3.5.78 \(\int x \sqrt {-9-4 x^2} \, dx\) [478]

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 (-4 x^2-9\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
gosper	\(-\frac {\left (-4 x^{2}-9\right )^{\frac {3}{2}}}{12}\)	\(12\)
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	\(-\frac {\left (4 x^{2}+9\right )^{2}}{12 \sqrt {-4 x^{2}-9}}\)	\(21\)
meijerg	\(-\frac {27 i \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2+\frac {8 x^{2}}{9}\right ) \sqrt {1+\frac {4 x^{2}}{9}}}{3}\right )}{16 \sqrt {\pi }}\)	\(34\)

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 = 0.28, 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 = 1.10, 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. 31 vs. \(2 (14) = 28\).
time = 0.07, size = 31, normalized size = 2.07 \begin {gather*} \frac {x^{2} \sqrt {- 4 x^{2} - 9}}{3} + \frac {3 \sqrt {- 4 x^{2} - 9}}{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(-4*x**2 - 9)/3 + 3*sqrt(-4*x**2 - 9)/4

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Giac [C] Result contains complex when optimal does not.
time = 0.53, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{12} i \, {\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*I*(4*x^2 + 9)^(3/2)

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Mupad [B]
time = 0.07, size = 11, normalized size = 0.73 \begin {gather*} -\frac {{\left (-4\,x^2-9\right )}^{3/2}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(- 4*x^2 - 9)^(1/2),x)

[Out]

-(- 4*x^2 - 9)^(3/2)/12

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