\[ \int \frac {(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A e +B d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c}-\frac {2 B \left (e x +d \right )^{\frac {5}{2}}}{5 c}+\frac {\arctanh \left (\frac {c^{\frac {1}{4}} \sqrt {e x +d}}{\sqrt {-e \sqrt {a}+d \sqrt {c}}}\right ) \left (B \sqrt {a}-A \sqrt {c}\right ) \left (-e \sqrt {a}+d \sqrt {c}\right )^{\frac {5}{2}}}{c^{\frac {9}{4}} \sqrt {a}}+\frac {\arctanh \left (\frac {c^{\frac {1}{4}} \sqrt {e x +d}}{\sqrt {e \sqrt {a}+d \sqrt {c}}}\right ) \left (B \sqrt {a}+A \sqrt {c}\right ) \left (e \sqrt {a}+d \sqrt {c}\right )^{\frac {5}{2}}}{c^{\frac {9}{4}} \sqrt {a}}-\frac {2 \left (2 A c d e +a B \,e^{2}+B c \,d^{2}\right ) \sqrt {e x +d}}{c^{2}} \]
command
integrate((B*x+A)*(e*x+d)**(5/2)/(-c*x**2+a),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 \[ \text {Timed out} \]_____________________________________________________