Optimal. Leaf size=23 \[ -\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]
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Rubi [A] time = 0.0283524, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[(-b^2 + 4*c)/(4*c) + b*x - c*x^2],x]
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Rubi in Sympy [A] time = 6.40854, size = 44, normalized size = 1.91 \[ - \frac{\operatorname{atan}{\left (\frac{4 b - 8 c x}{4 \sqrt{c} \sqrt{- \frac{b^{2}}{c} + 4 b x - 4 c x^{2} + 4}} \right )}}{\sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(2/((-b**2+4*c)/c+4*b*x-4*c*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0447288, size = 23, normalized size = 1. \[ -\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[(-b^2 + 4*c)/(4*c) + b*x - c*x^2],x]
[Out]
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Maple [B] time = 0.018, size = 44, normalized size = 1.9 \[{1\arctan \left ( 2\,{\sqrt{c} \left ( x-1/2\,{\frac{b}{c}} \right ){\frac{1}{\sqrt{-4\,c{x}^{2}+4\,bx-{\frac{{b}^{2}-4\,c}{c}}}}}} \right ){\frac{1}{\sqrt{c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(2/((-b^2+4*c)/c+4*b*x-4*c*x^2)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2/sqrt(-4*c*x^2 + 4*b*x - (b^2 - 4*c)/c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236473, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left ({\left (4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 2 \, c\right )} \sqrt{-c} +{\left (2 \, c^{2} x - b c\right )} \sqrt{-\frac{4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}{c}}\right )}{2 \, \sqrt{-c}}, \frac{\arctan \left (\frac{2 \, c x - b}{\sqrt{c} \sqrt{-\frac{4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}{c}}}\right )}{\sqrt{c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2/sqrt(-4*c*x^2 + 4*b*x - (b^2 - 4*c)/c),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ 2 \int \frac{1}{\sqrt{- \frac{b^{2}}{c} + 4 b x - 4 c x^{2} + 4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2/((-b**2+4*c)/c+4*b*x-4*c*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.229464, size = 72, normalized size = 3.13 \[ -\frac{{\rm ln}\left ({\left |{\left (2 \, \sqrt{-c} x - \sqrt{-4 \, c x^{2} + 4 \, b x - \frac{b^{2} - 4 \, c}{c}}\right )} \sqrt{-c} + b \right |}\right )}{\sqrt{-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2/sqrt(-4*c*x^2 + 4*b*x - (b^2 - 4*c)/c),x, algorithm="giac")
[Out]