3.65 \(\int \sqrt{4+12 x+9 x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{6} (3 x+2) \sqrt{9 x^2+12 x+4} \]

[Out]

((2 + 3*x)*Sqrt[4 + 12*x + 9*x^2])/6

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Rubi [A]  time = 0.00965837, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{1}{6} (3 x+2) \sqrt{9 x^2+12 x+4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[4 + 12*x + 9*x^2],x]

[Out]

((2 + 3*x)*Sqrt[4 + 12*x + 9*x^2])/6

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Rubi in Sympy [A]  time = 1.24647, size = 19, normalized size = 0.83 \[ \frac{\left (18 x + 12\right ) \sqrt{9 x^{2} + 12 x + 4}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((9*x**2+12*x+4)**(1/2),x)

[Out]

(18*x + 12)*sqrt(9*x**2 + 12*x + 4)/36

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Mathematica [A]  time = 0.00949838, size = 25, normalized size = 1.09 \[ \frac{x \sqrt{(3 x+2)^2} (3 x+4)}{6 x+4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[4 + 12*x + 9*x^2],x]

[Out]

(x*Sqrt[(2 + 3*x)^2]*(4 + 3*x))/(4 + 6*x)

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Maple [A]  time = 0.004, size = 25, normalized size = 1.1 \[{\frac{x \left ( 3\,x+4 \right ) }{4+6\,x}\sqrt{ \left ( 2+3\,x \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((9*x^2+12*x+4)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.769286, size = 41, normalized size = 1.78 \[ \frac{1}{2} \, \sqrt{9 \, x^{2} + 12 \, x + 4} x + \frac{1}{3} \, \sqrt{9 \, x^{2} + 12 \, x + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 + 12*x + 4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.219046, size = 12, normalized size = 0.52 \[ \frac{3}{2} \, x^{2} + 2 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 + 12*x + 4),x, algorithm="fricas")

[Out]

3/2*x^2 + 2*x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{9 x^{2} + 12 x + 4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((9*x**2+12*x+4)**(1/2),x)

[Out]

Integral(sqrt(9*x**2 + 12*x + 4), x)

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GIAC/XCAS [A]  time = 0.210094, size = 35, normalized size = 1.52 \[ \frac{1}{2} \,{\left (3 \, x^{2} + 4 \, x\right )}{\rm sign}\left (3 \, x + 2\right ) + \frac{2}{3} \,{\rm sign}\left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 + 12*x + 4),x, algorithm="giac")

[Out]

1/2*(3*x^2 + 4*x)*sign(3*x + 2) + 2/3*sign(3*x + 2)