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

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

[Out]

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

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Rubi [A]  time = 0.0073577, 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} \[ \frac{8 \sqrt{4 x^2-9}}{243 x}+\frac{\sqrt{4 x^2-9}}{27 x^3} \]

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 = 0.0038396, size = 25, normalized size = 0.68 \[ \frac{\sqrt{4 x^2-9} \left (8 x^2+9\right )}{243 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[-9 + 4*x^2]),x]

[Out]

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

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Maple [A]  time = 0.003, 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 = 3.32919, 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 = 1.2183, size = 68, normalized size = 1.84 \begin{align*} \frac{16 \, x^{3} +{\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*(16*x^3 + (8*x^2 + 9)*sqrt(4*x^2 - 9))/x^3

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Sympy [A]  time = 1.5154, size = 76, normalized size = 2.05 \begin{align*} \begin{cases} \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{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\\frac{16 \sqrt{1 - \frac{9}{4 x^{2}}}}{243} + \frac{2 \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*I*sqrt(-1 + 9/(4*x**2))/243 + 2*I*sqrt(-1 + 9/(4*x**2))/(27*x**2), 9/(4*Abs(x**2)) > 1), (16*sqr
t(1 - 9/(4*x**2))/243 + 2*sqrt(1 - 9/(4*x**2))/(27*x**2), True))

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Giac [A]  time = 2.17388, size = 57, normalized size = 1.54 \begin{align*} \frac{32 \,{\left ({\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2} + 3\right )}}{{\left ({\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2} + 9\right )}^{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]

32*((2*x - sqrt(4*x^2 - 9))^2 + 3)/((2*x - sqrt(4*x^2 - 9))^2 + 9)^3