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]

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 = {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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {\left (-4 x^2-9\right )^{3/2}}{27 x^3} \]

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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fricas [A]  time = 0.89, size = 14, normalized size = 0.78 \[ \frac {{\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{27 \, x^{3}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-4 \, x^{2} - 9}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-4*x^2 - 9)/x^4, x)

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \[ \frac {\left (-4 x^{2}-9\right )^{\frac {3}{2}}}{27 x^{3}} \]

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 = 2.89, size = 14, normalized size = 0.78 \[ \frac {{\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{27 \, x^{3}} \]

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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mupad [B]  time = 4.74, size = 31, normalized size = 1.72 \[ -\frac {4\,x^2\,\sqrt {-4\,x^2-9}+9\,\sqrt {-4\,x^2-9}}{27\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- 4*x^2 - 9)^(1/2)/x^4,x)

[Out]

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

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sympy [C]  time = 1.00, size = 37, normalized size = 2.06 \[ - \frac {8 i \sqrt {1 + \frac {9}{4 x^{2}}}}{27} - \frac {2 i \sqrt {1 + \frac {9}{4 x^{2}}}}{3 x^{2}} \]

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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