3.126 \(\int \frac{1}{\sqrt{\frac{-b^2+4 c}{4 c}+b x-c x^2}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]

[Out]

-(ArcSin[(b - 2*c*x)/(2*Sqrt[c])]/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]

[Out]

-(ArcSin[(b - 2*c*x)/(2*Sqrt[c])]/Sqrt[c])

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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]

-atan((4*b - 8*c*x)/(4*sqrt(c)*sqrt(-b**2/c + 4*b*x - 4*c*x**2 + 4)))/sqrt(c)

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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]

-(ArcSin[(b - 2*c*x)/(2*Sqrt[c])]/Sqrt[c])

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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)

[Out]

1/c^(1/2)*arctan(2*c^(1/2)*(x-1/2*b/c)/(-4*c*x^2+4*b*x-(b^2-4*c)/c)^(1/2))

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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]

Exception raised: ValueError

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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")

[Out]

[1/2*log((4*c^2*x^2 - 4*b*c*x + b^2 - 2*c)*sqrt(-c) + (2*c^2*x - b*c)*sqrt(-(4*c
^2*x^2 - 4*b*c*x + b^2 - 4*c)/c))/sqrt(-c), arctan((2*c*x - b)/(sqrt(c)*sqrt(-(4
*c^2*x^2 - 4*b*c*x + b^2 - 4*c)/c)))/sqrt(c)]

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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]

2*Integral(1/sqrt(-b**2/c + 4*b*x - 4*c*x**2 + 4), x)

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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]

-ln(abs((2*sqrt(-c)*x - sqrt(-4*c*x^2 + 4*b*x - (b^2 - 4*c)/c))*sqrt(-c) + b))/s
qrt(-c)