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]

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

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Rubi [A]  time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ 2 \sinh ^{-1}\left (\frac {2 x}{3}\right )-\frac {\sqrt {4 x^2+9}}{x} \]

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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fricas [A]  time = 0.73, size = 35, normalized size = 1.40 \[ -\frac {2 \, x \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) + 2 \, x + \sqrt {4 \, x^{2} + 9}}{x} \]

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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giac [A]  time = 1.15, size = 40, normalized size = 1.60 \[ \frac {36}{{\left (2 \, x - \sqrt {4 \, x^{2} + 9}\right )}^{2} - 9} - 2 \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \]

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))

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maple [A]  time = 0.00, size = 34, normalized size = 1.36 \[ \frac {4 \sqrt {4 x^{2}+9}\, x}{9}+2 \arcsinh \left (\frac {2 x}{3}\right )-\frac {\left (4 x^{2}+9\right )^{\frac {3}{2}}}{9 x} \]

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

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maxima [A]  time = 2.94, size = 21, normalized size = 0.84 \[ -\frac {\sqrt {4 \, x^{2} + 9}}{x} + 2 \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]

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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mupad [B]  time = 0.03, size = 19, normalized size = 0.76 \[ 2\,\mathrm {asinh}\left (\frac {2\,x}{3}\right )-\frac {2\,\sqrt {x^2+\frac {9}{4}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 + 9)^(1/2)/x^2,x)

[Out]

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

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sympy [A]  time = 0.25, size = 19, normalized size = 0.76 \[ 2 \operatorname {asinh}{\left (\frac {2 x}{3} \right )} - \frac {\sqrt {4 x^{2} + 9}}{x} \]

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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