\[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx \]
Optimal antiderivative \[ \frac {b \arctan \left (\frac {x \sqrt {-a f +b e}}{\sqrt {a}\, \sqrt {f \,x^{2}+e}}\right )}{\left (-a d +b c \right ) \sqrt {a}\, \sqrt {-a f +b e}}-\frac {d \arctan \left (\frac {x \sqrt {-c f +d e}}{\sqrt {c}\, \sqrt {f \,x^{2}+e}}\right )}{\left (-a d +b c \right ) \sqrt {c}\, \sqrt {-c f +d e}} \]
command
integrate(1/(b*x^2+a)/(d*x^2+c)/(f*x^2+e)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [-\frac {{\left (b c^{2} f - b c d e\right )} \sqrt {a^{2} f - a b e} \log \left (\frac {8 \, a^{2} f^{2} x^{4} - 4 \, {\left (2 \, a f x^{3} - {\left (b x^{3} - a x\right )} e\right )} \sqrt {a^{2} f - a b e} \sqrt {f x^{2} + e} + {\left (b^{2} x^{4} - 6 \, a b x^{2} + a^{2}\right )} e^{2} - 8 \, {\left (a b f x^{4} - a^{2} f x^{2}\right )} e}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + {\left (a^{2} d f - a b d e\right )} \sqrt {c^{2} f - c d e} \log \left (\frac {8 \, c^{2} f^{2} x^{4} + 4 \, {\left (2 \, c f x^{3} - {\left (d x^{3} - c x\right )} e\right )} \sqrt {c^{2} f - c d e} \sqrt {f x^{2} + e} + {\left (d^{2} x^{4} - 6 \, c d x^{2} + c^{2}\right )} e^{2} - 8 \, {\left (c d f x^{4} - c^{2} f x^{2}\right )} e}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}, \frac {2 \, {\left (a^{2} d f - a b d e\right )} \sqrt {-c^{2} f + c d e} \arctan \left (\frac {{\left (2 \, c f x^{2} - {\left (d x^{2} - c\right )} e\right )} \sqrt {-c^{2} f + c d e} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} f^{2} x^{3} - c d x e^{2} - {\left (c d f x^{3} - c^{2} f x\right )} e\right )}}\right ) - {\left (b c^{2} f - b c d e\right )} \sqrt {a^{2} f - a b e} \log \left (\frac {8 \, a^{2} f^{2} x^{4} - 4 \, {\left (2 \, a f x^{3} - {\left (b x^{3} - a x\right )} e\right )} \sqrt {a^{2} f - a b e} \sqrt {f x^{2} + e} + {\left (b^{2} x^{4} - 6 \, a b x^{2} + a^{2}\right )} e^{2} - 8 \, {\left (a b f x^{4} - a^{2} f x^{2}\right )} e}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}, -\frac {2 \, {\left (b c^{2} f - b c d e\right )} \sqrt {-a^{2} f + a b e} \arctan \left (\frac {{\left (2 \, a f x^{2} - {\left (b x^{2} - a\right )} e\right )} \sqrt {-a^{2} f + a b e} \sqrt {f x^{2} + e}}{2 \, {\left (a^{2} f^{2} x^{3} - a b x e^{2} - {\left (a b f x^{3} - a^{2} f x\right )} e\right )}}\right ) + {\left (a^{2} d f - a b d e\right )} \sqrt {c^{2} f - c d e} \log \left (\frac {8 \, c^{2} f^{2} x^{4} + 4 \, {\left (2 \, c f x^{3} - {\left (d x^{3} - c x\right )} e\right )} \sqrt {c^{2} f - c d e} \sqrt {f x^{2} + e} + {\left (d^{2} x^{4} - 6 \, c d x^{2} + c^{2}\right )} e^{2} - 8 \, {\left (c d f x^{4} - c^{2} f x^{2}\right )} e}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}, -\frac {{\left (b c^{2} f - b c d e\right )} \sqrt {-a^{2} f + a b e} \arctan \left (\frac {{\left (2 \, a f x^{2} - {\left (b x^{2} - a\right )} e\right )} \sqrt {-a^{2} f + a b e} \sqrt {f x^{2} + e}}{2 \, {\left (a^{2} f^{2} x^{3} - a b x e^{2} - {\left (a b f x^{3} - a^{2} f x\right )} e\right )}}\right ) - {\left (a^{2} d f - a b d e\right )} \sqrt {-c^{2} f + c d e} \arctan \left (\frac {{\left (2 \, c f x^{2} - {\left (d x^{2} - c\right )} e\right )} \sqrt {-c^{2} f + c d e} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} f^{2} x^{3} - c d x e^{2} - {\left (c d f x^{3} - c^{2} f x\right )} e\right )}}\right )}{2 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]