\[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^2} \, dx \]
Optimal antiderivative \[ -\frac {a \left (-9 a d +4 b c \right ) \left (d x +c \right )^{\frac {3}{2}}}{3 b^{4}}-\frac {a \left (-9 a d +4 b c \right ) \left (d x +c \right )^{\frac {5}{2}}}{5 b^{3} \left (-a d +b c \right )}+\frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 b^{2} d}-\frac {a^{2} \left (d x +c \right )^{\frac {7}{2}}}{b^{2} \left (-a d +b c \right ) \left (b x +a \right )}+\frac {a \left (-9 a d +4 b c \right ) \left (-a d +b c \right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {b}\, \sqrt {d x +c}}{\sqrt {-a d +b c}}\right )}{b^{\frac {11}{2}}}-\frac {a \left (-9 a d +4 b c \right ) \left (-a d +b c \right ) \sqrt {d x +c}}{b^{5}} \]
command
integrate(x**2*(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 \[ \text {Timed out} \]_____________________________________________________