3.575 \(\int \frac{1}{x^5 \sqrt{-9-4 x^2}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{\sqrt{-4 x^2-9}}{54 x^2}+\frac{\sqrt{-4 x^2-9}}{36 x^4}+\frac{2}{81} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]

[Out]

Sqrt[-9 - 4*x^2]/(36*x^4) - Sqrt[-9 - 4*x^2]/(54*x^2) + (2*ArcTan[Sqrt[-9 - 4*x^2]/3])/81

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Rubi [A]  time = 0.0232741, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 204} \[ -\frac{\sqrt{-4 x^2-9}}{54 x^2}+\frac{\sqrt{-4 x^2-9}}{36 x^4}+\frac{2}{81} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*Sqrt[-9 - 4*x^2]),x]

[Out]

Sqrt[-9 - 4*x^2]/(36*x^4) - Sqrt[-9 - 4*x^2]/(54*x^2) + (2*ArcTan[Sqrt[-9 - 4*x^2]/3])/81

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^5 \sqrt{-9-4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x^3} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9-4 x^2}}{36 x^4}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x^2} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9-4 x^2}}{36 x^4}-\frac{\sqrt{-9-4 x^2}}{54 x^2}+\frac{1}{27} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9-4 x^2}}{36 x^4}-\frac{\sqrt{-9-4 x^2}}{54 x^2}-\frac{1}{54} \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}}{36 x^4}-\frac{\sqrt{-9-4 x^2}}{54 x^2}+\frac{2}{81} \tan ^{-1}\left (\frac{1}{3} \sqrt{-9-4 x^2}\right )\\ \end{align*}

Mathematica [C]  time = 0.0047161, size = 32, normalized size = 0.56 \[ \frac{16}{729} \sqrt{-4 x^2-9} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{4 x^2}{9}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*Sqrt[-9 - 4*x^2]),x]

[Out]

(16*Sqrt[-9 - 4*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (4*x^2)/9])/729

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Maple [A]  time = 0.004, size = 44, normalized size = 0.8 \begin{align*}{\frac{1}{36\,{x}^{4}}\sqrt{-4\,{x}^{2}-9}}-{\frac{1}{54\,{x}^{2}}\sqrt{-4\,{x}^{2}-9}}-{\frac{2}{81}\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^5/(-4*x^2-9)^(1/2),x)

[Out]

1/36*(-4*x^2-9)^(1/2)/x^4-1/54*(-4*x^2-9)^(1/2)/x^2-2/81*arctan(3/(-4*x^2-9)^(1/2))

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Maxima [C]  time = 3.63359, size = 73, normalized size = 1.28 \begin{align*} -\frac{\sqrt{-4 \, x^{2} - 9}}{54 \, x^{2}} + \frac{\sqrt{-4 \, x^{2} - 9}}{36 \, x^{4}} - \frac{2}{81} 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^5/(-4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

-1/54*sqrt(-4*x^2 - 9)/x^2 + 1/36*sqrt(-4*x^2 - 9)/x^4 - 2/81*I*log(6*sqrt(4*x^2 + 9)/abs(x) + 18/abs(x))

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Fricas [C]  time = 1.32893, size = 192, normalized size = 3.37 \begin{align*} \frac{-4 i \, x^{4} \log \left (-\frac{4 \,{\left (i \, \sqrt{-4 \, x^{2} - 9} + 3\right )}}{81 \, x}\right ) + 4 i \, x^{4} \log \left (-\frac{4 \,{\left (-i \, \sqrt{-4 \, x^{2} - 9} + 3\right )}}{81 \, x}\right ) - 3 \,{\left (2 \, x^{2} - 3\right )} \sqrt{-4 \, x^{2} - 9}}{324 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

1/324*(-4*I*x^4*log(-4/81*(I*sqrt(-4*x^2 - 9) + 3)/x) + 4*I*x^4*log(-4/81*(-I*sqrt(-4*x^2 - 9) + 3)/x) - 3*(2*
x^2 - 3)*sqrt(-4*x^2 - 9))/x^4

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Sympy [C]  time = 4.46695, size = 65, normalized size = 1.14 \begin{align*} \frac{2 i \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{81} - \frac{i}{27 x \sqrt{1 + \frac{9}{4 x^{2}}}} - \frac{i}{36 x^{3} \sqrt{1 + \frac{9}{4 x^{2}}}} + \frac{i}{8 x^{5} \sqrt{1 + \frac{9}{4 x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(-4*x**2-9)**(1/2),x)

[Out]

2*I*asinh(3/(2*x))/81 - I/(27*x*sqrt(1 + 9/(4*x**2))) - I/(36*x**3*sqrt(1 + 9/(4*x**2))) + I/(8*x**5*sqrt(1 +
9/(4*x**2)))

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Giac [C]  time = 2.87814, size = 58, normalized size = 1.02 \begin{align*} -\frac{i \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} - 15 i \, \sqrt{4 \, x^{2} + 9}}{216 \, x^{4}} + \frac{2}{81} \, \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^5/(-4*x^2-9)^(1/2),x, algorithm="giac")

[Out]

-1/216*(I*(4*x^2 + 9)^(3/2) - 15*I*sqrt(4*x^2 + 9))/x^4 + 2/81*arctan(1/3*I*sqrt(4*x^2 + 9))