3.573 \(\int \frac{1}{x^3 \sqrt{-9-4 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{\sqrt{-4 x^2-9}}{18 x^2}-\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]

[Out]

Sqrt[-9 - 4*x^2]/(18*x^2) - (2*ArcTan[Sqrt[-9 - 4*x^2]/3])/27

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Rubi [A]  time = 0.0160702, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 204} \[ \frac{\sqrt{-4 x^2-9}}{18 x^2}-\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[-9 - 4*x^2]),x]

[Out]

Sqrt[-9 - 4*x^2]/(18*x^2) - (2*ArcTan[Sqrt[-9 - 4*x^2]/3])/27

Rule 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^3 \sqrt{-9-4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x^2} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9-4 x^2}}{18 x^2}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9-4 x^2}}{18 x^2}+\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{-\frac{9}{4}-\frac{x^2}{4}} \, dx,x,\sqrt{-9-4 x^2}\right )\\ &=\frac{\sqrt{-9-4 x^2}}{18 x^2}-\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{-9-4 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0215214, size = 54, normalized size = 1.38 \[ -\frac{4}{81} \sqrt{-4 x^2-9} \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{4 x^2}{9}+1}\right )}{2 \sqrt{\frac{4 x^2}{9}+1}}-\frac{9}{8 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[-9 - 4*x^2]),x]

[Out]

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

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Maple [A]  time = 0.004, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{18\,{x}^{2}}\sqrt{-4\,{x}^{2}-9}}+{\frac{2}{27}\arctan \left ( 3\,{\frac{1}{\sqrt{-4\,{x}^{2}-9}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(-4*x^2-9)^(1/2),x)

[Out]

1/18*(-4*x^2-9)^(1/2)/x^2+2/27*arctan(3/(-4*x^2-9)^(1/2))

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Maxima [C]  time = 2.95667, size = 54, normalized size = 1.38 \begin{align*} \frac{\sqrt{-4 \, x^{2} - 9}}{18 \, x^{2}} + \frac{2}{27} i \, \log \left (\frac{6 \, \sqrt{4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

1/18*sqrt(-4*x^2 - 9)/x^2 + 2/27*I*log(6*sqrt(4*x^2 + 9)/abs(x) + 18/abs(x))

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Fricas [C]  time = 1.19176, size = 174, normalized size = 4.46 \begin{align*} \frac{-2 i \, x^{2} \log \left (-\frac{4 \,{\left (i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{27 \, x}\right ) + 2 i \, x^{2} \log \left (-\frac{4 \,{\left (-i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{27 \, x}\right ) + 3 \, \sqrt{-4 \, x^{2} - 9}}{54 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

1/54*(-2*I*x^2*log(-4/27*(I*sqrt(-4*x^2 - 9) - 3)/x) + 2*I*x^2*log(-4/27*(-I*sqrt(-4*x^2 - 9) - 3)/x) + 3*sqrt
(-4*x^2 - 9))/x^2

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Sympy [C]  time = 2.20793, size = 46, normalized size = 1.18 \begin{align*} - \frac{2 i \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{27} + \frac{i}{9 x \sqrt{1 + \frac{9}{4 x^{2}}}} + \frac{i}{4 x^{3} \sqrt{1 + \frac{9}{4 x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(-4*x**2-9)**(1/2),x)

[Out]

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

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Giac [C]  time = 2.67715, size = 39, normalized size = 1. \begin{align*} \frac{i \, \sqrt{4 \, x^{2} + 9}}{18 \, x^{2}} - \frac{2}{27} \, \arctan \left (\frac{1}{3} i \, \sqrt{4 \, x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-4*x^2-9)^(1/2),x, algorithm="giac")

[Out]

1/18*I*sqrt(4*x^2 + 9)/x^2 - 2/27*arctan(1/3*I*sqrt(4*x^2 + 9))