\[ \int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx \]
Optimal antiderivative \[ \frac {12 e \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}{35 c^{2}}+\frac {2 e \left (e x +d \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}}{7 c}+\frac {2 e \left (71 c^{2} d^{2}+24 b^{2} e^{2}-c e \left (25 a e +71 b d \right )\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{105 c^{3}}+\frac {8 \left (-b e +2 c d \right ) \left (11 c^{2} d^{2}+6 b^{2} e^{2}-c e \left (13 a e +11 b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{105 c^{4} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (71 c^{2} d^{2}+24 b^{2} e^{2}-c e \left (25 a e +71 b d \right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{105 c^{4} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((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 (139 \, c^{4} d^{4} - 278 \, b c^{3} d^{3} e + {\left (347 \, b^{2} c^{2} - 554 \, a c^{3}\right )} d^{2} e^{2} - 2 \, {\left (104 \, b^{3} c - 277 \, a b c^{2}\right )} d e^{3} + {\left (48 \, b^{4} - 176 \, a b^{2} c + 75 \, a^{2} c^{2}\right )} e^{4}\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 ) - 24 \, {\left (22 \, c^{4} d^{3} e - 33 \, b c^{3} d^{2} e^{2} + {\left (23 \, b^{2} c^{2} - 26 \, a c^{3}\right )} d e^{3} - {\left (6 \, b^{3} c - 13 \, a b c^{2}\right )} e^{4}\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 (122 \, c^{4} d^{2} e^{2} + {\left (15 \, c^{4} x^{2} - 18 \, b c^{3} x + 24 \, b^{2} c^{2} - 25 \, a c^{3}\right )} e^{4} + {\left (66 \, c^{4} d x - 89 \, b c^{3} d\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{315 \, c^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]