3.5.61 \(\int \frac {\sqrt {9-4 x^2}}{x^4} \, dx\) [461]

Optimal. Leaf size=18 \[ -\frac {\left (9-4 x^2\right )^{3/2}}{27 x^3} \]

[Out]

-1/27*(-4*x^2+9)^(3/2)/x^3

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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 {\left (9-4 x^2\right )^{3/2}}{27 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[9 - 4*x^2]/x^4,x]

[Out]

-1/27*(9 - 4*x^2)^(3/2)/x^3

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 {\sqrt {9-4 x^2}}{x^4} \, dx &=-\frac {\left (9-4 x^2\right )^{3/2}}{27 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\left (9-4 x^2\right )^{3/2}}{27 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[9 - 4*x^2]/x^4,x]

[Out]

-1/27*(9 - 4*x^2)^(3/2)/x^3

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Maple [A]
time = 0.05, size = 15, normalized size = 0.83

method	result	size
default	\(-\frac {\left (-4 x^{2}+9\right )^{\frac {3}{2}}}{27 x^{3}}\)	\(15\)
meijerg	\(-\frac {\left (1-\frac {4 x^{2}}{9}\right )^{\frac {3}{2}}}{x^{3}}\)	\(15\)
trager	\(\frac {\left (4 x^{2}-9\right ) \sqrt {-4 x^{2}+9}}{27 x^{3}}\)	\(22\)
gosper	\(\frac {\left (2 x -3\right ) \left (2 x +3\right ) \sqrt {-4 x^{2}+9}}{27 x^{3}}\)	\(25\)
risch	\(-\frac {16 x^{4}-72 x^{2}+81}{27 x^{3} \sqrt {-4 x^{2}+9}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(-4*x^2+9)^(3/2)/x^3

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Maxima [A]
time = 0.59, size = 14, normalized size = 0.78 \begin {gather*} -\frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{27 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(-4*x^2 + 9)^(3/2)/x^3

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Fricas [A]
time = 1.02, size = 21, normalized size = 1.17 \begin {gather*} \frac {{\left (4 \, x^{2} - 9\right )} \sqrt {-4 \, x^{2} + 9}}{27 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(4*x^2 - 9)*sqrt(-4*x^2 + 9)/x^3

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Sympy [C] Result contains complex when optimal does not.
time = 0.48, size = 76, normalized size = 4.22 \begin {gather*} \begin {cases} \frac {8 \sqrt {-1 + \frac {9}{4 x^{2}}}}{27} - \frac {2 \sqrt {-1 + \frac {9}{4 x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {4}{9} \\\frac {8 i \sqrt {1 - \frac {9}{4 x^{2}}}}{27} - \frac {2 i \sqrt {1 - \frac {9}{4 x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((8*sqrt(-1 + 9/(4*x**2))/27 - 2*sqrt(-1 + 9/(4*x**2))/(3*x**2), 1/Abs(x**2) > 4/9), (8*I*sqrt(1 - 9/
(4*x**2))/27 - 2*I*sqrt(1 - 9/(4*x**2))/(3*x**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (14) = 28\).
time = 0.52, size = 73, normalized size = 4.06 \begin {gather*} -\frac {2 \, x^{3} {\left (\frac {3 \, {\left (\sqrt {-4 \, x^{2} + 9} - 3\right )}^{2}}{x^{2}} - 4\right )}}{27 \, {\left (\sqrt {-4 \, x^{2} + 9} - 3\right )}^{3}} + \frac {\sqrt {-4 \, x^{2} + 9} - 3}{18 \, x} - \frac {{\left (\sqrt {-4 \, x^{2} + 9} - 3\right )}^{3}}{216 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/27*x^3*(3*(sqrt(-4*x^2 + 9) - 3)^2/x^2 - 4)/(sqrt(-4*x^2 + 9) - 3)^3 + 1/18*(sqrt(-4*x^2 + 9) - 3)/x - 1/21
6*(sqrt(-4*x^2 + 9) - 3)^3/x^3

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Mupad [B]
time = 0.03, size = 31, normalized size = 1.72 \begin {gather*} \frac {8\,x^2\,\sqrt {\frac {9}{4}-x^2}-18\,\sqrt {\frac {9}{4}-x^2}}{27\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9 - 4*x^2)^(1/2)/x^4,x)

[Out]

(8*x^2*(9/4 - x^2)^(1/2) - 18*(9/4 - x^2)^(1/2))/(27*x^3)

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