\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{3 e \left (e x +d \right )^{\frac {3}{2}}}+\frac {10 \left (2 c e x -7 b e +16 c d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{21 e^{3} \sqrt {e x +d}}+\frac {2 \left (128 c^{2} d^{2}+51 b^{2} e^{2}-4 c e \left (-5 a e +44 b d \right )-48 c e \left (-b e +2 c d \right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{21 e^{5}}-\frac {\left (-b e +2 c d \right ) \left (128 c^{2} d^{2}+3 b^{2} e^{2}-4 c e \left (-29 a e +32 b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{21 c \,e^{6} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (128 c^{2} d^{2}+27 b^{2} e^{2}-4 c e \left (-5 a e +32 b d \right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{21 c \,e^{6} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (256 \, c^{4} d^{6} - {\left (3 \, b^{4} - 46 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} x^{2} e^{6} - 2 \, {\left ({\left (11 \, b^{3} c + 212 \, a b c^{2}\right )} d x^{2} + {\left (3 \, b^{4} - 46 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} d x\right )} e^{5} + {\left (2 \, {\left (139 \, b^{2} c^{2} + 212 \, a c^{3}\right )} d^{2} x^{2} - 4 \, {\left (11 \, b^{3} c + 212 \, a b c^{2}\right )} d^{2} x - {\left (3 \, b^{4} - 46 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} d^{2}\right )} e^{4} - 2 \, {\left (256 \, b c^{3} d^{3} x^{2} - 2 \, {\left (139 \, b^{2} c^{2} + 212 \, a c^{3}\right )} d^{3} x + {\left (11 \, b^{3} c + 212 \, a b c^{2}\right )} d^{3}\right )} e^{3} + 2 \, {\left (128 \, c^{4} d^{4} x^{2} - 512 \, b c^{3} d^{4} x + {\left (139 \, b^{2} c^{2} + 212 \, a c^{3}\right )} d^{4}\right )} e^{2} + 512 \, {\left (c^{4} d^{5} x - b c^{3} d^{5}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (256 \, c^{4} d^{5} e - {\left (3 \, b^{3} c + 116 \, a b c^{2}\right )} x^{2} e^{6} + 2 \, {\left ({\left (67 \, b^{2} c^{2} + 116 \, a c^{3}\right )} d x^{2} - {\left (3 \, b^{3} c + 116 \, a b c^{2}\right )} d x\right )} e^{5} - {\left (384 \, b c^{3} d^{2} x^{2} - 4 \, {\left (67 \, b^{2} c^{2} + 116 \, a c^{3}\right )} d^{2} x + {\left (3 \, b^{3} c + 116 \, a b c^{2}\right )} d^{2}\right )} e^{4} + 2 \, {\left (128 \, c^{4} d^{3} x^{2} - 384 \, b c^{3} d^{3} x + {\left (67 \, b^{2} c^{2} + 116 \, a c^{3}\right )} d^{3}\right )} e^{3} + 128 \, {\left (4 \, c^{4} d^{4} x - 3 \, b c^{3} d^{4}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (128 \, c^{4} d^{4} e^{2} + {\left (3 \, c^{4} x^{4} + 9 \, b c^{3} x^{3} - 49 \, a b c^{2} x - 7 \, a^{2} c^{2} + {\left (9 \, b^{2} c^{2} + 16 \, a c^{3}\right )} x^{2}\right )} e^{6} - {\left (6 \, c^{4} d x^{3} + 25 \, b c^{3} d x^{2} + 35 \, a b c^{2} d - {\left (67 \, b^{2} c^{2} + 130 \, a c^{3}\right )} d x\right )} e^{5} + {\left (16 \, c^{4} d^{2} x^{2} - 224 \, b c^{3} d^{2} x + {\left (51 \, b^{2} c^{2} + 100 \, a c^{3}\right )} d^{2}\right )} e^{4} + 16 \, {\left (10 \, c^{4} d^{3} x - 11 \, b c^{3} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{63 \, {\left (c^{2} x^{2} e^{9} + 2 \, c^{2} d x e^{8} + c^{2} d^{2} e^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]