\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {c^{\frac {1}{4}} \sqrt {e x +d}}{\sqrt {-e \sqrt {a}+d \sqrt {c}}}\right ) \left (a B e \left (-5 e \sqrt {a}+2 d \sqrt {c}\right )+3 A \left (4 c^{\frac {3}{2}} d^{2}-10 c d e \sqrt {a}+7 a \,e^{2} \sqrt {c}\right )\right )}{32 a^{\frac {5}{2}} c^{\frac {3}{4}} \left (-e \sqrt {a}+d \sqrt {c}\right )^{\frac {5}{2}}}+\frac {\arctanh \left (\frac {c^{\frac {1}{4}} \sqrt {e x +d}}{\sqrt {e \sqrt {a}+d \sqrt {c}}}\right ) \left (a B e \left (5 e \sqrt {a}+2 d \sqrt {c}\right )+3 A \left (4 c^{\frac {3}{2}} d^{2}+10 c d e \sqrt {a}+7 a \,e^{2} \sqrt {c}\right )\right )}{32 a^{\frac {5}{2}} c^{\frac {3}{4}} \left (e \sqrt {a}+d \sqrt {c}\right )^{\frac {5}{2}}}+\frac {\left (a \left (-A e +B d \right )+\left (A c d -a B e \right ) x \right ) \sqrt {e x +d}}{4 a \left (-a \,e^{2}+c \,d^{2}\right ) \left (-c \,x^{2}+a \right )^{2}}-\frac {\left (a e \left (-7 A a \,e^{2}+A c \,d^{2}+6 a B d e \right )-\left (6 A c d \left (-2 a \,e^{2}+c \,d^{2}\right )+a B e \left (5 a \,e^{2}+c \,d^{2}\right )\right ) x \right ) \sqrt {e x +d}}{16 a^{2} \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (-c \,x^{2}+a \right )} \]
command
integrate((B*x+A)/(e*x+d)^(1/2)/(-c*x^2+a)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: TypeError} \]_______________________________________________________