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

Optimal. Leaf size=42 \[ \frac{1}{12} \left (4 x^2+12 x+9\right )^{3/2}-\frac{3}{8} (2 x+3) \sqrt{4 x^2+12 x+9} \]

[Out]

(-3*(3 + 2*x)*Sqrt[9 + 12*x + 4*x^2])/8 + (9 + 12*x + 4*x^2)^(3/2)/12

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Rubi [A]  time = 0.031837, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{12} \left (4 x^2+12 x+9\right )^{3/2}-\frac{3}{8} (2 x+3) \sqrt{4 x^2+12 x+9} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[9 + 12*x + 4*x^2],x]

[Out]

(-3*(3 + 2*x)*Sqrt[9 + 12*x + 4*x^2])/8 + (9 + 12*x + 4*x^2)^(3/2)/12

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Rubi in Sympy [A]  time = 3.70171, size = 36, normalized size = 0.86 \[ - \frac{3 \left (8 x + 12\right ) \sqrt{4 x^{2} + 12 x + 9}}{32} + \frac{\left (4 x^{2} + 12 x + 9\right )^{\frac{3}{2}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(8*x + 12)*sqrt(4*x**2 + 12*x + 9)/32 + (4*x**2 + 12*x + 9)**(3/2)/12

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Mathematica [A]  time = 0.0125517, size = 30, normalized size = 0.71 \[ \frac{x^2 \sqrt{(2 x+3)^2} (4 x+9)}{6 (2 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[(3 + 2*x)^2]*(9 + 4*x))/(6*(3 + 2*x))

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Maple [A]  time = 0.004, size = 27, normalized size = 0.6 \[{\frac{{x}^{2} \left ( 4\,x+9 \right ) }{12\,x+18}\sqrt{ \left ( 2\,x+3 \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(4*x^2+12*x+9)^(1/2),x)

[Out]

1/6*x^2*(4*x+9)*((2*x+3)^2)^(1/2)/(2*x+3)

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Maxima [A]  time = 0.825536, size = 59, normalized size = 1.4 \[ \frac{1}{12} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} - \frac{3}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} x - \frac{9}{8} \, \sqrt{4 \, x^{2} + 12 \, x + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(4*x^2 + 12*x + 9)^(3/2) - 3/4*sqrt(4*x^2 + 12*x + 9)*x - 9/8*sqrt(4*x^2 +
12*x + 9)

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Fricas [A]  time = 0.217213, size = 15, normalized size = 0.36 \[ \frac{2}{3} \, x^{3} + \frac{3}{2} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*x^3 + 3/2*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{\left (2 x + 3\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt((2*x + 3)**2), x)

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GIAC/XCAS [A]  time = 0.207888, size = 42, normalized size = 1. \[ \frac{2}{3} \, x^{3}{\rm sign}\left (2 \, x + 3\right ) + \frac{3}{2} \, x^{2}{\rm sign}\left (2 \, x + 3\right ) - \frac{9}{8} \,{\rm sign}\left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*x^3*sign(2*x + 3) + 3/2*x^2*sign(2*x + 3) - 9/8*sign(2*x + 3)