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

Optimal. Leaf size=14 \[ -\frac{1}{4 \left (x^2+2 x+2\right )^2} \]

[Out]

-1/(4*(2 + 2*x + x^2)^2)

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Rubi [A]  time = 0.00632382, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{1}{4 \left (x^2+2 x+2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*(2 + 2*x + x^2)^2)

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Rubi in Sympy [A]  time = 1.23509, size = 14, normalized size = 1. \[ - \frac{1}{4 \left (x^{2} + 2 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(4*(x**2 + 2*x + 2)**2)

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Mathematica [A]  time = 0.00674588, size = 14, normalized size = 1. \[ -\frac{1}{4 \left (x^2+2 x+2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*(2 + 2*x + x^2)^2)

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Maple [A]  time = 0.172, size = 13, normalized size = 0.9 \[ -{\frac{1}{4\, \left ({x}^{2}+2\,x+2 \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(x^2+2*x+2)^3,x)

[Out]

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

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Maxima [A]  time = 1.34274, size = 16, normalized size = 1.14 \[ -\frac{1}{4 \,{\left (x^{2} + 2 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/(x^2 + 2*x + 2)^2

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Fricas [A]  time = 0.218837, size = 30, normalized size = 2.14 \[ -\frac{1}{4 \,{\left (x^{4} + 4 \, x^{3} + 8 \, x^{2} + 8 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/(x^4 + 4*x^3 + 8*x^2 + 8*x + 4)

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Sympy [A]  time = 0.132066, size = 22, normalized size = 1.57 \[ - \frac{1}{4 x^{4} + 16 x^{3} + 32 x^{2} + 32 x + 16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(4*x**4 + 16*x**3 + 32*x**2 + 32*x + 16)

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GIAC/XCAS [A]  time = 0.221687, size = 16, normalized size = 1.14 \[ -\frac{1}{4 \,{\left (x^{2} + 2 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/(x^2 + 2*x + 2)^2