3.208 \(\int x \left (9+12 x+4 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=42 \[ \frac{1}{28} \left (4 x^2+12 x+9\right )^{7/2}-\frac{1}{8} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} \]

[Out]

-((3 + 2*x)*(9 + 12*x + 4*x^2)^(5/2))/8 + (9 + 12*x + 4*x^2)^(7/2)/28

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Rubi [A]  time = 0.0325436, 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 \[ \frac{1}{28} \left (4 x^2+12 x+9\right )^{7/2}-\frac{1}{8} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[x*(9 + 12*x + 4*x^2)^(5/2),x]

[Out]

-((3 + 2*x)*(9 + 12*x + 4*x^2)^(5/2))/8 + (9 + 12*x + 4*x^2)^(7/2)/28

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Rubi in Sympy [A]  time = 3.83275, size = 34, normalized size = 0.81 \[ - \frac{\left (8 x + 12\right ) \left (4 x^{2} + 12 x + 9\right )^{\frac{5}{2}}}{32} + \frac{\left (4 x^{2} + 12 x + 9\right )^{\frac{7}{2}}}{28} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(4*x**2+12*x+9)**(5/2),x)

[Out]

-(8*x + 12)*(4*x**2 + 12*x + 9)**(5/2)/32 + (4*x**2 + 12*x + 9)**(7/2)/28

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Mathematica [A]  time = 0.0263512, size = 47, normalized size = 1.12 \[ \frac{x^2 \sqrt{(2 x+3)^2} \left (64 x^5+560 x^4+2016 x^3+3780 x^2+3780 x+1701\right )}{28 x+42} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(9 + 12*x + 4*x^2)^(5/2),x]

[Out]

(x^2*Sqrt[(3 + 2*x)^2]*(1701 + 3780*x + 3780*x^2 + 2016*x^3 + 560*x^4 + 64*x^5))
/(42 + 28*x)

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Maple [A]  time = 0.005, size = 47, normalized size = 1.1 \[{\frac{{x}^{2} \left ( 64\,{x}^{5}+560\,{x}^{4}+2016\,{x}^{3}+3780\,{x}^{2}+3780\,x+1701 \right ) }{14\, \left ( 2\,x+3 \right ) ^{5}} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(4*x^2+12*x+9)^(5/2),x)

[Out]

1/14*x^2*(64*x^5+560*x^4+2016*x^3+3780*x^2+3780*x+1701)*((2*x+3)^2)^(5/2)/(2*x+3
)^5

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Maxima [A]  time = 0.883183, size = 59, normalized size = 1.4 \[ \frac{1}{28} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{7}{2}} - \frac{1}{4} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} x - \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + 12*x + 9)^(5/2)*x,x, algorithm="maxima")

[Out]

1/28*(4*x^2 + 12*x + 9)^(7/2) - 1/4*(4*x^2 + 12*x + 9)^(5/2)*x - 3/8*(4*x^2 + 12
*x + 9)^(5/2)

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Fricas [A]  time = 0.215693, size = 42, normalized size = 1. \[ \frac{32}{7} \, x^{7} + 40 \, x^{6} + 144 \, x^{5} + 270 \, x^{4} + 270 \, x^{3} + \frac{243}{2} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + 12*x + 9)^(5/2)*x,x, algorithm="fricas")

[Out]

32/7*x^7 + 40*x^6 + 144*x^5 + 270*x^4 + 270*x^3 + 243/2*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \left (\left (2 x + 3\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(4*x**2+12*x+9)**(5/2),x)

[Out]

Integral(x*((2*x + 3)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.20779, size = 101, normalized size = 2.4 \[ \frac{32}{7} \, x^{7}{\rm sign}\left (2 \, x + 3\right ) + 40 \, x^{6}{\rm sign}\left (2 \, x + 3\right ) + 144 \, x^{5}{\rm sign}\left (2 \, x + 3\right ) + 270 \, x^{4}{\rm sign}\left (2 \, x + 3\right ) + 270 \, x^{3}{\rm sign}\left (2 \, x + 3\right ) + \frac{243}{2} \, x^{2}{\rm sign}\left (2 \, x + 3\right ) - \frac{729}{56} \,{\rm sign}\left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 + 12*x + 9)^(5/2)*x,x, algorithm="giac")

[Out]

32/7*x^7*sign(2*x + 3) + 40*x^6*sign(2*x + 3) + 144*x^5*sign(2*x + 3) + 270*x^4*
sign(2*x + 3) + 270*x^3*sign(2*x + 3) + 243/2*x^2*sign(2*x + 3) - 729/56*sign(2*
x + 3)