3.57 \(\int \frac{1+2 x}{\sqrt{2+x-x^2}} \, dx\)

Optimal. Leaf size=27 \[ -2 \sqrt{-x^2+x+2}-2 \sin ^{-1}\left (\frac{1}{3} (1-2 x)\right ) \]

[Out]

-2*Sqrt[2 + x - x^2] - 2*ArcSin[(1 - 2*x)/3]

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Rubi [A]  time = 0.0275121, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -2 \sqrt{-x^2+x+2}-2 \sin ^{-1}\left (\frac{1}{3} (1-2 x)\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x)/Sqrt[2 + x - x^2],x]

[Out]

-2*Sqrt[2 + x - x^2] - 2*ArcSin[(1 - 2*x)/3]

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Rubi in Sympy [A]  time = 2.21394, size = 32, normalized size = 1.19 \[ - 2 \sqrt{- x^{2} + x + 2} - 2 \operatorname{atan}{\left (\frac{- 2 x + 1}{2 \sqrt{- x^{2} + x + 2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x)/(-x**2+x+2)**(1/2),x)

[Out]

-2*sqrt(-x**2 + x + 2) - 2*atan((-2*x + 1)/(2*sqrt(-x**2 + x + 2)))

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Mathematica [A]  time = 0.0186192, size = 27, normalized size = 1. \[ -2 \sqrt{-x^2+x+2}-2 \sin ^{-1}\left (\frac{1}{3} (1-2 x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x)/Sqrt[2 + x - x^2],x]

[Out]

-2*Sqrt[2 + x - x^2] - 2*ArcSin[(1 - 2*x)/3]

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Maple [A]  time = 0.008, size = 22, normalized size = 0.8 \[ 2\,\arcsin \left ( -1/3+2/3\,x \right ) -2\,\sqrt{-{x}^{2}+x+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x)/(-x^2+x+2)^(1/2),x)

[Out]

2*arcsin(-1/3+2/3*x)-2*(-x^2+x+2)^(1/2)

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Maxima [A]  time = 1.52555, size = 28, normalized size = 1.04 \[ -2 \, \sqrt{-x^{2} + x + 2} - 2 \, \arcsin \left (-\frac{2}{3} \, x + \frac{1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)/sqrt(-x^2 + x + 2),x, algorithm="maxima")

[Out]

-2*sqrt(-x^2 + x + 2) - 2*arcsin(-2/3*x + 1/3)

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Fricas [A]  time = 0.226653, size = 45, normalized size = 1.67 \[ -2 \, \sqrt{-x^{2} + x + 2} + 2 \, \arctan \left (\frac{2 \, x - 1}{2 \, \sqrt{-x^{2} + x + 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)/sqrt(-x^2 + x + 2),x, algorithm="fricas")

[Out]

-2*sqrt(-x^2 + x + 2) + 2*arctan(1/2*(2*x - 1)/sqrt(-x^2 + x + 2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 x + 1}{\sqrt{- \left (x - 2\right ) \left (x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+2*x)/(-x**2+x+2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219947, size = 28, normalized size = 1.04 \[ -2 \, \sqrt{-x^{2} + x + 2} + 2 \, \arcsin \left (\frac{2}{3} \, x - \frac{1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)/sqrt(-x^2 + x + 2),x, algorithm="giac")

[Out]

-2*sqrt(-x^2 + x + 2) + 2*arcsin(2/3*x - 1/3)