\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-7 B e x -9 A e +8 B d \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x +d}}{63 e^{2}}-\frac {4 \left (32 B c \,d^{3}-36 A c \,d^{2} e +33 a B d \,e^{2}-45 a A \,e^{3}-3 e \left (-9 A c d e +7 a B \,e^{2}+8 B c \,d^{2}\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{315 e^{4}}+\frac {8 \left (36 A c d e \left (2 a \,e^{2}+c \,d^{2}\right )-B \left (21 a^{2} e^{4}+57 a c \,d^{2} e^{2}+32 c^{2} d^{4}\right )\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{315 e^{5} \sqrt {c}\, \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}+\frac {8 \left (a \,e^{2}+c \,d^{2}\right ) \left (-45 a A \,e^{3}-36 A c \,d^{2} e +33 a B d \,e^{2}+32 B c \,d^{3}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{315 e^{5} \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)*(c*x^2+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
\[ -\frac {2 \, {\left (4 \, {\left (32 \, B c^{2} d^{5} - 36 \, A c^{2} d^{4} e + 81 \, B a c d^{3} e^{2} - 99 \, A a c d^{2} e^{3} + 57 \, B a^{2} d e^{4} - 135 \, A a^{2} e^{5}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 12 \, {\left (32 \, B c^{2} d^{4} e - 36 \, A c^{2} d^{3} e^{2} + 57 \, B a c d^{2} e^{3} - 72 \, A a c d e^{4} + 21 \, B a^{2} e^{5}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) + 3 \, {\left (64 \, B c^{2} d^{3} e^{2} - {\left (35 \, B c^{2} x^{3} + 45 \, A c^{2} x^{2} + 77 \, B a c x + 135 \, A a c\right )} e^{5} + 2 \, {\left (20 \, B c^{2} d x^{2} + 27 \, A c^{2} d x + 53 \, B a c d\right )} e^{4} - 24 \, {\left (2 \, B c^{2} d^{2} x + 3 \, A c^{2} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )} e^{\left (-6\right )}}{945 \, c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt {c x^{2} + a}}{\sqrt {e x + d}}, x\right ) \]