3.209 \(\int x \left (9+12 x+4 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=42 \[ \frac{1}{20} \left (4 x^2+12 x+9\right )^{5/2}-\frac{3}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} \]

[Out]

(-3*(3 + 2*x)*(9 + 12*x + 4*x^2)^(3/2))/16 + (9 + 12*x + 4*x^2)^(5/2)/20

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Rubi [A]  time = 0.0320089, 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 \[ \frac{1}{20} \left (4 x^2+12 x+9\right )^{5/2}-\frac{3}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x*(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

(-3*(3 + 2*x)*(9 + 12*x + 4*x^2)^(3/2))/16 + (9 + 12*x + 4*x^2)^(5/2)/20

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Rubi in Sympy [A]  time = 3.66127, size = 36, normalized size = 0.86 \[ - \frac{3 \left (8 x + 12\right ) \left (4 x^{2} + 12 x + 9\right )^{\frac{3}{2}}}{64} + \frac{\left (4 x^{2} + 12 x + 9\right )^{\frac{5}{2}}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(4*x**2+12*x+9)**(3/2),x)

[Out]

-3*(8*x + 12)*(4*x**2 + 12*x + 9)**(3/2)/64 + (4*x**2 + 12*x + 9)**(5/2)/20

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Mathematica [A]  time = 0.018169, size = 37, normalized size = 0.88 \[ \frac{x^2 \sqrt{(2 x+3)^2} \left (16 x^3+90 x^2+180 x+135\right )}{20 x+30} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

(x^2*Sqrt[(3 + 2*x)^2]*(135 + 180*x + 90*x^2 + 16*x^3))/(30 + 20*x)

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Maple [A]  time = 0.006, size = 37, normalized size = 0.9 \[{\frac{{x}^{2} \left ( 16\,{x}^{3}+90\,{x}^{2}+180\,x+135 \right ) }{10\, \left ( 2\,x+3 \right ) ^{3}} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(4*x^2+12*x+9)^(3/2),x)

[Out]

1/10*x^2*(16*x^3+90*x^2+180*x+135)*((2*x+3)^2)^(3/2)/(2*x+3)^3

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Maxima [A]  time = 0.832145, size = 59, normalized size = 1.4 \[ \frac{1}{20} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} - \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} x - \frac{9}{16} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + 12*x + 9)^(3/2)*x,x, algorithm="maxima")

[Out]

1/20*(4*x^2 + 12*x + 9)^(5/2) - 3/8*(4*x^2 + 12*x + 9)^(3/2)*x - 9/16*(4*x^2 + 1
2*x + 9)^(3/2)

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Fricas [A]  time = 0.214659, size = 28, normalized size = 0.67 \[ \frac{8}{5} \, x^{5} + 9 \, x^{4} + 18 \, x^{3} + \frac{27}{2} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + 12*x + 9)^(3/2)*x,x, algorithm="fricas")

[Out]

8/5*x^5 + 9*x^4 + 18*x^3 + 27/2*x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(4*x**2+12*x+9)**(3/2),x)

[Out]

Integral(x*((2*x + 3)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.20616, size = 72, normalized size = 1.71 \[ \frac{8}{5} \, x^{5}{\rm sign}\left (2 \, x + 3\right ) + 9 \, x^{4}{\rm sign}\left (2 \, x + 3\right ) + 18 \, x^{3}{\rm sign}\left (2 \, x + 3\right ) + \frac{27}{2} \, x^{2}{\rm sign}\left (2 \, x + 3\right ) - \frac{243}{80} \,{\rm sign}\left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + 12*x + 9)^(3/2)*x,x, algorithm="giac")

[Out]

8/5*x^5*sign(2*x + 3) + 9*x^4*sign(2*x + 3) + 18*x^3*sign(2*x + 3) + 27/2*x^2*si
gn(2*x + 3) - 243/80*sign(2*x + 3)