3.456 \(\int x \sqrt {9-4 x^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac {1}{12} \left (9-4 x^2\right )^{3/2} \]

[Out]

-1/12*(-4*x^2+9)^(3/2)

________________________________________________________________________________________

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} \[ -\frac {1}{12} \left (9-4 x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[9 - 4*x^2],x]

[Out]

-(9 - 4*x^2)^(3/2)/12

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 {9-4 x^2} \, dx &=-\frac {1}{12} \left (9-4 x^2\right )^{3/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ -\frac {1}{12} \left (9-4 x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[9 - 4*x^2],x]

[Out]

-1/12*(9 - 4*x^2)^(3/2)

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 18, normalized size = 1.20 \[ \frac {1}{12} \, {\left (4 \, x^{2} - 9\right )} \sqrt {-4 \, x^{2} + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

1/12*(4*x^2 - 9)*sqrt(-4*x^2 + 9)

________________________________________________________________________________________

giac [A]  time = 1.08, size = 11, normalized size = 0.73 \[ -\frac {1}{12} \, {\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

-1/12*(-4*x^2 + 9)^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 22, normalized size = 1.47 \[ \frac {\left (2 x -3\right ) \left (2 x +3\right ) \sqrt {-4 x^{2}+9}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2+9)^(1/2)*x,x)

[Out]

1/12*(2*x-3)*(2*x+3)*(-4*x^2+9)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.28, size = 11, normalized size = 0.73 \[ -\frac {1}{12} \, {\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

-1/12*(-4*x^2 + 9)^(3/2)

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 18, normalized size = 1.20 \[ \frac {\sqrt {\frac {9}{4}-x^2}\,\left (\frac {4\,x^2}{3}-3\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(9 - 4*x^2)^(1/2),x)

[Out]

((9/4 - x^2)^(1/2)*((4*x^2)/3 - 3))/2

________________________________________________________________________________________

sympy [B]  time = 0.20, size = 27, normalized size = 1.80 \[ \frac {x^{2} \sqrt {9 - 4 x^{2}}}{3} - \frac {3 \sqrt {9 - 4 x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-4*x**2+9)**(1/2),x)

[Out]

x**2*sqrt(9 - 4*x**2)/3 - 3*sqrt(9 - 4*x**2)/4

________________________________________________________________________________________