\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx \]
Optimal antiderivative \[ \frac {2 \EllipticF \left (\frac {\sqrt {1-e}\, \sqrt {b x +a}}{\sqrt {a}}, \sqrt {-\frac {a d}{\left (-a d +b c \right ) \left (1-e \right )}}\right ) \sqrt {a}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}}{b \sqrt {1-e}\, \sqrt {d x +c}} \]
command
integrate(1/(b*x+a)^(1/2)/(d*x+c)^(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
\[ \frac {2 \, a \sqrt {\frac {b^{2} d e - b^{2} d}{a}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e^{2} - {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2}\right )} e\right )}}{3 \, {\left (b^{2} d^{2} e^{2} - 2 \, b^{2} d^{2} e + b^{2} d^{2}\right )}}, \frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{3} + 3 \, {\left (2 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{2} - 3 \, {\left (2 \, b^{3} c^{3} - 4 \, a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} e\right )}}{27 \, {\left (b^{3} d^{3} e^{3} - 3 \, b^{3} d^{3} e^{2} + 3 \, b^{3} d^{3} e - b^{3} d^{3}\right )}}, -\frac {3 \, b d x + b c + a d - {\left (3 \, b d x + b c + 2 \, a d\right )} e}{3 \, {\left (b d e - b d\right )}}\right )}{b^{2} d e - b^{2} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {d x + c} a \sqrt {\frac {a e + {\left (b e - b\right )} x}{a}}}{a^{2} c e + {\left (b^{2} d e - b^{2} d\right )} x^{3} - {\left (b^{2} c + a b d - {\left (b^{2} c + 2 \, a b d\right )} e\right )} x^{2} - {\left (a b c - {\left (2 \, a b c + a^{2} d\right )} e\right )} x}, x\right ) \]