\[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (b d -2 a e +\left (-b e +2 c d \right ) x \right )}{\left (-4 a c +b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {2 e \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}{c \left (-4 a c +b^{2}\right )}+\frac {4 e \left (3 c^{2} d^{2}+2 b^{2} e^{2}-c e \left (5 a e +3 b d \right )\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{3 c^{2} \left (-4 a c +b^{2}\right )}+\frac {\left (-b e +2 c d \right ) \left (3 c^{2} d^{2}+8 b^{2} e^{2}-c e \left (29 a e +3 b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 c^{3} \sqrt {-4 a c +b^{2}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (3 c^{2} d^{2}+2 b^{2} e^{2}-c e \left (5 a e +3 b d \right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{3 c^{3} \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (6 \, c^{5} d^{4} x^{2} + 6 \, b c^{4} d^{4} x + 6 \, a c^{4} d^{4} - {\left (8 \, a b^{4} - 41 \, a^{2} b^{2} c + 30 \, a^{3} c^{2} + {\left (8 \, b^{4} c - 41 \, a b^{2} c^{2} + 30 \, a^{2} c^{3}\right )} x^{2} + {\left (8 \, b^{5} - 41 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} x\right )} e^{4} + {\left ({\left (23 \, b^{3} c^{2} - 104 \, a b c^{3}\right )} d x^{2} + {\left (23 \, b^{4} c - 104 \, a b^{2} c^{2}\right )} d x + {\left (23 \, a b^{3} c - 104 \, a^{2} b c^{2}\right )} d\right )} e^{3} - {\left ({\left (17 \, b^{2} c^{3} - 104 \, a c^{4}\right )} d^{2} x^{2} + {\left (17 \, b^{3} c^{2} - 104 \, a b c^{3}\right )} d^{2} x + {\left (17 \, a b^{2} c^{2} - 104 \, a^{2} c^{3}\right )} d^{2}\right )} e^{2} - 12 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x + a b c^{3} d^{3}\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 (8 \, a b^{3} c - 29 \, a^{2} b c^{2} + {\left (8 \, b^{3} c^{2} - 29 \, a b c^{3}\right )} x^{2} + {\left (8 \, b^{4} c - 29 \, a b^{2} c^{2}\right )} x\right )} e^{4} - {\left ({\left (19 \, b^{2} c^{3} - 58 \, a c^{4}\right )} d x^{2} + {\left (19 \, b^{3} c^{2} - 58 \, a b c^{3}\right )} d x + {\left (19 \, a b^{2} c^{2} - 58 \, a^{2} c^{3}\right )} d\right )} e^{3} + 9 \, {\left (b c^{4} d^{2} x^{2} + b^{2} c^{3} d^{2} x + a b c^{3} d^{2}\right )} e^{2} - 6 \, {\left (c^{5} d^{3} x^{2} + b c^{4} d^{3} x + a c^{4} d^{3}\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 (4 \, a b^{2} c^{2} - 10 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (4 \, b^{3} c^{2} - 13 \, a b c^{3}\right )} x\right )} e^{4} - 9 \, {\left (a b c^{3} d + {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} d x\right )} e^{3} + 9 \, {\left (b c^{4} d^{2} x + 2 \, a c^{4} d^{2}\right )} e^{2} - 3 \, {\left (2 \, c^{5} d^{3} x + b c^{4} d^{3}\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{9 \, {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{2} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]