\[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx \]
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 )}{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}} \sqrt {e x +d}}-\frac {2 \left (5 a c e \left (-b e +2 c d \right )^{2}-\left (2 a c e -b^{2} e +b c d \right ) \left (8 c^{2} d^{2}-4 b^{2} e^{2}-c e \left (-14 a e +3 b d \right )\right )-4 c \left (-b e +2 c d \right ) \left (2 c^{2} d^{2}-b^{2} e^{2}-2 c e \left (-3 a e +b d \right )\right ) x \right )}{3 \left (-4 a c +b^{2}\right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}+\frac {2 e \left (16 c^{4} d^{4}-8 b^{4} e^{4}-4 c^{3} d^{2} e \left (-15 a e +8 b d \right )+b^{2} c \,e^{3} \left (57 a e +7 b d \right )+3 c^{2} e^{2} \left (-28 a^{2} e^{2}-20 a b d e +3 b^{2} d^{2}\right )\right ) \sqrt {c \,x^{2}+b x +a}}{3 \left (-4 a c +b^{2}\right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {e x +d}}-\frac {\left (16 c^{4} d^{4}-8 b^{4} e^{4}-4 c^{3} d^{2} e \left (-15 a e +8 b d \right )+b^{2} c \,e^{3} \left (57 a e +7 b d \right )+3 c^{2} e^{2} \left (-28 a^{2} e^{2}-20 a b d e +3 b^{2} d^{2}\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 (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \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 {8 \left (-b e +2 c d \right ) \left (2 c^{2} d^{2}-b^{2} e^{2}-2 c e \left (-3 a e +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 (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(5/2),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 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{3} e^{2} x^{8} + {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{7} + {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x^{6} + {\left (3 \, b c^{2} d^{2} + 6 \, {\left (b^{2} c + a c^{2}\right )} d e + {\left (b^{3} + 6 \, a b c\right )} e^{2}\right )} x^{5} + a^{3} d^{2} + {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d e + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{2}\right )} x^{4} + {\left (3 \, a^{2} b e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{2} + 6 \, {\left (a b^{2} + a^{2} c\right )} d e\right )} x^{3} + {\left (6 \, a^{2} b d e + a^{3} e^{2} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{2} + {\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x}, x\right ) \]