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

Optimal. Leaf size=21 \[ \frac{2 \sqrt{-x^2+x+2}}{3 (x-2)} \]

[Out]

(2*Sqrt[2 + x - x^2])/(3*(-2 + x))

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Rubi [A]  time = 0.0054853, antiderivative size = 23, normalized size of antiderivative = 1.1, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {650} \[ -\frac{2 \sqrt{-x^2+x+2}}{3 (2-x)} \]

Antiderivative was successfully verified.

[In]

Int[1/((-2 + x)*Sqrt[2 + x - x^2]),x]

[Out]

(-2*Sqrt[2 + x - x^2])/(3*(2 - x))

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(-2+x) \sqrt{2+x-x^2}} \, dx &=-\frac{2 \sqrt{2+x-x^2}}{3 (2-x)}\\ \end{align*}

Mathematica [A]  time = 0.0042338, size = 21, normalized size = 1. \[ -\frac{2 \sqrt{-x^2+x+2}}{6-3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2 + x - x^2])/(6 - 3*x)

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \begin{align*} -{\frac{2\,x+2}{3}{\frac{1}{\sqrt{-{x}^{2}+x+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-2+x)/(-x^2+x+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.41472, size = 23, normalized size = 1.1 \begin{align*} \frac{2 \, \sqrt{-x^{2} + x + 2}}{3 \,{\left (x - 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2+x)/(-x^2+x+2)^(1/2),x, algorithm="maxima")

[Out]

2/3*sqrt(-x^2 + x + 2)/(x - 2)

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Fricas [A]  time = 1.74288, size = 43, normalized size = 2.05 \begin{align*} \frac{2 \, \sqrt{-x^{2} + x + 2}}{3 \,{\left (x - 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2+x)/(-x^2+x+2)^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(-x^2 + x + 2)/(x - 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2+x)/(-x**2+x+2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.05579, size = 38, normalized size = 1.81 \begin{align*} -\frac{4}{3 \,{\left (\frac{2 \, \sqrt{-x^{2} + x + 2} - 3}{2 \, x - 1} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2+x)/(-x^2+x+2)^(1/2),x, algorithm="giac")

[Out]

-4/3/((2*sqrt(-x^2 + x + 2) - 3)/(2*x - 1) + 1)