Optimal. Leaf size=475 \[ \frac {\sqrt {2} (d+e x) \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )} \sqrt {\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) (e f-d g)}{(d+e x) \left (2 c f-g \left (\sqrt {b^2-4 a c}+b\right )\right )}} \sqrt {\frac {\left (x \left (\sqrt {b^2-4 a c}+b\right )+2 a\right ) (e f-d g)}{(d+e x) \left (f \sqrt {b^2-4 a c}-2 a g+b f\right )}} \Pi \left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g};\sin ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right )|\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{g \sqrt {\frac {2 a c}{\sqrt {b^2-4 a c}+b}+c x} \sqrt {a+b x+c x^2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \]
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Rubi [A] time = 0.42, antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {926} \[ \frac {\sqrt {2} (d+e x) \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )} \sqrt {\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) (e f-d g)}{(d+e x) \left (2 c f-g \left (\sqrt {b^2-4 a c}+b\right )\right )}} \sqrt {\frac {\left (x \left (\sqrt {b^2-4 a c}+b\right )+2 a\right ) (e f-d g)}{(d+e x) \left (f \sqrt {b^2-4 a c}-2 a g+b f\right )}} \Pi \left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g};\sin ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right )|\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{g \sqrt {\frac {2 a c}{\sqrt {b^2-4 a c}+b}+c x} \sqrt {a+b x+c x^2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \]
Antiderivative was successfully verified.
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Rule 926
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx &=\frac {\sqrt {2} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {\frac {(e f-d g) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b f+\sqrt {b^2-4 a c} f-2 a g\right ) (d+e x)}} (d+e x) \Pi \left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g};\sin ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right )|\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} g \sqrt {\frac {2 a c}{b+\sqrt {b^2-4 a c}}+c x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [B] time = 9.47, size = 1118, normalized size = 2.35 \[ -\frac {\sqrt {2} \sqrt {-\frac {g \left (c f^2+g (a g-b f)\right ) (d+e x)}{\left (-2 a e g^2-2 c d f g+b (e f+d g) g-d \sqrt {\left (b^2-4 a c\right ) g^2} g+e f \sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} (f+g x)^{3/2} \left (\frac {2 e f \sqrt {\left (b^2-4 a c\right ) g^2} \sqrt {-\frac {\left (c f^2+g (a g-b f)\right ) (a+x (b+c x))}{\left (b^2-4 a c\right ) (f+g x)^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {-2 a g^2+2 c f x g+b (f-g x) g+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}}}{\sqrt {2}}\right )|\frac {2 \sqrt {\left (b^2-4 a c\right ) g^2} (d g-e f)}{2 a e g^2+2 c d f g-b (e f+d g) g+d \sqrt {\left (b^2-4 a c\right ) g^2} g-e f \sqrt {\left (b^2-4 a c\right ) g^2}}\right )}{c f^2+g (a g-b f)}+\frac {d g \left (2 a g^2-2 c f x g-\sqrt {\left (b^2-4 a c\right ) g^2} x g+b (g x-f) g-f \sqrt {\left (b^2-4 a c\right ) g^2}\right ) \sqrt {\frac {2 a g^2-2 c f x g+b (g x-f) g+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {-2 a g^2+2 c f x g+b (f-g x) g+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}}}{\sqrt {2}}\right )|\frac {2 \sqrt {\left (b^2-4 a c\right ) g^2} (d g-e f)}{2 a e g^2+2 c d f g-b (e f+d g) g+d \sqrt {\left (b^2-4 a c\right ) g^2} g-e f \sqrt {\left (b^2-4 a c\right ) g^2}}\right )}{\left (c f^2+g (a g-b f)\right ) (f+g x) \sqrt {\frac {-2 a g^2+2 c f x g+b (f-g x) g+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}}}-\frac {4 e \sqrt {\left (b^2-4 a c\right ) g^2} \sqrt {-\frac {\left (c f^2+g (a g-b f)\right ) (a+x (b+c x))}{\left (b^2-4 a c\right ) (f+g x)^2}} \Pi \left (\frac {2 \sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}};\sin ^{-1}\left (\frac {\sqrt {\frac {-2 a g^2+2 c f x g+b (f-g x) g+\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt {\left (b^2-4 a c\right ) g^2} (f+g x)}}}{\sqrt {2}}\right )|\frac {2 \sqrt {\left (b^2-4 a c\right ) g^2} (d g-e f)}{2 a e g^2+2 c d f g-b (e f+d g) g+d \sqrt {\left (b^2-4 a c\right ) g^2} g-e f \sqrt {\left (b^2-4 a c\right ) g^2}}\right )}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )}{g^2 \sqrt {d+e x} \sqrt {a+x (b+c x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 645, normalized size = 1.36 \[ \frac {4 \sqrt {e x +d}\, \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {\left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) \left (g x +f \right )}{\left (b g -2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (d g -e f \right ) \left (-2 c x -b +\sqrt {-4 a c +b^{2}}\right )}{\left (-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (d g -e f \right ) \left (2 c x +b +\sqrt {-4 a c +b^{2}}\right )}{\left (b g -2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) \left (e x +d \right )}}\, \left (b \,e^{2} g \,x^{2}-2 c \,e^{2} f \,x^{2}+2 b d e g x -4 c d e f x +\sqrt {-4 a c +b^{2}}\, e^{2} g \,x^{2}+b \,d^{2} g -2 c \,d^{2} f +2 \sqrt {-4 a c +b^{2}}\, d e g x +\sqrt {-4 a c +b^{2}}\, d^{2} g \right ) \EllipticPi \left (\sqrt {\frac {\left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) \left (g x +f \right )}{\left (b g -2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) \left (e x +d \right )}}, \frac {\left (b g -2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) e}{\left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) g}, \sqrt {\frac {\left (-b e +2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) \left (b g -2 c f +\sqrt {-4 a c +b^{2}}\, g \right )}{\left (-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) \left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right )}}\right )}{\sqrt {-\frac {\left (g x +f \right ) \left (e x +d \right ) \left (-2 c x -b +\sqrt {-4 a c +b^{2}}\right ) \left (2 c x +b +\sqrt {-4 a c +b^{2}}\right )}{c}}\, \left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) \sqrt {c e g \,x^{4}+b e g \,x^{3}+c d g \,x^{3}+c e f \,x^{3}+a e g \,x^{2}+b d g \,x^{2}+b e f \,x^{2}+c d f \,x^{2}+a d g x +a e f x +b d f x +a d f}\, g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {d+e\,x}}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d + e x}}{\sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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