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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 0.80 \[ -\sqrt {-((x-2) x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x)/Sqrt[2*x - x^2],x]

[Out]

-Sqrt[-((-2 + x)*x)]

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fricas [A]  time = 0.61, size = 13, normalized size = 0.87 \[ -\sqrt {-x^{2} + 2 \, x} \]

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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giac [A]  time = 0.60, size = 13, normalized size = 0.87 \[ -\sqrt {-x^{2} + 2 \, x} \]

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)

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maple [A]  time = 0.00, size = 17, normalized size = 1.13 \[ \frac {\left (x -2\right ) x}{\sqrt {-x^{2}+2 x}} \]

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 = 0.87, size = 13, normalized size = 0.87 \[ -\sqrt {-x^{2} + 2 \, x} \]

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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mupad [B]  time = 3.64, size = 10, normalized size = 0.67 \[ -\sqrt {-x\,\left (x-2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/(2*x - x^2)^(1/2),x)

[Out]

-(-x*(x - 2))^(1/2)

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sympy [A]  time = 0.14, size = 10, normalized size = 0.67 \[ - \sqrt {- x^{2} + 2 x} \]

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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