\[ \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {3 e^{2} \left (-b e +2 c d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}+\frac {-\left (-4 a c +b^{2}\right ) \left (-b e +c d \right )+c \left (-4 a c +b^{2}\right ) e x}{\left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (c \,x^{2}+b x +a \right ) \sqrt {e x +d}}+\frac {3 e \arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {c}\, \left (2 c^{2} d^{2}+b \,e^{2} \left (b +\sqrt {-4 a c +b^{2}}\right )-2 c e \left (b d +a e +d \sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {-4 a c +b^{2}}\, \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}-\frac {3 e \arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {c}\, \left (2 c^{2} d^{2}+b \,e^{2} \left (b -\sqrt {-4 a c +b^{2}}\right )-2 c e \left (b d +a e -d \sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {-4 a c +b^{2}}\, \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}} \]
command
integrate((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]