3.2.27 \(\int x \sqrt {4-x^2} \, dx\) [127]

Optimal. Leaf size=15 \[ -\frac {1}{3} \left (4-x^2\right )^{3/2} \]

[Out]

-1/3*(-x^2+4)^(3/2)

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \begin {gather*} -\frac {1}{3} \left (4-x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[4 - x^2],x]

[Out]

-1/3*(4 - x^2)^(3/2)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \sqrt {4-x^2} \, dx &=-\frac {1}{3} \left (4-x^2\right )^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \left (4-x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[4 - x^2],x]

[Out]

-1/3*(4 - x^2)^(3/2)

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Mathics [A]
time = 1.90, size = 11, normalized size = 0.73 \begin {gather*} -\frac {{\left (4-x^2\right )}^{\frac {3}{2}}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x*Sqrt[4 - x^2],x]')

[Out]

-(4 - x ^ 2) ^ (3 / 2) / 3

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Maple [A]
time = 0.04, size = 12, normalized size = 0.80

method	result	size
derivativedivides	\(-\frac {\left (-x^{2}+4\right )^{\frac {3}{2}}}{3}\)	\(12\)
default	\(-\frac {\left (-x^{2}+4\right )^{\frac {3}{2}}}{3}\)	\(12\)
gosper	\(\frac {\left (-2+x \right ) \left (2+x \right ) \sqrt {-x^{2}+4}}{3}\)	\(18\)
trager	\(\left (\frac {x^{2}}{3}-\frac {4}{3}\right ) \sqrt {-x^{2}+4}\)	\(18\)
risch	\(-\frac {\left (x^{2}-4\right )^{2}}{3 \sqrt {-x^{2}+4}}\)	\(19\)
meijerg	\(\frac {\frac {8 \sqrt {\pi }}{3}-\frac {4 \sqrt {\pi }\, \left (-\frac {x^{2}}{2}+2\right ) \sqrt {1-\frac {x^{2}}{4}}}{3}}{\sqrt {\pi }}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-x^2+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(-x^2+4)^(3/2)

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Maxima [A]
time = 0.26, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{3} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+4)^(1/2),x, algorithm="maxima")

[Out]

-1/3*(-x^2 + 4)^(3/2)

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Fricas [A]
time = 0.34, size = 16, normalized size = 1.07 \begin {gather*} \frac {1}{3} \, {\left (x^{2} - 4\right )} \sqrt {-x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+4)^(1/2),x, algorithm="fricas")

[Out]

1/3*(x^2 - 4)*sqrt(-x^2 + 4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (10) = 20\)
time = 0.08, size = 24, normalized size = 1.60 \begin {gather*} \frac {x^{2} \sqrt {4 - x^{2}}}{3} - \frac {4 \sqrt {4 - x^{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x**2+4)**(1/2),x)

[Out]

x**2*sqrt(4 - x**2)/3 - 4*sqrt(4 - x**2)/3

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Giac [A]
time = 0.00, size = 20, normalized size = 1.33 \begin {gather*} -\frac {1}{3} \sqrt {-x^{2}+4} \left (-x^{2}+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+4)^(1/2),x)

[Out]

-1/3*(-x^2 + 4)^(3/2)

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Mupad [B]
time = 0.03, size = 11, normalized size = 0.73 \begin {gather*} -\frac {{\left (4-x^2\right )}^{3/2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(4 - x^2)^(1/2),x)

[Out]

-(4 - x^2)^(3/2)/3

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