3.858 \(\int \sqrt{1-x} \, dx\)

Optimal. Leaf size=13 \[ -\frac{2}{3} (1-x)^{3/2} \]

[Out]

(-2*(1 - x)^(3/2))/3

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Rubi [A]  time = 0.0008786, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {32} \[ -\frac{2}{3} (1-x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x],x]

[Out]

(-2*(1 - x)^(3/2))/3

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0030709, size = 13, normalized size = 1. \[ -\frac{2}{3} (1-x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x],x]

[Out]

(-2*(1 - x)^(3/2))/3

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Maple [A]  time = 0.003, size = 10, normalized size = 0.8 \begin{align*} -{\frac{2}{3} \left ( 1-x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15072, size = 12, normalized size = 0.92 \begin{align*} -\frac{2}{3} \,{\left (-x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-x + 1)^(3/2)

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Fricas [A]  time = 1.41788, size = 35, normalized size = 2.69 \begin{align*} \frac{2}{3} \,{\left (x - 1\right )} \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x - 1)*sqrt(-x + 1)

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Sympy [A]  time = 0.052573, size = 10, normalized size = 0.77 \begin{align*} - \frac{2 \left (1 - x\right )^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(1 - x)**(3/2)/3

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Giac [A]  time = 1.10317, size = 12, normalized size = 0.92 \begin{align*} -\frac{2}{3} \,{\left (-x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-x + 1)^(3/2)