3.327 \(\int \frac{4+8 x+5 x^2+2 x^3}{(2+2 x+x^2)^2} \, dx\)

Optimal. Leaf size=28 \[ -\frac{1}{x^2+2 x+2}+\log \left (x^2+2 x+2\right )-\tan ^{-1}(x+1) \]

[Out]

-(2 + 2*x + x^2)^(-1) - ArcTan[1 + x] + Log[2 + 2*x + x^2]

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Rubi [A]  time = 0.0219627, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1660, 634, 617, 204, 628} \[ -\frac{1}{x^2+2 x+2}+\log \left (x^2+2 x+2\right )-\tan ^{-1}(x+1) \]

Antiderivative was successfully verified.

[In]

Int[(4 + 8*x + 5*x^2 + 2*x^3)/(2 + 2*x + x^2)^2,x]

[Out]

-(2 + 2*x + x^2)^(-1) - ArcTan[1 + x] + Log[2 + 2*x + x^2]

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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

Rubi steps

\begin{align*} \int \frac{4+8 x+5 x^2+2 x^3}{\left (2+2 x+x^2\right )^2} \, dx &=-\frac{1}{2+2 x+x^2}+\frac{1}{4} \int \frac{4+8 x}{2+2 x+x^2} \, dx\\ &=-\frac{1}{2+2 x+x^2}-\int \frac{1}{2+2 x+x^2} \, dx+\int \frac{2+2 x}{2+2 x+x^2} \, dx\\ &=-\frac{1}{2+2 x+x^2}+\log \left (2+2 x+x^2\right )+\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+x\right )\\ &=-\frac{1}{2+2 x+x^2}-\tan ^{-1}(1+x)+\log \left (2+2 x+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0096268, size = 28, normalized size = 1. \[ -\frac{1}{x^2+2 x+2}+\log \left (x^2+2 x+2\right )-\tan ^{-1}(x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + 8*x + 5*x^2 + 2*x^3)/(2 + 2*x + x^2)^2,x]

[Out]

-(2 + 2*x + x^2)^(-1) - ArcTan[1 + x] + Log[2 + 2*x + x^2]

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Maple [A]  time = 0.004, size = 29, normalized size = 1. \begin{align*} - \left ({x}^{2}+2\,x+2 \right ) ^{-1}-\arctan \left ( 1+x \right ) +\ln \left ({x}^{2}+2\,x+2 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3+5*x^2+8*x+4)/(x^2+2*x+2)^2,x)

[Out]

-1/(x^2+2*x+2)-arctan(1+x)+ln(x^2+2*x+2)

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Maxima [A]  time = 1.68009, size = 38, normalized size = 1.36 \begin{align*} -\frac{1}{x^{2} + 2 \, x + 2} - \arctan \left (x + 1\right ) + \log \left (x^{2} + 2 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+5*x^2+8*x+4)/(x^2+2*x+2)^2,x, algorithm="maxima")

[Out]

-1/(x^2 + 2*x + 2) - arctan(x + 1) + log(x^2 + 2*x + 2)

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Fricas [A]  time = 1.54265, size = 123, normalized size = 4.39 \begin{align*} -\frac{{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) -{\left (x^{2} + 2 \, x + 2\right )} \log \left (x^{2} + 2 \, x + 2\right ) + 1}{x^{2} + 2 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+5*x^2+8*x+4)/(x^2+2*x+2)^2,x, algorithm="fricas")

[Out]

-((x^2 + 2*x + 2)*arctan(x + 1) - (x^2 + 2*x + 2)*log(x^2 + 2*x + 2) + 1)/(x^2 + 2*x + 2)

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Sympy [A]  time = 0.121162, size = 24, normalized size = 0.86 \begin{align*} \log{\left (x^{2} + 2 x + 2 \right )} - \operatorname{atan}{\left (x + 1 \right )} - \frac{1}{x^{2} + 2 x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3+5*x**2+8*x+4)/(x**2+2*x+2)**2,x)

[Out]

log(x**2 + 2*x + 2) - atan(x + 1) - 1/(x**2 + 2*x + 2)

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Giac [A]  time = 1.22375, size = 38, normalized size = 1.36 \begin{align*} -\frac{1}{x^{2} + 2 \, x + 2} - \arctan \left (x + 1\right ) + \log \left (x^{2} + 2 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+5*x^2+8*x+4)/(x^2+2*x+2)^2,x, algorithm="giac")

[Out]

-1/(x^2 + 2*x + 2) - arctan(x + 1) + log(x^2 + 2*x + 2)