\[ \int (A+B x) \sqrt {d+e x} \sqrt {a+c x^2} \, dx \]
Optimal antiderivative \[ \frac {2 B \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x +d}}{7 c}-\frac {2 \left (4 B c \,d^{2}-7 A c d e +5 a B \,e^{2}-3 c e \left (7 A e +B d \right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{105 c \,e^{2}}-\frac {4 \left (21 a A \,e^{3}-7 A c \,d^{2} e +8 a B d \,e^{2}+4 B c \,d^{3}\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{105 e^{3} \sqrt {c}\, \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}+\frac {4 \left (a \,e^{2}+c \,d^{2}\right ) \left (-7 A c d e +5 a B \,e^{2}+4 B c \,d^{2}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{105 c^{\frac {3}{2}} e^{3} \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, {\left (4 \, B c^{2} d^{4} - 7 \, A c^{2} d^{3} e + 11 \, B a c d^{2} e^{2} - 63 \, A a c d e^{3} + 15 \, B a^{2} e^{4}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 6 \, {\left (4 \, B c^{2} d^{3} e - 7 \, A c^{2} d^{2} e^{2} + 8 \, B a c d e^{3} + 21 \, A a c e^{4}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) + 3 \, {\left (4 \, B c^{2} d^{2} e^{2} - {\left (15 \, B c^{2} x^{2} + 21 \, A c^{2} x + 10 \, B a c\right )} e^{4} - {\left (3 \, B c^{2} d x + 7 \, A c^{2} d\right )} e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )} e^{\left (-4\right )}}{315 \, c^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x + d}, x\right ) \]