\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx \]
Optimal antiderivative \[ \frac {2 \EllipticF \left (\frac {\sqrt {1-c}\, \sqrt {b x +a}}{\sqrt {a}}, \sqrt {\frac {1-e}{1-c}}\right ) \sqrt {a}}{b \sqrt {1-c}} \]
command
integrate(1/(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
\[ -\frac {2 \, a^{2} \sqrt {-\frac {b^{3} c - b^{3} - {\left (b^{3} c - b^{3}\right )} e}{a^{2}}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} c^{2} - a^{2} c + a^{2} e^{2} + a^{2} - {\left (a^{2} c + a^{2}\right )} e\right )}}{3 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2} + {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e^{2} - 2 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e\right )}}, \frac {4 \, {\left (2 \, a^{3} c^{3} - 3 \, a^{3} c^{2} - 3 \, a^{3} c + 2 \, a^{3} e^{3} + 2 \, a^{3} - 3 \, {\left (a^{3} c + a^{3}\right )} e^{2} - 3 \, {\left (a^{3} c^{2} - 4 \, a^{3} c + a^{3}\right )} e\right )}}{27 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3} - {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{2} - 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e\right )}}, \frac {2 \, a c + 3 \, {\left (b c - b\right )} x - {\left (3 \, a c + 3 \, {\left (b c - b\right )} x - 2 \, a\right )} e - a}{3 \, {\left (b c - {\left (b c - b\right )} e - b\right )}}\right )}{b^{3} c - b^{3} - {\left (b^{3} c - b^{3}\right )} e} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x + a} a^{2} \sqrt {\frac {a c + {\left (b c - b\right )} x}{a}} \sqrt {\frac {a e + {\left (b e - b\right )} x}{a}}}{a^{3} c e - {\left (b^{3} c - b^{3} - {\left (b^{3} c - b^{3}\right )} e\right )} x^{3} - {\left (2 \, a b^{2} c - a b^{2} - {\left (3 \, a b^{2} c - 2 \, a b^{2}\right )} e\right )} x^{2} - {\left (a^{2} b c - {\left (3 \, a^{2} b c - a^{2} b\right )} e\right )} x}, x\right ) \]