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 = 0.0526654, 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 \[ \frac{5}{6} \log \left (3 x^2-4 x+11\right )-\frac{103 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{29}}\right )}{3 \sqrt{29}} \]

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

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Rubi in Sympy [A]  time = 6.5666, size = 37, normalized size = 1. \[ \frac{5 \log{\left (3 x^{2} - 4 x + 11 \right )}}{6} + \frac{103 \sqrt{29} \operatorname{atan}{\left (\sqrt{29} \left (\frac{3 x}{29} - \frac{2}{29}\right ) \right )}}{87} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0211698, 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 = 0.006, size = 31, normalized size = 0.8 \[{\frac{5\,\ln \left ( 3\,{x}^{2}-4\,x+11 \right ) }{6}}+{\frac{103\,\sqrt{29}}{87}\arctan \left ({\frac{ \left ( 6\,x-4 \right ) \sqrt{29}}{58}} \right ) } \]

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 = 0.855363, size = 41, normalized size = 1.11 \[ \frac{103}{87} \, \sqrt{29} \arctan \left (\frac{1}{29} \, \sqrt{29}{\left (3 \, x - 2\right )}\right ) + \frac{5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 31)/(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 = 0.264668, size = 47, normalized size = 1.27 \[ \frac{1}{174} \, \sqrt{29}{\left (5 \, \sqrt{29} \log \left (3 \, x^{2} - 4 \, x + 11\right ) + 206 \, \arctan \left (\frac{1}{29} \, \sqrt{29}{\left (3 \, x - 2\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/174*sqrt(29)*(5*sqrt(29)*log(3*x^2 - 4*x + 11) + 206*arctan(1/29*sqrt(29)*(3*x
- 2)))

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Sympy [A]  time = 0.252585, size = 44, normalized size = 1.19 \[ \frac{5 \log{\left (x^{2} - \frac{4 x}{3} + \frac{11}{3} \right )}}{6} + \frac{103 \sqrt{29} \operatorname{atan}{\left (\frac{3 \sqrt{29} x}{29} - \frac{2 \sqrt{29}}{29} \right )}}{87} \]

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/XCAS [A]  time = 0.261712, size = 41, normalized size = 1.11 \[ \frac{103}{87} \, \sqrt{29} \arctan \left (\frac{1}{29} \, \sqrt{29}{\left (3 \, x - 2\right )}\right ) + \frac{5}{6} \,{\rm ln}\left (3 \, x^{2} - 4 \, x + 11\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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