\[ \int \frac {(a+b x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a e +b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3 b^{3}}+\frac {2 \left (-a e +b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5 b^{2}}+\frac {2 \left (e x +d \right )^{\frac {7}{2}}}{7 b}-\frac {2 \left (-a e +b d \right )^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {b}\, \sqrt {e x +d}}{\sqrt {-a e +b d}}\right )}{b^{\frac {9}{2}}}+\frac {2 \left (-a e +b d \right )^{3} \sqrt {e x +d}}{b^{4}} \]
command
integrate((b*x+a)*(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {2 \left (d + e x\right )^{\frac {7}{2}}}{7 b} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 2 a e + 2 b d\right )}{5 b^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 a^{2} e^{2} - 4 a b d e + 2 b^{2} d^{2}\right )}{3 b^{3}} + \frac {\sqrt {d + e x} \left (- 2 a^{3} e^{3} + 6 a^{2} b d e^{2} - 6 a b^{2} d^{2} e + 2 b^{3} d^{3}\right )}{b^{4}} + \frac {2 \left (a e - b d\right )^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{5} \sqrt {\frac {a e - b d}{b}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________