3.1.5 \(\int \frac {1+x}{(2+2 x+x^2)^3} \, dx\) [5]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = {643} \begin {gather*} -\frac {1}{4 \left (x^2+2 x+2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)/(2 + 2*x + x^2)^3,x]

[Out]

-1/4*1/(2 + 2*x + x^2)^2

Rule 643

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*}

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Mathematica [A]
time = 0.02, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{4 \left (2+2 x+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*1/(2 + 2*x + x^2)^2

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Mathics [A]
time = 1.73, size = 24, normalized size = 1.71 \begin {gather*} -\frac {1}{16+32 x+32 x^2+16 x^3+4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.13, size = 13, normalized size = 0.93

method	result	size
gosper	\(-\frac {1}{4 \left (x^{2}+2 x +2\right )^{2}}\)	\(13\)
default	\(-\frac {1}{4 \left (x^{2}+2 x +2\right )^{2}}\)	\(13\)
norman	\(-\frac {1}{4 \left (x^{2}+2 x +2\right )^{2}}\)	\(13\)
risch	\(-\frac {1}{4 \left (x^{2}+2 x +2\right )^{2}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(x^2+2*x+2)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{4 \, {\left (x^{2} + 2 \, x + 2\right )}^{2}} \end {gather*}

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 = 0.33, size = 22, normalized size = 1.57 \begin {gather*} -\frac {1}{4 \, {\left (x^{4} + 4 \, x^{3} + 8 \, x^{2} + 8 \, x + 4\right )}} \end {gather*}

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 = 0.06, size = 22, normalized size = 1.57 \begin {gather*} - \frac {1}{4 x^{4} + 16 x^{3} + 32 x^{2} + 32 x + 16} \end {gather*}

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 = 0.00, size = 16, normalized size = 1.14 \begin {gather*} -\frac {1}{16 \left (x+\frac {x^{2}}{2}+1\right )^{2}} \end {gather*}

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^2 + 2*x + 2)^2

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Mupad [B]
time = 0.06, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{4\,{\left (x^2+2\,x+2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(2*x + x^2 + 2)^3,x)

[Out]

-1/(4*(2*x + x^2 + 2)^2)

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