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

Optimal. Leaf size=25 \[ 2 \sinh ^{-1}\left (\frac{2 x}{3}\right )-\frac{\sqrt{4 x^2+9}}{x} \]

[Out]

-(Sqrt[9 + 4*x^2]/x) + 2*ArcSinh[(2*x)/3]

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Rubi [A]  time = 0.0048838, antiderivative size = 25, 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 = {277, 215} \[ 2 \sinh ^{-1}\left (\frac{2 x}{3}\right )-\frac{\sqrt{4 x^2+9}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[9 + 4*x^2]/x^2,x]

[Out]

-(Sqrt[9 + 4*x^2]/x) + 2*ArcSinh[(2*x)/3]

Rule 277

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{\sqrt{9+4 x^2}}{x^2} \, dx &=-\frac{\sqrt{9+4 x^2}}{x}+4 \int \frac{1}{\sqrt{9+4 x^2}} \, dx\\ &=-\frac{\sqrt{9+4 x^2}}{x}+2 \sinh ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0057339, size = 25, normalized size = 1. \[ 2 \sinh ^{-1}\left (\frac{2 x}{3}\right )-\frac{\sqrt{4 x^2+9}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[9 + 4*x^2]/x^2,x]

[Out]

-(Sqrt[9 + 4*x^2]/x) + 2*ArcSinh[(2*x)/3]

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Maple [A]  time = 0.002, size = 34, normalized size = 1.4 \begin{align*} -{\frac{1}{9\,x} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}+{\frac{4\,x}{9}\sqrt{4\,{x}^{2}+9}}+2\,{\it Arcsinh} \left ( 2/3\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 3.04903, size = 28, normalized size = 1.12 \begin{align*} -\frac{\sqrt{4 \, x^{2} + 9}}{x} + 2 \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(4*x^2 + 9)/x + 2*arcsinh(2/3*x)

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Fricas [A]  time = 1.45516, size = 84, normalized size = 3.36 \begin{align*} -\frac{2 \, x \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) + 2 \, x + \sqrt{4 \, x^{2} + 9}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.23143, size = 19, normalized size = 0.76 \begin{align*} 2 \operatorname{asinh}{\left (\frac{2 x}{3} \right )} - \frac{\sqrt{4 x^{2} + 9}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*asinh(2*x/3) - sqrt(4*x**2 + 9)/x

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Giac [A]  time = 1.96634, size = 54, normalized size = 2.16 \begin{align*} \frac{36}{{\left (2 \, x - \sqrt{4 \, x^{2} + 9}\right )}^{2} - 9} - 2 \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

36/((2*x - sqrt(4*x^2 + 9))^2 - 9) - 2*log(-2*x + sqrt(4*x^2 + 9))