\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 e \sqrt {c \,x^{2}+b x +a}}{5 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}-\frac {8 e \left (-b e +2 c d \right ) \sqrt {c \,x^{2}+b x +a}}{15 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 e \left (23 c^{2} d^{2}+8 b^{2} e^{2}-c e \left (9 a e +23 b d \right )\right ) \sqrt {c \,x^{2}+b x +a}}{15 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {e x +d}}+\frac {\left (23 c^{2} d^{2}+8 b^{2} e^{2}-c e \left (9 a e +23 b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{15 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {8 \left (-b e +2 c d \right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{15 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (22 \, c^{3} d^{6} - {\left (8 \, b^{3} - 21 \, a b c\right )} x^{3} e^{6} + 3 \, {\left ({\left (9 \, b^{2} c - 14 \, a c^{2}\right )} d x^{3} - {\left (8 \, b^{3} - 21 \, a b c\right )} d x^{2}\right )} e^{5} - 3 \, {\left (11 \, b c^{2} d^{2} x^{3} - 3 \, {\left (9 \, b^{2} c - 14 \, a c^{2}\right )} d^{2} x^{2} + {\left (8 \, b^{3} - 21 \, a b c\right )} d^{2} x\right )} e^{4} + {\left (22 \, c^{3} d^{3} x^{3} - 99 \, b c^{2} d^{3} x^{2} + 9 \, {\left (9 \, b^{2} c - 14 \, a c^{2}\right )} d^{3} x - {\left (8 \, b^{3} - 21 \, a b c\right )} d^{3}\right )} e^{3} + 3 \, {\left (22 \, c^{3} d^{4} x^{2} - 33 \, b c^{2} d^{4} x + {\left (9 \, b^{2} c - 14 \, a c^{2}\right )} d^{4}\right )} e^{2} + 33 \, {\left (2 \, c^{3} d^{5} x - b c^{2} 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 (23 \, c^{3} d^{5} e + {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} x^{3} e^{6} - {\left (23 \, b c^{2} d x^{3} - 3 \, {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} d x^{2}\right )} e^{5} + {\left (23 \, c^{3} d^{2} x^{3} - 69 \, b c^{2} d^{2} x^{2} + 3 \, {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} d^{2} x\right )} e^{4} + {\left (69 \, c^{3} d^{3} x^{2} - 69 \, b c^{2} d^{3} x + {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} d^{3}\right )} e^{3} + 23 \, {\left (3 \, c^{3} d^{4} x - b c^{2} 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 (34 \, c^{3} d^{4} e^{2} - {\left (4 \, a b c x - 3 \, a^{2} c - {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} x^{2}\right )} e^{6} - {\left (23 \, b c^{2} d x^{2} + 10 \, a b c d - 10 \, {\left (2 \, b^{2} c - a c^{2}\right )} d x\right )} e^{5} + {\left (23 \, c^{3} d^{2} x^{2} - 58 \, b c^{2} d^{2} x + 5 \, {\left (3 \, b^{2} c + a c^{2}\right )} d^{2}\right )} e^{4} + {\left (54 \, c^{3} d^{3} x - 41 \, b c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{45 \, {\left (c^{4} d^{9} e + a^{3} c x^{3} e^{10} - 3 \, {\left (a^{2} b c d x^{3} - a^{3} c d x^{2}\right )} e^{9} - 3 \, {\left (3 \, a^{2} b c d^{2} x^{2} - a^{3} c d^{2} x - {\left (a b^{2} c + a^{2} c^{2}\right )} d^{2} x^{3}\right )} e^{8} - {\left (9 \, a^{2} b c d^{3} x - a^{3} c d^{3} + {\left (b^{3} c + 6 \, a b c^{2}\right )} d^{3} x^{3} - 9 \, {\left (a b^{2} c + a^{2} c^{2}\right )} d^{3} x^{2}\right )} e^{7} - 3 \, {\left (a^{2} b c d^{4} - {\left (b^{2} c^{2} + a c^{3}\right )} d^{4} x^{3} + {\left (b^{3} c + 6 \, a b c^{2}\right )} d^{4} x^{2} - 3 \, {\left (a b^{2} c + a^{2} c^{2}\right )} d^{4} x\right )} e^{6} - 3 \, {\left (b c^{3} d^{5} x^{3} - 3 \, {\left (b^{2} c^{2} + a c^{3}\right )} d^{5} x^{2} + {\left (b^{3} c + 6 \, a b c^{2}\right )} d^{5} x - {\left (a b^{2} c + a^{2} c^{2}\right )} d^{5}\right )} e^{5} + {\left (c^{4} d^{6} x^{3} - 9 \, b c^{3} d^{6} x^{2} + 9 \, {\left (b^{2} c^{2} + a c^{3}\right )} d^{6} x - {\left (b^{3} c + 6 \, a b c^{2}\right )} d^{6}\right )} e^{4} + 3 \, {\left (c^{4} d^{7} x^{2} - 3 \, b c^{3} d^{7} x + {\left (b^{2} c^{2} + a c^{3}\right )} d^{7}\right )} e^{3} + 3 \, {\left (c^{4} d^{8} x - b c^{3} d^{8}\right )} e^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c e^{4} x^{6} + {\left (4 \, c d e^{3} + b e^{4}\right )} x^{5} + a d^{4} + {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{4} + 2 \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{3} + {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{2} + {\left (b d^{4} + 4 \, a d^{3} e\right )} x}, x\right ) \]