Optimal. Leaf size=475 \[ \frac {\sqrt {2} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {\frac {(e f-d g) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {\frac {(e f-d g) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b f+\sqrt {b^2-4 a c} f-2 a g\right ) (d+e x)}} (d+e x) \Pi \left (\frac {e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g};\sin ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right )|\frac {\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} g \sqrt {\frac {2 a c}{b+\sqrt {b^2-4 a c}}+c x} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.26, 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 = {940}
\begin {gather*} \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};\text {ArcSin}\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 )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 940
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] Leaf count is larger than twice the leaf count of optimal. \(1118\) vs. \(2(475)=950\).
time = 26.60, size = 1118, normalized size = 2.35 \begin {gather*} -\frac {\sqrt {2} \sqrt {-\frac {g \left (c f^2+g (-b f+a g)\right ) (d+e x)}{\left (-2 c d f g-2 a e g^2+e f \sqrt {\left (b^2-4 a c\right ) g^2}-d g \sqrt {\left (b^2-4 a c\right ) g^2}+b g (e f+d g)\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 (-b f+a g)\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 g x+b g (f-g x)+\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} (-e f+d g)}{2 c d f g+2 a e g^2-e f \sqrt {\left (b^2-4 a c\right ) g^2}+d g \sqrt {\left (b^2-4 a c\right ) g^2}-b g (e f+d g)}\right )}{c f^2+g (-b f+a g)}+\frac {d g \left (2 a g^2-f \sqrt {\left (b^2-4 a c\right ) g^2}-2 c f g x-g \sqrt {\left (b^2-4 a c\right ) g^2} x+b g (-f+g x)\right ) \sqrt {\frac {2 a g^2-2 c f g x+b g (-f+g x)+\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 g x+b g (f-g x)+\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} (-e f+d g)}{2 c d f g+2 a e g^2-e f \sqrt {\left (b^2-4 a c\right ) g^2}+d g \sqrt {\left (b^2-4 a c\right ) g^2}-b g (e f+d g)}\right )}{\left (c f^2+g (-b f+a g)\right ) (f+g x) \sqrt {\frac {-2 a g^2+2 c f g x+b g (f-g x)+\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 (-b f+a g)\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 g x+b g (f-g x)+\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} (-e f+d g)}{2 c d f g+2 a e g^2-e f \sqrt {\left (b^2-4 a c\right ) g^2}+d g \sqrt {\left (b^2-4 a c\right ) g^2}-b g (e f+d g)}\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)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.21, size = 590, normalized size = 1.24
method | result | size |
default | \(\frac {4 \sqrt {c \,x^{2}+b x +a}\, \sqrt {g x +f}\, \sqrt {e x +d}\, \sqrt {\frac {\left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \left (g x +f \right )}{\left (g \sqrt {-4 a c +b^{2}}+b g -2 c f \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (d g -e f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (2 c f -b g +g \sqrt {-4 a c +b^{2}}\right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (d g -e f \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (g \sqrt {-4 a c +b^{2}}+b g -2 c f \right ) \left (e x +d \right )}}\, \EllipticPi \left (\sqrt {\frac {\left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \left (g x +f \right )}{\left (g \sqrt {-4 a c +b^{2}}+b g -2 c f \right ) \left (e x +d \right )}}, \frac {\left (g \sqrt {-4 a c +b^{2}}+b g -2 c f \right ) e}{g \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right )}, \sqrt {\frac {\left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (g \sqrt {-4 a c +b^{2}}+b g -2 c f \right )}{\left (2 c f -b g +g \sqrt {-4 a c +b^{2}}\right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right )}}\right ) \left (\sqrt {-4 a c +b^{2}}\, e^{2} g \,x^{2}+b \,e^{2} g \,x^{2}-2 c \,e^{2} f \,x^{2}+2 \sqrt {-4 a c +b^{2}}\, d e g x +2 b d e g x -4 c d e f x +\sqrt {-4 a c +b^{2}}\, d^{2} g +b \,d^{2} g -2 c \,d^{2} f \right )}{g \sqrt {-\frac {\left (g x +f \right ) \left (e x +d \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}{c}}\, \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \sqrt {\left (g x +f \right ) \left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}}\) | \(590\) |
elliptic | \(\text {Expression too large to display}\) | \(1330\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {d + e x}}{\sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {d+e\,x}}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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