3.472 \(\int \frac{31+5 x}{11-4 x+3 x^2} \, dx\)

Optimal. Leaf size=37 \[ \frac{5}{6} \log \left (3 x^2-4 x+11\right )-\frac{103 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{29}}\right )}{3 \sqrt{29}} \]

[Out]

(-103*ArcTan[(2 - 3*x)/Sqrt[29]])/(3*Sqrt[29]) + (5*Log[11 - 4*x + 3*x^2])/6

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Rubi [A]  time = 0.0223993, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {634, 618, 204, 628} \[ \frac{5}{6} \log \left (3 x^2-4 x+11\right )-\frac{103 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{29}}\right )}{3 \sqrt{29}} \]

Antiderivative was successfully verified.

[In]

Int[(31 + 5*x)/(11 - 4*x + 3*x^2),x]

[Out]

(-103*ArcTan[(2 - 3*x)/Sqrt[29]])/(3*Sqrt[29]) + (5*Log[11 - 4*x + 3*x^2])/6

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{31+5 x}{11-4 x+3 x^2} \, dx &=\frac{5}{6} \int \frac{-4+6 x}{11-4 x+3 x^2} \, dx+\frac{103}{3} \int \frac{1}{11-4 x+3 x^2} \, dx\\ &=\frac{5}{6} \log \left (11-4 x+3 x^2\right )-\frac{206}{3} \operatorname{Subst}\left (\int \frac{1}{-116-x^2} \, dx,x,-4+6 x\right )\\ &=-\frac{103 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{29}}\right )}{3 \sqrt{29}}+\frac{5}{6} \log \left (11-4 x+3 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0136187, size = 37, normalized size = 1. \[ \frac{5}{6} \log \left (3 x^2-4 x+11\right )+\frac{103 \tan ^{-1}\left (\frac{3 x-2}{\sqrt{29}}\right )}{3 \sqrt{29}} \]

Antiderivative was successfully verified.

[In]

Integrate[(31 + 5*x)/(11 - 4*x + 3*x^2),x]

[Out]

(103*ArcTan[(-2 + 3*x)/Sqrt[29]])/(3*Sqrt[29]) + (5*Log[11 - 4*x + 3*x^2])/6

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Maple [A]  time = 0.004, size = 31, normalized size = 0.8 \begin{align*}{\frac{5\,\ln \left ( 3\,{x}^{2}-4\,x+11 \right ) }{6}}+{\frac{103\,\sqrt{29}}{87}\arctan \left ({\frac{ \left ( 6\,x-4 \right ) \sqrt{29}}{58}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((31+5*x)/(3*x^2-4*x+11),x)

[Out]

5/6*ln(3*x^2-4*x+11)+103/87*29^(1/2)*arctan(1/58*(6*x-4)*29^(1/2))

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Maxima [A]  time = 1.65553, size = 41, normalized size = 1.11 \begin{align*} \frac{103}{87} \, \sqrt{29} \arctan \left (\frac{1}{29} \, \sqrt{29}{\left (3 \, x - 2\right )}\right ) + \frac{5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((31+5*x)/(3*x^2-4*x+11),x, algorithm="maxima")

[Out]

103/87*sqrt(29)*arctan(1/29*sqrt(29)*(3*x - 2)) + 5/6*log(3*x^2 - 4*x + 11)

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Fricas [A]  time = 1.16902, size = 104, normalized size = 2.81 \begin{align*} \frac{103}{87} \, \sqrt{29} \arctan \left (\frac{1}{29} \, \sqrt{29}{\left (3 \, x - 2\right )}\right ) + \frac{5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((31+5*x)/(3*x^2-4*x+11),x, algorithm="fricas")

[Out]

103/87*sqrt(29)*arctan(1/29*sqrt(29)*(3*x - 2)) + 5/6*log(3*x^2 - 4*x + 11)

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Sympy [A]  time = 0.111909, size = 44, normalized size = 1.19 \begin{align*} \frac{5 \log{\left (x^{2} - \frac{4 x}{3} + \frac{11}{3} \right )}}{6} + \frac{103 \sqrt{29} \operatorname{atan}{\left (\frac{3 \sqrt{29} x}{29} - \frac{2 \sqrt{29}}{29} \right )}}{87} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((31+5*x)/(3*x**2-4*x+11),x)

[Out]

5*log(x**2 - 4*x/3 + 11/3)/6 + 103*sqrt(29)*atan(3*sqrt(29)*x/29 - 2*sqrt(29)/29)/87

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Giac [A]  time = 1.13874, size = 41, normalized size = 1.11 \begin{align*} \frac{103}{87} \, \sqrt{29} \arctan \left (\frac{1}{29} \, \sqrt{29}{\left (3 \, x - 2\right )}\right ) + \frac{5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((31+5*x)/(3*x^2-4*x+11),x, algorithm="giac")

[Out]

103/87*sqrt(29)*arctan(1/29*sqrt(29)*(3*x - 2)) + 5/6*log(3*x^2 - 4*x + 11)