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