3.40 \(\int \frac{2+3 x+5 x^2}{3-x+2 x^2} \, dx\)

Optimal. Leaf size=42 \[ \frac{11}{8} \log \left (2 x^2-x+3\right )+\frac{5 x}{2}+\frac{33 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{23}} \]

[Out]

(5*x)/2 + (33*ArcTan[(1 - 4*x)/Sqrt[23]])/(4*Sqrt[23]) + (11*Log[3 - x + 2*x^2])
/8

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Rubi [A]  time = 0.0672588, antiderivative size = 42, 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 \[ \frac{11}{8} \log \left (2 x^2-x+3\right )+\frac{5 x}{2}+\frac{33 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2),x]

[Out]

(5*x)/2 + (33*ArcTan[(1 - 4*x)/Sqrt[23]])/(4*Sqrt[23]) + (11*Log[3 - x + 2*x^2])
/8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{11 \log{\left (2 x^{2} - x + 3 \right )}}{8} - \frac{33 \sqrt{23} \operatorname{atan}{\left (\sqrt{23} \left (\frac{4 x}{23} - \frac{1}{23}\right ) \right )}}{92} + \int \frac{5}{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11*log(2*x**2 - x + 3)/8 - 33*sqrt(23)*atan(sqrt(23)*(4*x/23 - 1/23))/92 + Integ
ral(5/2, x)

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Mathematica [A]  time = 0.0197414, size = 42, normalized size = 1. \[ \frac{11}{8} \log \left (2 x^2-x+3\right )+\frac{5 x}{2}-\frac{33 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{4 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2),x]

[Out]

(5*x)/2 - (33*ArcTan[(-1 + 4*x)/Sqrt[23]])/(4*Sqrt[23]) + (11*Log[3 - x + 2*x^2]
)/8

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Maple [A]  time = 0.004, size = 34, normalized size = 0.8 \[{\frac{5\,x}{2}}+{\frac{11\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{8}}-{\frac{33\,\sqrt{23}}{92}\arctan \left ({\frac{ \left ( 4\,x-1 \right ) \sqrt{23}}{23}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+3*x+2)/(2*x^2-x+3),x)

[Out]

5/2*x+11/8*ln(2*x^2-x+3)-33/92*23^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))

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Maxima [A]  time = 0.769842, size = 45, normalized size = 1.07 \[ -\frac{33}{92} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{5}{2} \, x + \frac{11}{8} \, \log \left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 3*x + 2)/(2*x^2 - x + 3),x, algorithm="maxima")

[Out]

-33/92*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 5/2*x + 11/8*log(2*x^2 - x + 3
)

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Fricas [A]  time = 0.268168, size = 55, normalized size = 1.31 \[ \frac{1}{184} \, \sqrt{23}{\left (20 \, \sqrt{23} x + 11 \, \sqrt{23} \log \left (2 \, x^{2} - x + 3\right ) - 66 \, \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/184*sqrt(23)*(20*sqrt(23)*x + 11*sqrt(23)*log(2*x^2 - x + 3) - 66*arctan(1/23*
sqrt(23)*(4*x - 1)))

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Sympy [A]  time = 0.135538, size = 46, normalized size = 1.1 \[ \frac{5 x}{2} + \frac{11 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{8} - \frac{33 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{92} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+3*x+2)/(2*x**2-x+3),x)

[Out]

5*x/2 + 11*log(x**2 - x/2 + 3/2)/8 - 33*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)
/23)/92

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GIAC/XCAS [A]  time = 0.264574, size = 45, normalized size = 1.07 \[ -\frac{33}{92} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{5}{2} \, x + \frac{11}{8} \,{\rm ln}\left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-33/92*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 5/2*x + 11/8*ln(2*x^2 - x + 3)