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]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {1}{12} \left (4 x^2+9\right )^{3/2} \]

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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fricas [A]  time = 0.94, size = 11, normalized size = 0.73 \[ \frac {1}{12} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} \]

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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giac [A]  time = 1.04, size = 11, normalized size = 0.73 \[ \frac {1}{12} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} \]

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)

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maple [A]  time = 0.00, size = 12, normalized size = 0.80 \[ \frac {\left (4 x^{2}+9\right )^{\frac {3}{2}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 11, normalized size = 0.73 \[ \frac {1}{12} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} \]

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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mupad [B]  time = 0.02, size = 16, normalized size = 1.07 \[ \frac {\sqrt {x^2+\frac {9}{4}}\,\left (\frac {4\,x^2}{3}+3\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(4*x^2 + 9)^(1/2),x)

[Out]

((x^2 + 9/4)^(1/2)*((4*x^2)/3 + 3))/2

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sympy [B]  time = 0.20, size = 27, normalized size = 1.80 \[ \frac {x^{2} \sqrt {4 x^{2} + 9}}{3} + \frac {3 \sqrt {4 x^{2} + 9}}{4} \]

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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