3.127 \(\int x \sqrt {4-x^2} \, dx\)

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 = {261} \[ -\frac {1}{3} \left (4-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[4 - x^2],x]

[Out]

-(4 - x^2)^(3/2)/3

Rule 261

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 \[ -\frac {1}{3} \left (4-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[4 - x^2],x]

[Out]

-1/3*(4 - x^2)^(3/2)

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fricas [A]  time = 0.41, size = 16, normalized size = 1.07 \[ \frac {1}{3} \, {\left (x^{2} - 4\right )} \sqrt {-x^{2} + 4} \]

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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giac [A]  time = 1.02, size = 11, normalized size = 0.73 \[ -\frac {1}{3} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-x^2 + 4)^(3/2)

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maple [A]  time = 0.00, size = 18, normalized size = 1.20 \[ \frac {\left (x -2\right ) \left (x +2\right ) \sqrt {-x^{2}+4}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 11, normalized size = 0.73 \[ -\frac {1}{3} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} \]

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

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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sympy [B]  time = 0.21, size = 24, normalized size = 1.60 \[ \frac {x^{2} \sqrt {4 - x^{2}}}{3} - \frac {4 \sqrt {4 - x^{2}}}{3} \]

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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