\[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b c d -b^{2} e +2 a c e +c \left (-b e +2 c d \right ) x \right )}{\left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}-\frac {4 e \left (3 c^{2} d^{2}+2 b^{2} e^{2}-c e \left (5 a e +3 b d \right )\right ) \sqrt {c \,x^{2}+b x +a}}{3 \left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 e \left (-b e +2 c d \right ) \left (3 c^{2} d^{2}+8 b^{2} e^{2}-c e \left (29 a e +3 b d \right )\right ) \sqrt {c \,x^{2}+b x +a}}{3 \left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {e x +d}}+\frac {\left (-b e +2 c d \right ) \left (3 c^{2} d^{2}+8 b^{2} e^{2}-c e \left (29 a e +3 b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {-4 a c +b^{2}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {4 \left (3 c^{2} d^{2}+2 b^{2} e^{2}-c e \left (5 a e +3 b d \right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),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 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{2} e^{3} x^{7} + {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{6} + {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{5} + a^{2} d^{3} + {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 2 \, a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{4} + {\left (2 \, b c d^{3} + 6 \, a b d e^{2} + a^{2} e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{3} + {\left (6 \, a b d^{2} e + 3 \, a^{2} d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{2} + {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x}, x\right ) \]