\[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {c \left (d x +c \right )^{\frac {3}{2}}}{3 a \,x^{3} \left (b x +a \right )^{\frac {3}{2}}}+\frac {5 \left (-a d +b c \right ) \left (a^{2} d^{2}-14 a b c d +21 b^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {c}\, \sqrt {b x +a}}{\sqrt {a}\, \sqrt {d x +c}}\right )}{8 a^{\frac {11}{2}} \sqrt {c}}-\frac {7 b \left (-7 a d +15 b c \right ) \left (-a d +b c \right ) \sqrt {d x +c}}{24 a^{4} \left (b x +a \right )^{\frac {3}{2}}}+\frac {3 c \left (-a d +b c \right ) \sqrt {d x +c}}{4 a^{2} x^{2} \left (b x +a \right )^{\frac {3}{2}}}-\frac {\left (-11 a d +21 b c \right ) \left (-a d +b c \right ) \sqrt {d x +c}}{8 a^{3} x \left (b x +a \right )^{\frac {3}{2}}}-\frac {b \left (113 a^{2} d^{2}-420 a b c d +315 b^{2} c^{2}\right ) \sqrt {d x +c}}{24 a^{5} \sqrt {b x +a}} \]
command
integrate((d*x+c)^(5/2)/x^4/(b*x+a)^(5/2),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 \[ \text {Timed out} \]_______________________________________________________