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]

_______________________________________________________________________________________

Rubi [A]  time = 0.0219201, 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 \[ 2 \sinh ^{-1}\left (\frac{2 x}{3}\right )-\frac{\sqrt{4 x^2+9}}{x} \]

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]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.29761, 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]  rubi_integrate((4*x**2+9)**(1/2)/x**2,x)

[Out]

2*asinh(2*x/3) - sqrt(4*x**2 + 9)/x

_______________________________________________________________________________________

Mathematica [A]  time = 0.0136399, 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]

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 34, normalized size = 1.4 \[ -{\frac{1}{9\,x} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}+{\frac{4\,x}{9}\sqrt{4\,{x}^{2}+9}}+2\,{\it Arcsinh} \left ( 2/3\,x \right ) \]

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)

_______________________________________________________________________________________

Maxima [A]  time = 1.49106, size = 28, normalized size = 1.12 \[ -\frac{\sqrt{4 \, x^{2} + 9}}{x} + 2 \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 + 9)/x^2,x, algorithm="maxima")

[Out]

-sqrt(4*x^2 + 9)/x + 2*arcsinh(2/3*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.231843, size = 78, normalized size = 3.12 \[ -\frac{2 \,{\left (2 \, x^{2} - \sqrt{4 \, x^{2} + 9} x\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) - 9}{2 \, x^{2} - \sqrt{4 \, x^{2} + 9} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 + 9)/x^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.662192, 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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205073, size = 54, normalized size = 2.16 \[ \frac{36}{{\left (2 \, x - \sqrt{4 \, x^{2} + 9}\right )}^{2} - 9} - 2 \,{\rm ln}\left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 + 9)/x^2,x, algorithm="giac")

[Out]

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