3.2839 \(\int \frac{2+x}{\left (1+(2+x)^2\right )^3} \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{4 \left ((x+2)^2+1\right )^2} \]

[Out]

-1/(4*(1 + (2 + x)^2)^2)

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Rubi [A]  time = 0.00808053, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{1}{4 \left ((x+2)^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(2 + x)/(1 + (2 + x)^2)^3,x]

[Out]

-1/(4*(1 + (2 + x)^2)^2)

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Rubi in Sympy [A]  time = 2.17161, size = 12, normalized size = 0.92 \[ - \frac{1}{4 \left (\left (x + 2\right )^{2} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+x)/(1+(2+x)**2)**3,x)

[Out]

-1/(4*((x + 2)**2 + 1)**2)

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Mathematica [A]  time = 0.00441033, size = 13, normalized size = 1. \[ -\frac{1}{4 \left ((x+2)^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x)/(1 + (2 + x)^2)^3,x]

[Out]

-1/(4*(1 + (2 + x)^2)^2)

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Maple [A]  time = 0., size = 13, normalized size = 1. \[ -{\frac{1}{4\, \left ({x}^{2}+4\,x+5 \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(1+(2+x)^2)^3,x)

[Out]

-1/4/(x^2+4*x+5)^2

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Maxima [A]  time = 1.42362, size = 15, normalized size = 1.15 \[ -\frac{1}{4 \,{\left ({\left (x + 2\right )}^{2} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 2)/((x + 2)^2 + 1)^3,x, algorithm="maxima")

[Out]

-1/4/((x + 2)^2 + 1)^2

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Fricas [A]  time = 0.201345, size = 30, normalized size = 2.31 \[ -\frac{1}{4 \,{\left (x^{4} + 8 \, x^{3} + 26 \, x^{2} + 40 \, x + 25\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 2)/((x + 2)^2 + 1)^3,x, algorithm="fricas")

[Out]

-1/4/(x^4 + 8*x^3 + 26*x^2 + 40*x + 25)

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Sympy [A]  time = 0.266931, size = 22, normalized size = 1.69 \[ - \frac{1}{4 x^{4} + 32 x^{3} + 104 x^{2} + 160 x + 100} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+x)/(1+(2+x)**2)**3,x)

[Out]

-1/(4*x**4 + 32*x**3 + 104*x**2 + 160*x + 100)

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GIAC/XCAS [A]  time = 0.217095, size = 16, normalized size = 1.23 \[ -\frac{1}{4 \,{\left (x^{2} + 4 \, x + 5\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 2)/((x + 2)^2 + 1)^3,x, algorithm="giac")

[Out]

-1/4/(x^2 + 4*x + 5)^2