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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {2}{3} (1-x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x],x]

[Out]

(-2*(1 - x)^(3/2))/3

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fricas [A]  time = 0.40, size = 12, normalized size = 0.92 \[ \frac {2}{3} \, {\left (x - 1\right )} \sqrt {-x + 1} \]

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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giac [A]  time = 0.35, size = 9, normalized size = 0.69 \[ -\frac {2}{3} \, {\left (-x + 1\right )}^{\frac {3}{2}} \]

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)

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maple [A]  time = 0.00, size = 10, normalized size = 0.77 \[ -\frac {2 \left (-x +1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.81, size = 9, normalized size = 0.69 \[ -\frac {2}{3} \, {\left (-x + 1\right )}^{\frac {3}{2}} \]

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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mupad [B]  time = 3.51, size = 9, normalized size = 0.69 \[ -\frac {2\,{\left (1-x\right )}^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(1/2),x)

[Out]

-(2*(1 - x)^(3/2))/3

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sympy [A]  time = 0.06, size = 10, normalized size = 0.77 \[ - \frac {2 \left (1 - x\right )^{\frac {3}{2}}}{3} \]

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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