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

Optimal. Leaf size=18 \[ -\frac{\sqrt{4 x^2+9}}{9 x} \]

[Out]

-Sqrt[9 + 4*x^2]/(9*x)

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Rubi [A]  time = 0.003062, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\sqrt{4 x^2+9}}{9 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[9 + 4*x^2]),x]

[Out]

-Sqrt[9 + 4*x^2]/(9*x)

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^2 \sqrt{9+4 x^2}} \, dx &=-\frac{\sqrt{9+4 x^2}}{9 x}\\ \end{align*}

Mathematica [A]  time = 0.0023532, size = 18, normalized size = 1. \[ -\frac{\sqrt{4 x^2+9}}{9 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[9 + 4*x^2]),x]

[Out]

-Sqrt[9 + 4*x^2]/(9*x)

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{9\,x}\sqrt{4\,{x}^{2}+9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(4*x^2+9)^(1/2)/x

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Maxima [A]  time = 2.12417, size = 19, normalized size = 1.06 \begin{align*} -\frac{\sqrt{4 \, x^{2} + 9}}{9 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(4*x^2 + 9)/x

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Fricas [A]  time = 1.21919, size = 43, normalized size = 2.39 \begin{align*} -\frac{2 \, x + \sqrt{4 \, x^{2} + 9}}{9 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(2*x + sqrt(4*x^2 + 9))/x

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Sympy [A]  time = 0.706813, size = 15, normalized size = 0.83 \begin{align*} - \frac{2 \sqrt{1 + \frac{9}{4 x^{2}}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(4*x**2+9)**(1/2),x)

[Out]

-2*sqrt(1 + 9/(4*x**2))/9

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Giac [A]  time = 2.26969, size = 31, normalized size = 1.72 \begin{align*} \frac{4}{{\left (2 \, x - \sqrt{4 \, x^{2} + 9}\right )}^{2} - 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/((2*x - sqrt(4*x^2 + 9))^2 - 9)