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

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

[Out]

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

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Rubi [A]  time = 0.0229079, 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, 207} \[ \frac{\sqrt{4 x^2+9}}{54 x^2}-\frac{\sqrt{4 x^2+9}}{36 x^4}-\frac{2}{81} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right ) \]

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*ArcTanh[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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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}{x^3 \sqrt{9+4 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{9+4 x^2}}{36 x^4}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{9+4 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}{27} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{9+4 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} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9+4 x^2}\right )\\ \end{align*}

Mathematica [C]  time = 0.004676, 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 = 0.003, 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}{\it Artanh} \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*arctanh(3/(4*x^2+9)^(1/2))

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Maxima [A]  time = 2.97595, size = 51, normalized size = 0.89 \begin{align*} \frac{\sqrt{4 \, x^{2} + 9}}{54 \, x^{2}} - \frac{\sqrt{4 \, x^{2} + 9}}{36 \, x^{4}} - \frac{2}{81} \, \operatorname{arsinh}\left (\frac{3}{2 \,{\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*arcsinh(3/2/abs(x))

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Fricas [A]  time = 1.36904, size = 167, normalized size = 2.93 \begin{align*} -\frac{8 \, x^{4} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} + 3\right ) - 8 \, x^{4} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} - 3\right ) - 3 \, \sqrt{4 \, x^{2} + 9}{\left (2 \, x^{2} - 3\right )}}{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*(8*x^4*log(-2*x + sqrt(4*x^2 + 9) + 3) - 8*x^4*log(-2*x + sqrt(4*x^2 + 9) - 3) - 3*sqrt(4*x^2 + 9)*(2*x
^2 - 3))/x^4

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Sympy [A]  time = 4.45956, size = 63, normalized size = 1.11 \begin{align*} - \frac{2 \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{81} + \frac{1}{27 x \sqrt{1 + \frac{9}{4 x^{2}}}} + \frac{1}{36 x^{3} \sqrt{1 + \frac{9}{4 x^{2}}}} - \frac{1}{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*asinh(3/(2*x))/81 + 1/(27*x*sqrt(1 + 9/(4*x**2))) + 1/(36*x**3*sqrt(1 + 9/(4*x**2))) - 1/(8*x**5*sqrt(1 + 9
/(4*x**2)))

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Giac [A]  time = 3.02201, size = 74, normalized size = 1.3 \begin{align*} \frac{{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} - 15 \, \sqrt{4 \, x^{2} + 9}}{216 \, x^{4}} - \frac{1}{81} \, \log \left (\sqrt{4 \, x^{2} + 9} + 3\right ) + \frac{1}{81} \, \log \left (\sqrt{4 \, x^{2} + 9} - 3\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*((4*x^2 + 9)^(3/2) - 15*sqrt(4*x^2 + 9))/x^4 - 1/81*log(sqrt(4*x^2 + 9) + 3) + 1/81*log(sqrt(4*x^2 + 9)
- 3)