3.5.56 \(\int x \sqrt {9-4 x^2} \, dx\) [456]

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

[Out]

-1/12*(-4*x^2+9)^(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}{12} \left (9-4 x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[9 - 4*x^2],x]

[Out]

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

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[9 - 4*x^2],x]

[Out]

-1/12*(9 - 4*x^2)^(3/2)

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

method	result	size
derivativedivides	\(-\frac {\left (-4 x^{2}+9\right )^{\frac {3}{2}}}{12}\)	\(12\)
default	\(-\frac {\left (-4 x^{2}+9\right )^{\frac {3}{2}}}{12}\)	\(12\)
trager	\(\left (\frac {x^{2}}{3}-\frac {3}{4}\right ) \sqrt {-4 x^{2}+9}\)	\(18\)
risch	\(-\frac {\left (4 x^{2}-9\right )^{2}}{12 \sqrt {-4 x^{2}+9}}\)	\(21\)
gosper	\(\frac {\left (2 x -3\right ) \left (2 x +3\right ) \sqrt {-4 x^{2}+9}}{12}\)	\(22\)
meijerg	\(\frac {\frac {9 \sqrt {\pi }}{4}-\frac {9 \sqrt {\pi }\, \left (-\frac {8 x^{2}}{9}+2\right ) \sqrt {1-\frac {4 x^{2}}{9}}}{8}}{\sqrt {\pi }}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{12} \, {\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} \end {gather*}

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)

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Fricas [A]
time = 0.72, size = 18, normalized size = 1.20 \begin {gather*} \frac {1}{12} \, {\left (4 \, x^{2} - 9\right )} \sqrt {-4 \, x^{2} + 9} \end {gather*}

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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).
time = 0.07, size = 27, normalized size = 1.80 \begin {gather*} \frac {x^{2} \sqrt {9 - 4 x^{2}}}{3} - \frac {3 \sqrt {9 - 4 x^{2}}}{4} \end {gather*}

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

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Giac [A]
time = 0.56, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{12} \, {\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} \end {gather*}

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)

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

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

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