\[ \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a f +b e \right )^{2}}{5 f^{2} \left (-c f +d e \right ) \left (f x +e \right )^{\frac {5}{2}}}-\frac {2 \left (-a f +b e \right ) \left (a d f -2 b c f +b d e \right )}{3 f^{2} \left (-c f +d e \right )^{2} \left (f x +e \right )^{\frac {3}{2}}}-\frac {2 \left (-a d +b c \right )^{2} \arctanh \left (\frac {\sqrt {d}\, \sqrt {f x +e}}{\sqrt {-c f +d e}}\right ) \sqrt {d}}{\left (-c f +d e \right )^{\frac {7}{2}}}+\frac {2 \left (-a d +b c \right )^{2}}{\left (-c f +d e \right )^{3} \sqrt {f x +e}} \]
command
integrate((b*x+a)**2/(d*x+c)/(f*x+e)**(7/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {2 \left (a d - b c\right )^{2}}{\sqrt {e + f x} \left (c f - d e\right )^{3}} - \frac {2 \left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{\sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )^{3}} + \frac {2 \left (a f - b e\right ) \left (a d f - 2 b c f + b d e\right )}{3 f^{2} \left (e + f x\right )^{\frac {3}{2}} \left (c f - d e\right )^{2}} - \frac {2 \left (a f - b e\right )^{2}}{5 f^{2} \left (e + f x\right )^{\frac {5}{2}} \left (c f - d e\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________