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