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

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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.10, 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*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ -\frac {2 \sqrt {-x^2+x+2}}{6-3 x} \]

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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IntegrateAlgebraic [A]  time = 0.18, size = 21, normalized size = 1.00 \[ \frac {2 \sqrt {-x^2+x+2}}{3 (x-2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[2 + x - x^2])/(3*(-2 + x))

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fricas [A]  time = 0.72, size = 17, normalized size = 0.81 \[ \frac {2 \, \sqrt {-x^{2} + x + 2}}{3 \, {\left (x - 2\right )}} \]

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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giac [A]  time = 0.78, size = 28, normalized size = 1.33 \[ -\frac {4}{3 \, {\left (\frac {2 \, \sqrt {-x^{2} + x + 2} - 3}{2 \, x - 1} + 1\right )}} \]

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)

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maple [A]  time = 0.30, size = 16, normalized size = 0.76

method	result	size
gosper	\(-\frac {2 \left (1+x \right )}{3 \sqrt {-x^{2}+x +2}}\)	\(16\)
risch	\(-\frac {2 \left (1+x \right )}{3 \sqrt {-x^{2}+x +2}}\)	\(16\)
trager	\(\frac {2 \sqrt {-x^{2}+x +2}}{3 \left (-2+x \right )}\)	\(18\)
default	\(\frac {2 \sqrt {-\left (-2+x \right )^{2}+6-3 x}}{3 \left (-2+x \right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-2+x)/(-x^2+x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.96, size = 17, normalized size = 0.81 \[ \frac {2 \, \sqrt {-x^{2} + x + 2}}{3 \, {\left (x - 2\right )}} \]

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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mupad [B]  time = 0.22, size = 19, normalized size = 0.90 \[ \frac {2\,\sqrt {-x^2+x+2}}{3\,\left (x-2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(x - x^2 + 2)^(1/2))/(3*(x - 2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (x - 2\right ) \left (x + 1\right )} \left (x - 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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