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

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

[Out]

(-9 - 4*x^2)^(3/2)/(27*x^3)

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Rubi [A]  time = 0.003333, 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} \[ \frac{\left (-4 x^2-9\right )^{3/2}}{27 x^3} \]

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 = 0.0024752, size = 18, normalized size = 1. \[ \frac{\left (-4 x^2-9\right )^{3/2}}{27 x^3} \]

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 = 15, normalized size = 0.8 \begin{align*}{\frac{1}{27\,{x}^{3}} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}} \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*(-4*x^2-9)^(3/2)/x^3

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Maxima [A]  time = 3.31359, 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 = 1.23472, size = 39, normalized size = 2.17 \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="fricas")

[Out]

1/27*(-4*x^2 - 9)^(3/2)/x^3

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Sympy [C]  time = 1.0267, size = 37, normalized size = 2.06 \begin{align*} - \frac{8 i \sqrt{1 + \frac{9}{4 x^{2}}}}{27} - \frac{2 i \sqrt{1 + \frac{9}{4 x^{2}}}}{3 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*I*sqrt(1 + 9/(4*x**2))/27 - 2*I*sqrt(1 + 9/(4*x**2))/(3*x**2)

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Giac [C]  time = 1.89349, size = 109, normalized size = 6.06 \begin{align*} -\frac{2 \, x^{3}{\left (\frac{3 \,{\left (-i \, \sqrt{4 \, x^{2} + 9} - 3 i\right )}^{2}}{x^{2}} - 4\right )}}{27 \,{\left (-i \, \sqrt{4 \, x^{2} + 9} - 3 i\right )}^{3}} - \frac{81 i \, \sqrt{4 \, x^{2} + 9} + 243 i}{1458 \, x} - \frac{{\left (-i \, \sqrt{4 \, x^{2} + 9} - 3 i\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*(-I*sqrt(4*x^2 + 9) - 3*I)^2/x^2 - 4)/(-I*sqrt(4*x^2 + 9) - 3*I)^3 - 1/1458*(81*I*sqrt(4*x^2 + 9)
+ 243*I)/x - 1/216*(-I*sqrt(4*x^2 + 9) - 3*I)^3/x^3