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

Optimal. Leaf size=44 \[ \frac{3}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{1}{4 \sqrt{4 x^2+12 x+9}} \]

[Out]

-1/(4*Sqrt[9 + 12*x + 4*x^2]) + 3/(8*(3 + 2*x)*Sqrt[9 + 12*x + 4*x^2])

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Rubi [A]  time = 0.0323912, 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 \[ \frac{3}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{1}{4 \sqrt{4 x^2+12 x+9}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*Sqrt[9 + 12*x + 4*x^2]) + 3/(8*(3 + 2*x)*Sqrt[9 + 12*x + 4*x^2])

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Rubi in Sympy [A]  time = 3.35039, size = 22, normalized size = 0.5 \[ \frac{x^{2} \left (8 x + 12\right )}{24 \left (4 x^{2} + 12 x + 9\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(8*x + 12)/(24*(4*x**2 + 12*x + 9)**(3/2))

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Mathematica [A]  time = 0.0120221, size = 27, normalized size = 0.61 \[ \frac{-4 x-3}{8 (2 x+3) \sqrt{(2 x+3)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 22, normalized size = 0.5 \[ -{\frac{ \left ( 2\,x+3 \right ) \left ( 3+4\,x \right ) }{8} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(4*x^2+12*x+9)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.836346, size = 32, normalized size = 0.73 \[ -\frac{1}{4 \, \sqrt{4 \, x^{2} + 12 \, x + 9}} + \frac{3}{8 \,{\left (2 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216035, size = 26, normalized size = 0.59 \[ -\frac{4 \, x + 3}{8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*x + 3)/(4*x^2 + 12*x + 9)

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

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 = 0.58403, 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)^(3/2),x, algorithm="giac")

[Out]

sage0*x