\[ \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e x +d \right )^{\frac {7}{2}}}{\sqrt {c \,x^{2}+b x +a}}+\frac {14 e^{2} \left (e x +d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}{5 c}+\frac {56 e^{2} \left (-b e +2 c d \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{15 c^{2}}+\frac {7 e \left (23 c^{2} d^{2}+8 b^{2} e^{2}-c e \left (9 a e +23 b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{15 c^{3} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {56 e \left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{15 c^{3} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*x+b)*(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 (7 \, {\left (22 \, c^{4} d^{3} x^{2} + 22 \, b c^{3} d^{3} x + 22 \, a c^{3} d^{3} - {\left (8 \, a b^{3} - 21 \, a^{2} b c + {\left (8 \, b^{3} c - 21 \, a b c^{2}\right )} x^{2} + {\left (8 \, b^{4} - 21 \, a b^{2} c\right )} x\right )} e^{3} + 3 \, {\left ({\left (9 \, b^{2} c^{2} - 14 \, a c^{3}\right )} d x^{2} + {\left (9 \, b^{3} c - 14 \, a b c^{2}\right )} d x + {\left (9 \, a b^{2} c - 14 \, a^{2} c^{2}\right )} d\right )} e^{2} - 33 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x + a b c^{2} d^{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 ) - 21 \, {\left ({\left (8 \, a b^{2} c - 9 \, a^{2} c^{2} + {\left (8 \, b^{2} c^{2} - 9 \, a c^{3}\right )} x^{2} + {\left (8 \, b^{3} c - 9 \, a b c^{2}\right )} x\right )} e^{3} - 23 \, {\left (b c^{3} d x^{2} + b^{2} c^{2} d x + a b c^{2} d\right )} e^{2} + 23 \, {\left (c^{4} d^{2} x^{2} + b c^{3} d^{2} x + a c^{3} d^{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 \, {\left (45 \, c^{4} d^{2} x e + 15 \, c^{4} d^{3} - {\left (6 \, c^{4} x^{3} - 7 \, b c^{3} x^{2} - 28 \, a b c^{2} - 7 \, {\left (4 \, b^{2} c^{2} - 3 \, a c^{3}\right )} x\right )} e^{3} - {\left (32 \, c^{4} d x^{2} + 77 \, b c^{3} d x + 77 \, a c^{3} d\right )} e^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{45 \, {\left (c^{5} x^{2} + b c^{4} x + a c^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (2 \, c e^{3} x^{4} + b d^{3} + {\left (6 \, c d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (2 \, c d^{2} e + b d e^{2}\right )} x^{2} + {\left (2 \, c d^{3} + 3 \, b d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]