3.574 \(\int \frac{1}{x^4 \sqrt{-9-4 x^2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{\sqrt{-4 x^2-9}}{27 x^3}-\frac{8 \sqrt{-4 x^2-9}}{243 x} \]

[Out]

Sqrt[-9 - 4*x^2]/(27*x^3) - (8*Sqrt[-9 - 4*x^2])/(243*x)

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Rubi [A]  time = 0.0072743, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{\sqrt{-4 x^2-9}}{27 x^3}-\frac{8 \sqrt{-4 x^2-9}}{243 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[-9 - 4*x^2]),x]

[Out]

Sqrt[-9 - 4*x^2]/(27*x^3) - (8*Sqrt[-9 - 4*x^2])/(243*x)

Rule 271

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

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{1}{x^4 \sqrt{-9-4 x^2}} \, dx &=\frac{\sqrt{-9-4 x^2}}{27 x^3}-\frac{8}{27} \int \frac{1}{x^2 \sqrt{-9-4 x^2}} \, dx\\ &=\frac{\sqrt{-9-4 x^2}}{27 x^3}-\frac{8 \sqrt{-9-4 x^2}}{243 x}\\ \end{align*}

Mathematica [A]  time = 0.0055563, size = 25, normalized size = 0.68 \[ \frac{\left (9-8 x^2\right ) \sqrt{-4 x^2-9}}{243 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[-9 - 4*x^2]),x]

[Out]

((9 - 8*x^2)*Sqrt[-9 - 4*x^2])/(243*x^3)

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Maple [A]  time = 0.003, size = 22, normalized size = 0.6 \begin{align*} -{\frac{8\,{x}^{2}-9}{243\,{x}^{3}}\sqrt{-4\,{x}^{2}-9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/243*(8*x^2-9)/x^3*(-4*x^2-9)^(1/2)

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Maxima [A]  time = 3.60023, size = 39, normalized size = 1.05 \begin{align*} -\frac{8 \, \sqrt{-4 \, x^{2} - 9}}{243 \, x} + \frac{\sqrt{-4 \, x^{2} - 9}}{27 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/243*sqrt(-4*x^2 - 9)/x + 1/27*sqrt(-4*x^2 - 9)/x^3

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Fricas [A]  time = 1.26705, size = 55, normalized size = 1.49 \begin{align*} -\frac{{\left (8 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} - 9}}{243 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/243*(8*x^2 - 9)*sqrt(-4*x^2 - 9)/x^3

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Sympy [C]  time = 1.47664, size = 36, normalized size = 0.97 \begin{align*} - \frac{16 i \sqrt{1 + \frac{9}{4 x^{2}}}}{243} + \frac{2 i \sqrt{1 + \frac{9}{4 x^{2}}}}{27 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*I*sqrt(1 + 9/(4*x**2))/243 + 2*I*sqrt(1 + 9/(4*x**2))/(27*x**2)

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Giac [C]  time = 2.5751, size = 109, normalized size = 2.95 \begin{align*} -\frac{2 \, x^{3}{\left (\frac{9 \,{\left (-i \, \sqrt{4 \, x^{2} + 9} - 3 i\right )}^{2}}{x^{2}} + 4\right )}}{243 \,{\left (-i \, \sqrt{4 \, x^{2} + 9} - 3 i\right )}^{3}} - \frac{19683 i \, \sqrt{4 \, x^{2} + 9} + 59049 i}{1062882 \, x} + \frac{{\left (-i \, \sqrt{4 \, x^{2} + 9} - 3 i\right )}^{3}}{1944 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/243*x^3*(9*(-I*sqrt(4*x^2 + 9) - 3*I)^2/x^2 + 4)/(-I*sqrt(4*x^2 + 9) - 3*I)^3 - 1/1062882*(19683*I*sqrt(4*x
^2 + 9) + 59049*I)/x + 1/1944*(-I*sqrt(4*x^2 + 9) - 3*I)^3/x^3