqrt {a+b x}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right )^{3} \left (3 a^{2} d^{2}+14 a b c d +63 b^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {c}\, \sqrt {b x +a}}{\sqrt {a}\, \sqrt {d x +c}}\right )}{128 a^{\frac {11}{2}} c^{\frac {5}{2}}}-\frac {c \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}{5 a \,x^{5}}+\frac {c \left (-13 a d +9 b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}{40 a^{2} x^{4}}-\frac {\left (93 a^{2} d^{2}-148 a b c d +63 b^{2} c^{2}\right ) \sqrt {b x +a}\, \sqrt {d x +c}}{240 a^{3} x^{3}}+\frac {\left (-15 a^{3} d^{3}+481 a^{2} b c \,d^{2}-749 a \,b^{2} c^{2} d +315 b^{3} c^{3}\right ) \sqrt {b x +a}\, \sqrt {d x +c}}{960 a^{4} c \,x^{2}}-\frac {\left (-45 a^{4} d^{4}-90 a^{3} b c \,d^{3}+1564 a^{2} b^{2} c^{2} d^{2}-2310 a \,b^{3} c^{3} d +945 b^{4} c^{4}\right ) \sqrt {b x +a}\, \sqrt {d x +c}}{1920 a^{5} c^{2} x} \]
command
integrate((d*x+c)^(5/2)/x^6/(b*x+a)^(1/2),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 \[ \text {Timed out} \]_______________________________________________________