3.200 \(\int \frac{7+5 x+4 x^2}{5+4 x+4 x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{8} \log \left (4 x^2+4 x+5\right )+x+\frac{3}{8} \tan ^{-1}\left (x+\frac{1}{2}\right ) \]

[Out]

x + (3*ArcTan[1/2 + x])/8 + Log[5 + 4*x + 4*x^2]/8

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Rubi [A]  time = 0.0247256, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {1657, 634, 618, 204, 628} \[ \frac{1}{8} \log \left (4 x^2+4 x+5\right )+x+\frac{3}{8} \tan ^{-1}\left (x+\frac{1}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(7 + 5*x + 4*x^2)/(5 + 4*x + 4*x^2),x]

[Out]

x + (3*ArcTan[1/2 + x])/8 + Log[5 + 4*x + 4*x^2]/8

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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{7+5 x+4 x^2}{5+4 x+4 x^2} \, dx &=\int \left (1+\frac{2+x}{5+4 x+4 x^2}\right ) \, dx\\ &=x+\int \frac{2+x}{5+4 x+4 x^2} \, dx\\ &=x+\frac{1}{8} \int \frac{4+8 x}{5+4 x+4 x^2} \, dx+\frac{3}{2} \int \frac{1}{5+4 x+4 x^2} \, dx\\ &=x+\frac{1}{8} \log \left (5+4 x+4 x^2\right )-3 \operatorname{Subst}\left (\int \frac{1}{-64-x^2} \, dx,x,4+8 x\right )\\ &=x+\frac{3}{8} \tan ^{-1}\left (\frac{1}{2}+x\right )+\frac{1}{8} \log \left (5+4 x+4 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0046953, size = 31, normalized size = 1.15 \[ \frac{1}{8} \log \left (4 x^2+4 x+5\right )+x+\frac{3}{8} \tan ^{-1}\left (\frac{1}{2} (2 x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(7 + 5*x + 4*x^2)/(5 + 4*x + 4*x^2),x]

[Out]

x + (3*ArcTan[(1 + 2*x)/2])/8 + Log[5 + 4*x + 4*x^2]/8

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Maple [A]  time = 0.004, size = 22, normalized size = 0.8 \begin{align*} x+{\frac{3}{8}\arctan \left ( x+{\frac{1}{2}} \right ) }+{\frac{\ln \left ( 4\,{x}^{2}+4\,x+5 \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2+5*x+7)/(4*x^2+4*x+5),x)

[Out]

x+3/8*arctan(x+1/2)+1/8*ln(4*x^2+4*x+5)

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Maxima [A]  time = 1.40412, size = 28, normalized size = 1.04 \begin{align*} x + \frac{3}{8} \, \arctan \left (x + \frac{1}{2}\right ) + \frac{1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+5*x+7)/(4*x^2+4*x+5),x, algorithm="maxima")

[Out]

x + 3/8*arctan(x + 1/2) + 1/8*log(4*x^2 + 4*x + 5)

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Fricas [A]  time = 1.93485, size = 70, normalized size = 2.59 \begin{align*} x + \frac{3}{8} \, \arctan \left (x + \frac{1}{2}\right ) + \frac{1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+5*x+7)/(4*x^2+4*x+5),x, algorithm="fricas")

[Out]

x + 3/8*arctan(x + 1/2) + 1/8*log(4*x^2 + 4*x + 5)

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Sympy [A]  time = 0.098302, size = 22, normalized size = 0.81 \begin{align*} x + \frac{\log{\left (x^{2} + x + \frac{5}{4} \right )}}{8} + \frac{3 \operatorname{atan}{\left (x + \frac{1}{2} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2+5*x+7)/(4*x**2+4*x+5),x)

[Out]

x + log(x**2 + x + 5/4)/8 + 3*atan(x + 1/2)/8

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Giac [A]  time = 1.04893, size = 28, normalized size = 1.04 \begin{align*} x + \frac{3}{8} \, \arctan \left (x + \frac{1}{2}\right ) + \frac{1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+5*x+7)/(4*x^2+4*x+5),x, algorithm="giac")

[Out]

x + 3/8*arctan(x + 1/2) + 1/8*log(4*x^2 + 4*x + 5)