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

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 = 0.0028358, 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, Rules used = {629} \[ -\frac{1}{4 \left (x^2+2 x+2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)/(2 + 2*x + x^2)^3,x]

[Out]

-1/(4*(2 + 2*x + x^2)^2)

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{1+x}{\left (2+2 x+x^2\right )^3} \, dx &=-\frac{1}{4 \left (2+2 x+x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0040282, 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 = 0.007, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{4\, \left ({x}^{2}+2\,x+2 \right ) ^{2}}} \end{align*}

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 = 0.983106, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{4 \,{\left (x^{2} + 2 \, x + 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(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 = 0.404062, size = 51, normalized size = 3.64 \begin{align*} -\frac{1}{4 \,{\left (x^{4} + 4 \, x^{3} + 8 \, x^{2} + 8 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(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 = 0.105645, size = 22, normalized size = 1.57 \begin{align*} - \frac{1}{4 x^{4} + 16 x^{3} + 32 x^{2} + 32 x + 16} \end{align*}

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 [A]  time = 1.0551, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{4 \,{\left (x^{2} + 2 \, x + 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(x^2 + 2*x + 2)^2