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