3.6.50 \(\int \frac {1}{x^2 \sqrt {9-4 x^2}} \, dx\) [550]

Optimal. Leaf size=18 \[ -\frac {\sqrt {9-4 x^2}}{9 x} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \begin {gather*} -\frac {\sqrt {9-4 x^2}}{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[9 - 4*x^2]),x]

[Out]

-1/9*Sqrt[9 - 4*x^2]/x

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \sqrt {9-4 x^2}} \, dx &=-\frac {\sqrt {9-4 x^2}}{9 x}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {9-4 x^2}}{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[9 - 4*x^2]),x]

[Out]

-1/9*Sqrt[9 - 4*x^2]/x

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Maple [A]
time = 0.04, size = 15, normalized size = 0.83

method	result	size
default	\(-\frac {\sqrt {-4 x^{2}+9}}{9 x}\)	\(15\)
trager	\(-\frac {\sqrt {-4 x^{2}+9}}{9 x}\)	\(15\)
meijerg	\(-\frac {\sqrt {1-\frac {4 x^{2}}{9}}}{3 x}\)	\(15\)
risch	\(\frac {4 x^{2}-9}{9 x \sqrt {-4 x^{2}+9}}\)	\(22\)
gosper	\(\frac {\left (2 x -3\right ) \left (2 x +3\right )}{9 x \sqrt {-4 x^{2}+9}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {-4 \, x^{2} + 9}}{9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(-4*x^2 + 9)/x

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Fricas [A]
time = 1.27, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {-4 \, x^{2} + 9}}{9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(-4*x^2 + 9)/x

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Sympy [C] Result contains complex when optimal does not.
time = 0.38, size = 36, normalized size = 2.00 \begin {gather*} \begin {cases} - \frac {i \sqrt {4 x^{2} - 9}}{9 x} & \text {for}\: \left |{x^{2}}\right | > \frac {9}{4} \\- \frac {\sqrt {9 - 4 x^{2}}}{9 x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*sqrt(4*x**2 - 9)/(9*x), Abs(x**2) > 9/4), (-sqrt(9 - 4*x**2)/(9*x), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
time = 1.21, size = 33, normalized size = 1.83 \begin {gather*} \frac {2 \, x}{9 \, {\left (\sqrt {-4 \, x^{2} + 9} - 3\right )}} - \frac {\sqrt {-4 \, x^{2} + 9} - 3}{18 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*x/(sqrt(-4*x^2 + 9) - 3) - 1/18*(sqrt(-4*x^2 + 9) - 3)/x

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Mupad [B]
time = 0.02, size = 14, normalized size = 0.78 \begin {gather*} -\frac {2\,\sqrt {\frac {9}{4}-x^2}}{9\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(9 - 4*x^2)^(1/2)),x)

[Out]

-(2*(9/4 - x^2)^(1/2))/(9*x)

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