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

Optimal. Leaf size=37 \[ \frac{8 \sqrt{4 x^2+9}}{243 x}-\frac{\sqrt{4 x^2+9}}{27 x^3} \]

[Out]

-Sqrt[9 + 4*x^2]/(27*x^3) + (8*Sqrt[9 + 4*x^2])/(243*x)

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Rubi [A]  time = 0.0071159, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{8 \sqrt{4 x^2+9}}{243 x}-\frac{\sqrt{4 x^2+9}}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[9 + 4*x^2]),x]

[Out]

-Sqrt[9 + 4*x^2]/(27*x^3) + (8*Sqrt[9 + 4*x^2])/(243*x)

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

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^4 \sqrt{9+4 x^2}} \, dx &=-\frac{\sqrt{9+4 x^2}}{27 x^3}-\frac{8}{27} \int \frac{1}{x^2 \sqrt{9+4 x^2}} \, dx\\ &=-\frac{\sqrt{9+4 x^2}}{27 x^3}+\frac{8 \sqrt{9+4 x^2}}{243 x}\\ \end{align*}

Mathematica [A]  time = 0.0041132, size = 27, normalized size = 0.73 \[ -\frac{\left (9-8 x^2\right ) \sqrt{\frac{4 x^2}{9}+1}}{81 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[9 + 4*x^2]),x]

[Out]

-((9 - 8*x^2)*Sqrt[1 + (4*x^2)/9])/(81*x^3)

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Maple [A]  time = 0.002, size = 22, normalized size = 0.6 \begin{align*}{\frac{8\,{x}^{2}-9}{243\,{x}^{3}}\sqrt{4\,{x}^{2}+9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/243*(4*x^2+9)^(1/2)*(8*x^2-9)/x^3

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Maxima [A]  time = 3.56123, size = 39, normalized size = 1.05 \begin{align*} \frac{8 \, \sqrt{4 \, x^{2} + 9}}{243 \, x} - \frac{\sqrt{4 \, x^{2} + 9}}{27 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/243*sqrt(4*x^2 + 9)/x - 1/27*sqrt(4*x^2 + 9)/x^3

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Fricas [A]  time = 1.25628, size = 68, normalized size = 1.84 \begin{align*} \frac{16 \, x^{3} +{\left (8 \, x^{2} - 9\right )} \sqrt{4 \, x^{2} + 9}}{243 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/243*(16*x^3 + (8*x^2 - 9)*sqrt(4*x^2 + 9))/x^3

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Sympy [A]  time = 1.45535, size = 32, normalized size = 0.86 \begin{align*} \frac{16 \sqrt{1 + \frac{9}{4 x^{2}}}}{243} - \frac{2 \sqrt{1 + \frac{9}{4 x^{2}}}}{27 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16*sqrt(1 + 9/(4*x**2))/243 - 2*sqrt(1 + 9/(4*x**2))/(27*x**2)

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Giac [A]  time = 1.93105, size = 57, normalized size = 1.54 \begin{align*} \frac{32 \,{\left ({\left (2 \, x - \sqrt{4 \, x^{2} + 9}\right )}^{2} - 3\right )}}{{\left ({\left (2 \, x - \sqrt{4 \, x^{2} + 9}\right )}^{2} - 9\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32*((2*x - sqrt(4*x^2 + 9))^2 - 3)/((2*x - sqrt(4*x^2 + 9))^2 - 9)^3