\[ \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e x +d \right )^{\frac {7}{2}}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {14 e \left (e x +d \right )^{\frac {3}{2}} \left (b d -2 a e +\left (-b e +2 c d \right ) x \right )}{3 \left (-4 a c +b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {14 e^{2} \left (-b e +2 c d \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{3 c \left (-4 a c +b^{2}\right )}+\frac {14 e \left (c^{2} d^{2}+b^{2} e^{2}-c e \left (3 a e +b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 c^{2} \sqrt {-4 a c +b^{2}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {14 e \left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{3 c^{2} \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*x+b)*(e*x+d)^(7/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (7 \, {\left (2 \, c^{5} d^{3} x^{4} + 4 \, b c^{4} d^{3} x^{3} + 4 \, a b c^{3} d^{3} x + 2 \, a^{2} c^{3} d^{3} + 2 \, {\left (b^{2} c^{3} + 2 \, a c^{4}\right )} d^{3} x^{2} + {\left (2 \, a^{2} b^{3} - 9 \, a^{3} b c + {\left (2 \, b^{3} c^{2} - 9 \, a b c^{3}\right )} x^{4} + 2 \, {\left (2 \, b^{4} c - 9 \, a b^{2} c^{2}\right )} x^{3} + {\left (2 \, b^{5} - 5 \, a b^{3} c - 18 \, a^{2} b c^{2}\right )} x^{2} + 2 \, {\left (2 \, a b^{4} - 9 \, a^{2} b^{2} c\right )} x\right )} e^{3} - 3 \, {\left ({\left (b^{2} c^{3} - 6 \, a c^{4}\right )} d x^{4} + 2 \, {\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} d x^{3} + {\left (b^{4} c - 4 \, a b^{2} c^{2} - 12 \, a^{2} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{3} c - 6 \, a^{2} b c^{2}\right )} d x + {\left (a^{2} b^{2} c - 6 \, a^{3} c^{2}\right )} d\right )} e^{2} - 3 \, {\left (b c^{4} d^{2} x^{4} + 2 \, b^{2} c^{3} d^{2} x^{3} + 2 \, a b^{2} c^{2} d^{2} x + a^{2} b c^{2} d^{2} + {\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} d^{2} x^{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 ) + 42 \, {\left ({\left (a^{2} b^{2} c - 3 \, a^{3} c^{2} + {\left (b^{2} c^{3} - 3 \, a c^{4}\right )} x^{4} + 2 \, {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} x^{3} + {\left (b^{4} c - a b^{2} c^{2} - 6 \, a^{2} c^{3}\right )} x^{2} + 2 \, {\left (a b^{3} c - 3 \, a^{2} b c^{2}\right )} x\right )} e^{3} - {\left (b c^{4} d x^{4} + 2 \, b^{2} c^{3} d x^{3} + 2 \, a b^{2} c^{2} d x + a^{2} b c^{2} d + {\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} d x^{2}\right )} e^{2} + {\left (c^{5} d^{2} x^{4} + 2 \, b c^{4} d^{2} x^{3} + 2 \, a b c^{3} d^{2} x + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} + 2 \, a c^{4}\right )} d^{2} x^{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 ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{3} + {\left (7 \, a^{2} b c^{2} + 2 \, {\left (4 \, b^{2} c^{3} - 9 \, a c^{4}\right )} x^{3} + 7 \, {\left (b^{3} c^{2} - a b c^{3}\right )} x^{2} + 14 \, {\left (a b^{2} c^{2} - a^{2} c^{3}\right )} x\right )} e^{3} - {\left (14 \, b c^{4} d x^{3} + 42 \, a b c^{3} d x + 28 \, a^{2} c^{3} d + {\left (11 \, b^{2} c^{3} + 40 \, a c^{4}\right )} d x^{2}\right )} e^{2} + {\left (14 \, c^{5} d^{2} x^{3} + 21 \, b c^{4} d^{2} x^{2} + 7 \, a b c^{3} d^{2} + 2 \, {\left (5 \, b^{2} c^{3} + a c^{4}\right )} d^{2} x\right )} e\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{9 \, {\left (a^{2} b^{2} c^{3} - 4 \, a^{3} c^{4} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3} + {\left (b^{4} c^{3} - 2 \, a b^{2} c^{4} - 8 \, a^{2} c^{5}\right )} x^{2} + 2 \, {\left (a b^{3} c^{3} - 4 \, a^{2} b c^{4}\right )} x\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (2 \, c e^{3} x^{4} + b d^{3} + {\left (6 \, c d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (2 \, c d^{2} e + b d e^{2}\right )} x^{2} + {\left (2 \, c d^{3} + 3 \, b d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]