\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-14 c e x -15 b e +16 c d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \sqrt {e x +d}}{63 e^{2}}-\frac {2 \left (128 c^{3} d^{3}-b^{3} e^{3}+3 b c \,e^{2} \left (-36 a e +37 b d \right )-12 c^{2} d e \left (-11 a e +20 b d \right )-3 c e \left (32 c^{2} d^{2}+b^{2} e^{2}-4 c e \left (-7 a e +8 b d \right )\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{315 c \,e^{4}}+\frac {2 \left (128 c^{4} d^{4}-b^{4} e^{4}-4 c^{3} d^{2} e \left (-57 a e +64 b d \right )-b^{2} c \,e^{3} \left (-15 a e +7 b d \right )+3 c^{2} e^{2} \left (28 a^{2} e^{2}-76 a b d e +45 b^{2} d^{2}\right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{315 c^{2} e^{5} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {2 \left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (128 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-33 a e +32 b d \right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{315 c^{2} e^{5} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*x+b)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (256 \, c^{5} d^{5} - 640 \, b c^{4} d^{4} e + 2 \, {\left (239 \, b^{2} c^{3} + 324 \, a c^{4}\right )} d^{3} e^{2} - {\left (77 \, b^{3} c^{2} + 972 \, a b c^{3}\right )} d^{2} e^{3} - {\left (13 \, b^{4} c - 258 \, a b^{2} c^{2} - 456 \, a^{2} c^{3}\right )} d e^{4} - {\left (2 \, b^{5} - 33 \, a b^{3} c + 228 \, a^{2} b c^{2}\right )} e^{5}\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 (128 \, c^{5} d^{4} e - 256 \, b c^{4} d^{3} e^{2} + 3 \, {\left (45 \, b^{2} c^{3} + 76 \, a c^{4}\right )} d^{2} e^{3} - {\left (7 \, b^{3} c^{2} + 228 \, a b c^{3}\right )} d e^{4} - {\left (b^{4} c - 15 \, a b^{2} c^{2} - 84 \, a^{2} c^{3}\right )} e^{5}\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 (128 \, c^{5} d^{3} e^{2} - {\left (70 \, c^{5} x^{3} + 145 \, b c^{4} x^{2} + b^{3} c^{2} + 183 \, a b c^{3} + 2 \, {\left (39 \, b^{2} c^{3} + 77 \, a c^{4}\right )} x\right )} e^{5} + {\left (80 \, c^{5} d x^{2} + 176 \, b c^{4} d x + {\left (111 \, b^{2} c^{3} + 212 \, a c^{4}\right )} d\right )} e^{4} - 48 \, {\left (2 \, c^{5} d^{2} x + 5 \, b c^{4} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )} e^{\left (-6\right )}}{945 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (2 \, c^{2} x^{3} + 3 \, b c x^{2} + a b + {\left (b^{2} + 2 \, a c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{\sqrt {e x + d}}, x\right ) \]