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

Optimal. Leaf size=15 \[ -\sqrt{2 x-x^2} \]

[Out]

-Sqrt[2*x - x^2]

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Rubi [A]  time = 0.0038994, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {629} \[ -\sqrt{2 x-x^2} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x)/Sqrt[2*x - x^2],x]

[Out]

-Sqrt[2*x - x^2]

Rule 629

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

Rubi steps

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

Mathematica [A]  time = 0.0061087, size = 12, normalized size = 0.8 \[ -\sqrt{-(x-2) x} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x)/Sqrt[2*x - x^2],x]

[Out]

-Sqrt[-((-2 + x)*x)]

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Maple [A]  time = 0.001, size = 17, normalized size = 1.1 \begin{align*}{x \left ( -2+x \right ){\frac{1}{\sqrt{-{x}^{2}+2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10355, size = 18, normalized size = 1.2 \begin{align*} -\sqrt{-x^{2} + 2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 2*x)

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Fricas [A]  time = 1.4182, size = 26, normalized size = 1.73 \begin{align*} -\sqrt{-x^{2} + 2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 2*x)

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Sympy [A]  time = 0.122126, size = 10, normalized size = 0.67 \begin{align*} - \sqrt{- x^{2} + 2 x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x**2 + 2*x)

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Giac [A]  time = 1.09644, size = 18, normalized size = 1.2 \begin{align*} -\sqrt{-x^{2} + 2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 2*x)