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