\[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx \]
Optimal antiderivative \[ -\frac {c \left (d x +c \right )^{\frac {3}{2}}}{3 a \,x^{3} \left (b x +a \right )}-\frac {\left (-3 a d +8 b c \right ) \left (-a d +b c \right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {b}\, \sqrt {d x +c}}{\sqrt {-a d +b c}}\right ) \sqrt {b}}{a^{5}}+\frac {\left (-5 a^{3} d^{3}+60 a^{2} b c \,d^{2}-120 a \,b^{2} c^{2} d +64 b^{3} c^{3}\right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 a^{5} \sqrt {c}}-\frac {b \left (19 a^{2} d^{2}-52 a b c d +32 b^{2} c^{2}\right ) \sqrt {d x +c}}{8 a^{4} \left (b x +a \right )}+\frac {c \left (-9 a d +8 b c \right ) \sqrt {d x +c}}{12 a^{2} x^{2} \left (b x +a \right )}-\frac {\left (33 a^{2} d^{2}-82 a b c d +48 b^{2} c^{2}\right ) \sqrt {d x +c}}{24 a^{3} x \left (b x +a \right )} \]
command
integrate((d*x+c)**(5/2)/x**4/(b*x+a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________