3.9.58 \(\int \sqrt {1-x} \, dx\) [858]

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

[Out]

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

________________________________________________________________________________________

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} \begin {gather*} -\frac {2}{3} (1-x)^{3/2} \end {gather*}

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 = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {2}{3} (1-x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x],x]

[Out]

(-2*(1 - x)^(3/2))/3

________________________________________________________________________________________

Maple [A]
time = 0.53, size = 10, normalized size = 0.77

method	result	size
gosper	\(-\frac {2 \left (1-x \right )^{\frac {3}{2}}}{3}\)	\(10\)
derivativedivides	\(-\frac {2 \left (1-x \right )^{\frac {3}{2}}}{3}\)	\(10\)
default	\(-\frac {2 \left (1-x \right )^{\frac {3}{2}}}{3}\)	\(10\)
trager	\(\left (\frac {2 x}{3}-\frac {2}{3}\right ) \sqrt {1-x}\)	\(14\)
risch	\(-\frac {2 \left (-1+x \right )^{2}}{3 \sqrt {1-x}}\)	\(15\)
meijerg	\(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2-2 x \right ) \sqrt {1-x}}{3}}{2 \sqrt {\pi }}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 9, normalized size = 0.69 \begin {gather*} -\frac {2}{3} \, {\left (-x + 1\right )}^{\frac {3}{2}} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 12, normalized size = 0.92 \begin {gather*} \frac {2}{3} \, {\left (x - 1\right )} \sqrt {-x + 1} \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 10, normalized size = 0.77 \begin {gather*} - \frac {2 \left (1 - x\right )^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(1 - x)**(3/2)/3

________________________________________________________________________________________

Giac [A]
time = 4.74, size = 9, normalized size = 0.69 \begin {gather*} -\frac {2}{3} \, {\left (-x + 1\right )}^{\frac {3}{2}} \end {gather*}

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(1/2),x)

[Out]

-(2*(1 - x)^(3/2))/3

________________________________________________________________________________________