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

Optimal. Leaf size=15 \[ \frac{1}{12} \left (4 x^2+9\right )^{3/2} \]

[Out]

(9 + 4*x^2)^(3/2)/12

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Rubi [A]  time = 0.0021563, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ \frac{1}{12} \left (4 x^2+9\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[9 + 4*x^2],x]

[Out]

(9 + 4*x^2)^(3/2)/12

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x \sqrt{9+4 x^2} \, dx &=\frac{1}{12} \left (9+4 x^2\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0018369, size = 15, normalized size = 1. \[ \frac{1}{12} \left (4 x^2+9\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[9 + 4*x^2],x]

[Out]

(9 + 4*x^2)^(3/2)/12

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Maple [A]  time = 0.001, size = 12, normalized size = 0.8 \begin{align*}{\frac{1}{12} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*x^2+9)^(3/2)

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Maxima [A]  time = 1.22952, size = 15, normalized size = 1. \begin{align*} \frac{1}{12} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*x^2 + 9)^(3/2)

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Fricas [A]  time = 1.49535, size = 32, normalized size = 2.13 \begin{align*} \frac{1}{12} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*x^2 + 9)^(3/2)

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Sympy [B]  time = 0.184337, size = 27, normalized size = 1.8 \begin{align*} \frac{x^{2} \sqrt{4 x^{2} + 9}}{3} + \frac{3 \sqrt{4 x^{2} + 9}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.55571, size = 15, normalized size = 1. \begin{align*} \frac{1}{12} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*x^2 + 9)^(3/2)