\[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^2} \, dx \]
Optimal antiderivative \[ \frac {\left (-7 a d +2 b c \right ) \left (d x +c \right )^{\frac {3}{2}}}{3 b^{3}}+\frac {\left (-7 a d +2 b c \right ) \left (d x +c \right )^{\frac {5}{2}}}{5 b^{2} \left (-a d +b c \right )}+\frac {a \left (d x +c \right )^{\frac {7}{2}}}{b \left (-a d +b c \right ) \left (b x +a \right )}-\frac {\left (-7 a d +2 b c \right ) \left (-a d +b c \right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {b}\, \sqrt {d x +c}}{\sqrt {-a d +b c}}\right )}{b^{\frac {9}{2}}}+\frac {\left (-7 a d +2 b c \right ) \left (-a d +b c \right ) \sqrt {d x +c}}{b^{4}} \]
command
integrate(x*(d*x+c)**(5/2)/(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 \[ \text {Timed out} \]_____________________________________________________