\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (16 c^{2} d^{3}-b \,e^{2} \left (-5 a e +2 b d \right )-c d e \left (-4 a e +11 b d \right )+e \left (26 c^{2} d^{2}+3 b^{2} e^{2}-2 c e \left (-7 a e +13 b d \right )\right ) x \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{63 e^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{9 e \left (e x +d \right )^{\frac {9}{2}}}-\frac {2 \left (128 c^{4} d^{5}-2 a \,b^{3} e^{5}-4 c^{3} d^{3} e \left (-49 a e +60 b d \right )-b c \,e^{3} \left (-24 a^{2} e^{2}+9 a b d e +b^{2} d^{2}\right )+3 c^{2} d \,e^{2} \left (12 a^{2} e^{2}-52 a b d e +37 b^{2} d^{2}\right )+e \left (160 c^{4} d^{4}-2 b^{4} e^{4}-4 c^{3} d^{2} e \left (-69 a e +80 b d \right )-b^{2} c \,e^{3} \left (-27 a e +11 b d \right )+3 c^{2} e^{2} \left (28 a^{2} e^{2}-92 a b d e +57 b^{2} d^{2}\right )\right ) x \right ) \sqrt {c \,x^{2}+b x +a}}{63 e^{5} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 \left (128 c^{4} d^{4}-b^{4} e^{4}-4 c^{3} d^{2} e \left (-57 a e +64 b d \right )-b^{2} c \,e^{3} \left (-15 a e +7 b d \right )+3 c^{2} e^{2} \left (28 a^{2} e^{2}-76 a b d e +45 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 {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{63 e^{6} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{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 \left (-b e +2 c d \right ) \left (128 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-33 a e +32 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 {-4 a c +b^{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 )}}}{63 e^{6} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(11/2),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 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{e^{6} x^{6} + 6 \, d e^{5} x^{5} + 15 \, d^{2} e^{4} x^{4} + 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 6 \, d^{5} e x + d^{6}}, x\right ) \]