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

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

[Out]

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

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Rubi [A]  time = 0.0070717, 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} \[ -\frac{8 \sqrt{9-4 x^2}}{243 x}-\frac{\sqrt{9-4 x^2}}{27 x^3} \]

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 = 0.003646, size = 27, normalized size = 0.73 \[ -\frac{\sqrt{1-\frac{4 x^2}{9}} \left (8 x^2+9\right )}{81 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[9 - 4*x^2]),x]

[Out]

-(Sqrt[1 - (4*x^2)/9]*(9 + 8*x^2))/(81*x^3)

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Maple [A]  time = 0.002, size = 32, normalized size = 0.9 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) \left ( 8\,{x}^{2}+9 \right ) }{243\,{x}^{3}}{\frac{1}{\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*(-3+2*x)*(3+2*x)*(8*x^2+9)/x^3/(-4*x^2+9)^(1/2)

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Maxima [A]  time = 3.45287, 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 = 1.21108, 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 [A]  time = 1.49926, size = 80, normalized size = 2.16 \begin{align*} \begin{cases} - \frac{16 \sqrt{-1 + \frac{9}{4 x^{2}}}}{243} - \frac{2 \sqrt{-1 + \frac{9}{4 x^{2}}}}{27 x^{2}} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\- \frac{16 i \sqrt{1 - \frac{9}{4 x^{2}}}}{243} - \frac{2 i \sqrt{1 - \frac{9}{4 x^{2}}}}{27 x^{2}} & \text{otherwise} \end{cases} \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]

Piecewise((-16*sqrt(-1 + 9/(4*x**2))/243 - 2*sqrt(-1 + 9/(4*x**2))/(27*x**2), 9/(4*Abs(x**2)) > 1), (-16*I*sqr
t(1 - 9/(4*x**2))/243 - 2*I*sqrt(1 - 9/(4*x**2))/(27*x**2), True))

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Giac [B]  time = 2.57631, size = 99, normalized size = 2.68 \begin{align*} \frac{2 \, x^{3}{\left (\frac{9 \,{\left (\sqrt{-4 \, x^{2} + 9} - 3\right )}^{2}}{x^{2}} + 4\right )}}{243 \,{\left (\sqrt{-4 \, x^{2} + 9} - 3\right )}^{3}} - \frac{\sqrt{-4 \, x^{2} + 9} - 3}{54 \, x} - \frac{{\left (\sqrt{-4 \, x^{2} + 9} - 3\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*(sqrt(-4*x^2 + 9) - 3)^2/x^2 + 4)/(sqrt(-4*x^2 + 9) - 3)^3 - 1/54*(sqrt(-4*x^2 + 9) - 3)/x - 1/19
44*(sqrt(-4*x^2 + 9) - 3)^3/x^3