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 = 0.0037743, 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, Rules used = {195, 215} \[ \frac{1}{2} \sqrt{4 x^2+9} x+\frac{9}{4} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

Mathematica [A]  time = 0.0060767, 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 = 0.002, size = 20, normalized size = 0.7 \begin{align*}{\frac{9}{4}{\it Arcsinh} \left ({\frac{2\,x}{3}} \right ) }+{\frac{x}{2}\sqrt{4\,{x}^{2}+9}} \end{align*}

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 = 2.2904, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{2} \, \sqrt{4 \, x^{2} + 9} x + \frac{9}{4} \, \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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.44501, size = 77, normalized size = 2.85 \begin{align*} \frac{1}{2} \, \sqrt{4 \, x^{2} + 9} x - \frac{9}{4} \, \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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.193213, size = 22, normalized size = 0.81 \begin{align*} \frac{x \sqrt{4 x^{2} + 9}}{2} + \frac{9 \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{4} \end{align*}

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 [A]  time = 1.33756, size = 39, normalized size = 1.44 \begin{align*} \frac{1}{2} \, \sqrt{4 \, x^{2} + 9} x - \frac{9}{4} \, \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, algorithm="giac")

[Out]

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