\[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (8 c^{2} d^{3}-c d e \left (-4 a e +5 b d \right )-b \,e^{2} \left (-3 a e +2 b d \right )+e \left (14 c^{2} d^{2}+b^{2} e^{2}-2 c e \left (-5 a e +7 b d \right )\right ) x \right ) \sqrt {c \,x^{2}+b x +a}}{15 e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {4 \left (-b e +2 c d \right ) \left (4 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-2 a e +b d \right )\right ) \sqrt {c \,x^{2}+b x +a}}{15 e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {2 \left (-b e +2 c d \right ) \left (4 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-2 a e +b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{15 e^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {2 \left (16 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-5 a e +4 b d \right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{15 e^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (16 \, c^{4} d^{7} + {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} x^{3} e^{7} + {\left ({\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d x^{3} + 3 \, {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} d x^{2}\right )} e^{6} + {\left ({\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{2} x^{3} + 3 \, {\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d^{2} x^{2} + 3 \, {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} d^{2} x\right )} e^{5} - {\left (32 \, b c^{3} d^{3} x^{3} - 3 \, {\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{3} x^{2} - 3 \, {\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d^{3} x - {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} d^{3}\right )} e^{4} + {\left (16 \, c^{4} d^{4} x^{3} - 96 \, b c^{3} d^{4} x^{2} + 3 \, {\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{4} x + {\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d^{4}\right )} e^{3} + {\left (48 \, c^{4} d^{5} x^{2} - 96 \, b c^{3} d^{5} x + {\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{5}\right )} e^{2} + 16 \, {\left (3 \, c^{4} d^{6} x - 2 \, b c^{3} d^{6}\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 ) + 6 \, {\left (8 \, c^{4} d^{6} e + {\left (b^{3} c - 8 \, a b c^{2}\right )} x^{3} e^{7} + {\left (2 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d x^{3} + 3 \, {\left (b^{3} c - 8 \, a b c^{2}\right )} d x^{2}\right )} e^{6} - 3 \, {\left (4 \, b c^{3} d^{2} x^{3} - 2 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{2} - {\left (b^{3} c - 8 \, a b c^{2}\right )} d^{2} x\right )} e^{5} + {\left (8 \, c^{4} d^{3} x^{3} - 36 \, b c^{3} d^{3} x^{2} + 6 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{3} x + {\left (b^{3} c - 8 \, a b c^{2}\right )} d^{3}\right )} e^{4} + 2 \, {\left (12 \, c^{4} d^{4} x^{2} - 18 \, b c^{3} d^{4} x + {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{4}\right )} e^{3} + 12 \, {\left (2 \, c^{4} d^{5} x - b c^{3} d^{5}\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 (8 \, c^{4} d^{5} e^{2} - {\left (3 \, a^{2} b c - 2 \, {\left (b^{3} c - 8 \, a b c^{2}\right )} x^{2} + {\left (a b^{2} c + 10 \, a^{2} c^{2}\right )} x\right )} e^{7} + {\left (4 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d x^{2} + {\left (5 \, b^{3} c - 8 \, a b c^{2}\right )} d x + {\left (5 \, a b^{2} c - 4 \, a^{2} c^{2}\right )} d\right )} e^{6} - {\left (24 \, b c^{3} d^{2} x^{2} + 10 \, a b c^{2} d^{2} + {\left (7 \, b^{2} c^{2} - 40 \, a c^{3}\right )} d^{2} x\right )} e^{5} + {\left (16 \, c^{4} d^{3} x^{2} - 20 \, b c^{3} d^{3} x + {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} d^{3}\right )} e^{4} + {\left (18 \, c^{4} d^{4} x - 11 \, b c^{3} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{45 \, {\left (c^{3} d^{7} e^{4} + a^{2} c x^{3} e^{11} - {\left (2 \, a b c d x^{3} - 3 \, a^{2} c d x^{2}\right )} e^{10} - {\left (6 \, a b c d^{2} x^{2} - 3 \, a^{2} c d^{2} x - {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} x^{3}\right )} e^{9} - {\left (2 \, b c^{2} d^{3} x^{3} + 6 \, a b c d^{3} x - a^{2} c d^{3} - 3 \, {\left (b^{2} c + 2 \, a c^{2}\right )} d^{3} x^{2}\right )} e^{8} + {\left (c^{3} d^{4} x^{3} - 6 \, b c^{2} d^{4} x^{2} - 2 \, a b c d^{4} + 3 \, {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x\right )} e^{7} + {\left (3 \, c^{3} d^{5} x^{2} - 6 \, b c^{2} d^{5} x + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{5}\right )} e^{6} + {\left (3 \, c^{3} d^{6} x - 2 \, b c^{2} d^{6}\right )} e^{5}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {e x + d}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]