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

Optimal. Leaf size=44 \[ \frac{3}{16 (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}}-\frac{1}{12 \left (4 x^2+12 x+9\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0313068, antiderivative size = 44, 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 \[ \frac{3}{16 (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}}-\frac{1}{12 \left (4 x^2+12 x+9\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.73285, size = 36, normalized size = 0.82 \[ \frac{3 \left (8 x + 12\right )}{64 \left (4 x^{2} + 12 x + 9\right )^{\frac{5}{2}}} - \frac{1}{12 \left (4 x^{2} + 12 x + 9\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(8*x + 12)/(64*(4*x**2 + 12*x + 9)**(5/2)) - 1/(12*(4*x**2 + 12*x + 9)**(3/2))

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Mathematica [A]  time = 0.0140831, size = 27, normalized size = 0.61 \[ \frac{-8 x-3}{48 (2 x+3)^3 \sqrt{(2 x+3)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-3 - 8*x)/(48*(3 + 2*x)^3*Sqrt[(3 + 2*x)^2])

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Maple [A]  time = 0.005, size = 22, normalized size = 0.5 \[ -{\frac{ \left ( 2\,x+3 \right ) \left ( 8\,x+3 \right ) }{48} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(4*x^2+12*x+9)^(5/2),x)

[Out]

-1/48*(2*x+3)*(8*x+3)/((2*x+3)^2)^(5/2)

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Maxima [A]  time = 0.771228, size = 32, normalized size = 0.73 \[ -\frac{1}{12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}}} + \frac{3}{16 \,{\left (2 \, x + 3\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21804, size = 39, normalized size = 0.89 \[ -\frac{8 \, x + 3}{48 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(8*x + 3)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

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

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 = 0.572296, size = 4, normalized size = 0.09 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x