\[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (\left (-4 a c +b^{2}\right ) \left (-b e +c d \right )-c \left (-4 a c +b^{2}\right ) e x \right ) \sqrt {e x +d}}{3 \left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {2 e \left (3 b^{2} c d e -8 a \,c^{2} d e -2 b^{3} e^{2}-b c \left (-7 a \,e^{2}+c \,d^{2}\right )-2 c \left (c^{2} d^{2}+b^{2} e^{2}-c e \left (3 a e +b d \right )\right ) x \right ) \sqrt {e x +d}}{3 \left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {2 e \left (c^{2} d^{2}+b^{2} e^{2}-c e \left (3 a e +b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {-4 a c +b^{2}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {2 e \left (-b e +2 c d \right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*x+b)/(c*x^2+b*x+a)^(5/2)/(e*x+d)^(1/2),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 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {e x + d}}{c^{3} e x^{7} + {\left (c^{3} d + 3 \, b c^{2} e\right )} x^{6} + 3 \, {\left (b c^{2} d + {\left (b^{2} c + a c^{2}\right )} e\right )} x^{5} + {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d + {\left (b^{3} + 6 \, a b c\right )} e\right )} x^{4} + a^{3} d + {\left ({\left (b^{3} + 6 \, a b c\right )} d + 3 \, {\left (a b^{2} + a^{2} c\right )} e\right )} x^{3} + 3 \, {\left (a^{2} b e + {\left (a b^{2} + a^{2} c\right )} d\right )} x^{2} + {\left (3 \, a^{2} b d + a^{3} e\right )} x}, x\right ) \]