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

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

[Out]

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

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Rubi [A]  time = 0.0165838, 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, 207} \[ \frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right )-\frac{\sqrt{4 x^2+9}}{18 x^2} \]

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

Mathematica [A]  time = 0.010559, size = 37, normalized size = 0.95 \[ \frac{1}{54} \left (4 \tanh ^{-1}\left (\sqrt{\frac{4 x^2}{9}+1}\right )-\frac{3 \sqrt{4 x^2+9}}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[9 + 4*x^2]),x]

[Out]

((-3*Sqrt[9 + 4*x^2])/x^2 + 4*ArcTanh[Sqrt[1 + (4*x^2)/9]])/54

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Maple [A]  time = 0.005, size = 30, normalized size = 0.8 \begin{align*} -{\frac{1}{18\,{x}^{2}}\sqrt{4\,{x}^{2}+9}}+{\frac{2}{27}{\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^3/(4*x^2+9)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.90796, size = 32, normalized size = 0.82 \begin{align*} -\frac{\sqrt{4 \, x^{2} + 9}}{18 \, x^{2}} + \frac{2}{27} \, \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^3/(4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.23201, size = 149, normalized size = 3.82 \begin{align*} \frac{4 \, x^{2} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} + 3\right ) - 4 \, x^{2} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} - 3\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*(4*x^2*log(-2*x + sqrt(4*x^2 + 9) + 3) - 4*x^2*log(-2*x + sqrt(4*x^2 + 9) - 3) - 3*sqrt(4*x^2 + 9))/x^2

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Sympy [A]  time = 2.17704, size = 44, normalized size = 1.13 \begin{align*} \frac{2 \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{27} - \frac{1}{9 x \sqrt{1 + \frac{9}{4 x^{2}}}} - \frac{1}{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*asinh(3/(2*x))/27 - 1/(9*x*sqrt(1 + 9/(4*x**2))) - 1/(4*x**3*sqrt(1 + 9/(4*x**2)))

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Giac [A]  time = 1.72825, size = 58, normalized size = 1.49 \begin{align*} -\frac{\sqrt{4 \, x^{2} + 9}}{18 \, x^{2}} + \frac{1}{27} \, \log \left (\sqrt{4 \, x^{2} + 9} + 3\right ) - \frac{1}{27} \, \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^3/(4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

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