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

Optimal. Leaf size=27 \[ \frac{1}{2} \sqrt{4 x^2+9} x+\frac{9}{4} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0136354, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2} \sqrt{4 x^2+9} x+\frac{9}{4} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[9 + 4*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.27449, size = 22, normalized size = 0.81 \[ \frac{x \sqrt{4 x^{2} + 9}}{2} + \frac{9 \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**2+9)**(1/2),x)

[Out]

x*sqrt(4*x**2 + 9)/2 + 9*asinh(2*x/3)/4

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Mathematica [A]  time = 0.0108145, size = 27, normalized size = 1. \[ \frac{1}{2} \sqrt{4 x^2+9} x+\frac{9}{4} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[9 + 4*x^2],x]

[Out]

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

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Maple [A]  time = 0.003, size = 20, normalized size = 0.7 \[{\frac{9}{4}{\it Arcsinh} \left ({\frac{2\,x}{3}} \right ) }+{\frac{x}{2}\sqrt{4\,{x}^{2}+9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^2+9)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.49607, size = 26, normalized size = 0.96 \[ \frac{1}{2} \, \sqrt{4 \, x^{2} + 9} x + \frac{9}{4} \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225448, size = 120, normalized size = 4.44 \[ -\frac{32 \, x^{4} + 72 \, x^{2} + 9 \,{\left (8 \, x^{2} - 4 \, \sqrt{4 \, x^{2} + 9} x + 9\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) - 2 \,{\left (8 \, x^{3} + 9 \, x\right )} \sqrt{4 \, x^{2} + 9}}{4 \,{\left (8 \, x^{2} - 4 \, \sqrt{4 \, x^{2} + 9} x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(32*x^4 + 72*x^2 + 9*(8*x^2 - 4*sqrt(4*x^2 + 9)*x + 9)*log(-2*x + sqrt(4*x^
2 + 9)) - 2*(8*x^3 + 9*x)*sqrt(4*x^2 + 9))/(8*x^2 - 4*sqrt(4*x^2 + 9)*x + 9)

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Sympy [A]  time = 0.485953, size = 22, normalized size = 0.81 \[ \frac{x \sqrt{4 x^{2} + 9}}{2} + \frac{9 \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**2+9)**(1/2),x)

[Out]

x*sqrt(4*x**2 + 9)/2 + 9*asinh(2*x/3)/4

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GIAC/XCAS [A]  time = 0.204013, size = 39, normalized size = 1.44 \[ \frac{1}{2} \, \sqrt{4 \, x^{2} + 9} x - \frac{9}{4} \,{\rm ln}\left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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