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

Optimal. Leaf size=48 \[ \frac{1}{4} \sqrt{4 x^2+12 x+9}-\frac{3 (2 x+3) \log (2 x+3)}{4 \sqrt{4 x^2+12 x+9}} \]

[Out]

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

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Rubi [A]  time = 0.0380652, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{4} \sqrt{4 x^2+12 x+9}-\frac{3 (2 x+3) \log (2 x+3)}{4 \sqrt{4 x^2+12 x+9}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[9 + 12*x + 4*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.14537, size = 42, normalized size = 0.88 \[ - \frac{3 \left (4 x + 6\right ) \log{\left (2 x + 3 \right )}}{8 \sqrt{4 x^{2} + 12 x + 9}} + \frac{\sqrt{4 x^{2} + 12 x + 9}}{4} \]

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*(4*x + 6)*log(2*x + 3)/(8*sqrt(4*x**2 + 12*x + 9)) + sqrt(4*x**2 + 12*x + 9)/
4

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Mathematica [A]  time = 0.0189488, size = 33, normalized size = 0.69 \[ \frac{(2 x+3) (2 x-3 \log (2 x+3)+3)}{4 \sqrt{(2 x+3)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 29, normalized size = 0.6 \[ -{\frac{ \left ( 2\,x+3 \right ) \left ( -2\,x+3\,\ln \left ( 2\,x+3 \right ) \right ) }{4}{\frac{1}{\sqrt{ \left ( 2\,x+3 \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(4*x^2+12*x+9)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.831064, size = 28, normalized size = 0.58 \[ \frac{1}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} - \frac{3}{4} \, \log \left (x + \frac{3}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218945, size = 16, normalized size = 0.33 \[ \frac{1}{2} \, x - \frac{3}{4} \, \log \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x - 3/4*log(2*x + 3)

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

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 = 0.208983, size = 49, normalized size = 1.02 \[ \frac{1}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} + \frac{3}{4} \,{\rm ln}\left ({\left | -2 \, x + \sqrt{4 \, x^{2} + 12 \, x + 9} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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