\[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{9 e}-\frac {2 \left (-b e +2 c d \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {e x +d}}{21 c e}-\frac {2 \left (-8 b^{4} e^{4}+7 b^{3} c d \,e^{3}+9 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{315 c^{\frac {5}{2}} e^{4} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {8 d \left (-b e +c d \right ) \left (-b e +2 c d \right ) \left (-b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{315 c^{\frac {5}{2}} e^{4} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (8 c^{3} d^{3}-15 b \,c^{2} d^{2} e +3 b^{2} c d \,e^{2}-4 b^{3} e^{3}-6 c e \left (2 b^{2} e^{2}-b c d e +c^{2} d^{2}\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}{315 c^{2} e^{3}} \]
command
integrate((c*x^2+b*x)^(3/2)*(e*x+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (16 \, c^{5} d^{5} - 40 \, b c^{4} d^{4} e + 22 \, b^{2} c^{3} d^{3} e^{2} + 7 \, b^{3} c^{2} d^{2} e^{3} + 11 \, b^{4} c d e^{4} - 8 \, b^{5} e^{5}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (16 \, c^{5} d^{4} e - 32 \, b c^{4} d^{3} e^{2} + 9 \, b^{2} c^{3} d^{2} e^{3} + 7 \, b^{3} c^{2} d e^{4} - 8 \, b^{4} c e^{5}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (8 \, c^{5} d^{3} e^{2} + {\left (35 \, c^{5} x^{3} + 50 \, b c^{4} x^{2} + 3 \, b^{2} c^{3} x - 4 \, b^{3} c^{2}\right )} e^{5} + {\left (5 \, c^{5} d x^{2} + 11 \, b c^{4} d x + 3 \, b^{2} c^{3} d\right )} e^{4} - 3 \, {\left (2 \, c^{5} d^{2} x + 5 \, b c^{4} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{945 \, c^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (c x^{2} + b x\right )}^{\frac {3}{2}} \sqrt {e x + d}, x\right ) \]