\[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {5 \left (-a d +b c \right )^{2} \left (-a^{2} d^{2}-14 a b c d +63 b^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {c}\, \sqrt {b x +a}}{\sqrt {a}\, \sqrt {d x +c}}\right )}{64 a^{\frac {11}{2}} c^{\frac {3}{2}}}-\frac {c \left (d x +c \right )^{\frac {3}{2}}}{4 a \,x^{4} \sqrt {b x +a}}+\frac {b \left (-15 a^{3} d^{3}+839 a^{2} b c \,d^{2}-1785 a \,b^{2} c^{2} d +945 b^{3} c^{3}\right ) \sqrt {d x +c}}{192 a^{5} c \sqrt {b x +a}}+\frac {c \left (-11 a d +9 b c \right ) \sqrt {d x +c}}{24 a^{2} x^{3} \sqrt {b x +a}}-\frac {\left (-59 a d +63 b c \right ) \left (-a d +b c \right ) \sqrt {d x +c}}{96 a^{3} x^{2} \sqrt {b x +a}}+\frac {\left (-a d +b c \right ) \left (15 a^{2} d^{2}-322 a b c d +315 b^{2} c^{2}\right ) \sqrt {d x +c}}{192 a^{4} c x \sqrt {b x +a}} \]
command
integrate((d*x+c)^(5/2)/x^5/(b*x+a)^(3/2),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 \[ \text {Timed out} \]_______________________________________________________