\[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (2 c x +b \right ) \sqrt {e x +d}}{3 \left (-4 a c +b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {2 \left (9 b^{2} c d e -4 a \,c^{2} d e -b^{3} e^{2}-4 b c \left (a \,e^{2}+2 c \,d^{2}\right )-c \left (16 c^{2} d^{2}+b^{2} e^{2}-4 c e \left (-3 a e +4 b d \right )\right ) x \right ) \sqrt {e x +d}}{3 \left (-4 a c +b^{2}\right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\left (16 c^{2} d^{2}+b^{2} e^{2}-4 c e \left (-3 a e +4 b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {16 \left (-b e +2 c d \right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{3 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((e*x+d)^(1/2)/(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
\[ \frac {2 \, {\left ({\left (16 \, c^{5} d^{3} x^{4} + 32 \, b c^{4} d^{3} x^{3} + 32 \, a b c^{3} d^{3} x + 16 \, a^{2} c^{3} d^{3} + 16 \, {\left (b^{2} c^{3} + 2 \, a c^{4}\right )} d^{3} x^{2} + {\left (a^{2} b^{3} - 12 \, a^{3} b c + {\left (b^{3} c^{2} - 12 \, a b c^{3}\right )} x^{4} + 2 \, {\left (b^{4} c - 12 \, a b^{2} c^{2}\right )} x^{3} + {\left (b^{5} - 10 \, a b^{3} c - 24 \, a^{2} b c^{2}\right )} x^{2} + 2 \, {\left (a b^{4} - 12 \, a^{2} b^{2} c\right )} x\right )} e^{3} + 6 \, {\left ({\left (b^{2} c^{3} + 4 \, a c^{4}\right )} d x^{4} + 2 \, {\left (b^{3} c^{2} + 4 \, a b c^{3}\right )} d x^{3} + {\left (b^{4} c + 6 \, a b^{2} c^{2} + 8 \, a^{2} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{3} c + 4 \, a^{2} b c^{2}\right )} d x + {\left (a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d\right )} e^{2} - 24 \, {\left (b c^{4} d^{2} x^{4} + 2 \, b^{2} c^{3} d^{2} x^{3} + 2 \, a b^{2} c^{2} d^{2} x + a^{2} b c^{2} d^{2} + {\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} d^{2} x^{2}\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 ({\left (a^{2} b^{2} c + 12 \, a^{3} c^{2} + {\left (b^{2} c^{3} + 12 \, a c^{4}\right )} x^{4} + 2 \, {\left (b^{3} c^{2} + 12 \, a b c^{3}\right )} x^{3} + {\left (b^{4} c + 14 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x^{2} + 2 \, {\left (a b^{3} c + 12 \, a^{2} b c^{2}\right )} x\right )} e^{3} - 16 \, {\left (b c^{4} d x^{4} + 2 \, b^{2} c^{3} d x^{3} + 2 \, a b^{2} c^{2} d x + a^{2} b c^{2} d + {\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} d x^{2}\right )} e^{2} + 16 \, {\left (c^{5} d^{2} x^{4} + 2 \, b c^{4} d^{2} x^{3} + 2 \, a b c^{3} d^{2} x + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} + 2 \, a c^{4}\right )} d^{2} x^{2}\right )} e\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 \, \sqrt {c x^{2} + b x + a} {\left ({\left (8 \, a^{2} b c^{2} + {\left (b^{2} c^{3} + 12 \, a c^{4}\right )} x^{3} + 2 \, {\left (b^{3} c^{2} + 8 \, a b c^{3}\right )} x^{2} + {\left (b^{4} c + 3 \, a b^{2} c^{2} + 20 \, a^{2} c^{3}\right )} x\right )} e^{3} - {\left (16 \, b c^{4} d x^{3} + {\left (25 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d x^{2} + {\left (7 \, b^{3} c^{2} + 20 \, a b c^{3}\right )} d x - {\left (b^{4} c - 13 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d\right )} e^{2} + {\left (16 \, c^{5} d^{2} x^{3} + 24 \, b c^{4} d^{2} x^{2} + 6 \, {\left (b^{2} c^{3} + 4 \, a c^{4}\right )} d^{2} x - {\left (b^{3} c^{2} - 12 \, a b c^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}\right )}}{9 \, {\left ({\left (a^{3} b^{4} c - 8 \, a^{4} b^{2} c^{2} + 16 \, a^{5} c^{3} + {\left (a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5}\right )} x^{4} + 2 \, {\left (a b^{5} c^{2} - 8 \, a^{2} b^{3} c^{3} + 16 \, a^{3} b c^{4}\right )} x^{3} + {\left (a b^{6} c - 6 \, a^{2} b^{4} c^{2} + 32 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (a^{2} b^{5} c - 8 \, a^{3} b^{3} c^{2} + 16 \, a^{4} b c^{3}\right )} x\right )} e^{3} - {\left ({\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d x^{4} + 2 \, {\left (b^{6} c^{2} - 8 \, a b^{4} c^{3} + 16 \, a^{2} b^{2} c^{4}\right )} d x^{3} + {\left (b^{7} c - 6 \, a b^{5} c^{2} + 32 \, a^{3} b c^{4}\right )} d x^{2} + 2 \, {\left (a b^{6} c - 8 \, a^{2} b^{4} c^{2} + 16 \, a^{3} b^{2} c^{3}\right )} d x + {\left (a^{2} b^{5} c - 8 \, a^{3} b^{3} c^{2} + 16 \, a^{4} b c^{3}\right )} d\right )} e^{2} + {\left ({\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{2} x^{4} + 2 \, {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{2} x^{3} + {\left (b^{6} c^{2} - 6 \, a b^{4} c^{3} + 32 \, a^{3} c^{5}\right )} d^{2} x^{2} + 2 \, {\left (a b^{5} c^{2} - 8 \, a^{2} b^{3} c^{3} + 16 \, a^{3} b c^{4}\right )} d^{2} x + {\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} d^{2}\right )} e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]