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

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

[Out]

-(9 - 4*x^2)^(3/2)/(27*x^3)

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Rubi [A]  time = 0.0033959, antiderivative size = 18, normalized size of antiderivative = 1., 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 = {264} \[ -\frac{\left (9-4 x^2\right )^{3/2}}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[9 - 4*x^2]/x^4,x]

[Out]

-(9 - 4*x^2)^(3/2)/(27*x^3)

Rule 264

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*}

Mathematica [A]  time = 0.0029203, size = 18, normalized size = 1. \[ -\frac{\left (9-4 x^2\right )^{3/2}}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[9 - 4*x^2]/x^4,x]

[Out]

-(9 - 4*x^2)^(3/2)/(27*x^3)

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Maple [A]  time = 0.003, size = 25, normalized size = 1.4 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) }{27\,{x}^{3}}\sqrt{-4\,{x}^{2}+9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 3.65625, size = 19, normalized size = 1.06 \begin{align*} -\frac{{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}}}{27 \, x^{3}} \end{align*}

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 = 1.45615, size = 53, normalized size = 2.94 \begin{align*} \frac{{\left (4 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} + 9}}{27 \, x^{3}} \end{align*}

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 [B]  time = 1.03708, size = 76, normalized size = 4.22 \begin{align*} \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{9}{4 \left |{x^{2}}\right |} > 1 \\\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{align*}

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), 9/(4*Abs(x**2)) > 1), (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]  time = 2.00095, size = 99, normalized size = 5.5 \begin{align*} -\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{align*}

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