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

Optimal. Leaf size=43 \[ -\frac{11 (3 x+5)}{46 \left (2 x^2-x+3\right )}-\frac{82 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{23 \sqrt{23}} \]

[Out]

(-11*(5 + 3*x))/(46*(3 - x + 2*x^2)) - (82*ArcTan[(1 - 4*x)/Sqrt[23]])/(23*Sqrt[
23])

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Rubi [A]  time = 0.0488336, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{11 (3 x+5)}{46 \left (2 x^2-x+3\right )}-\frac{82 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{23 \sqrt{23}} \]

Antiderivative was successfully verified.

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

[Out]

(-11*(5 + 3*x))/(46*(3 - x + 2*x^2)) - (82*ArcTan[(1 - 4*x)/Sqrt[23]])/(23*Sqrt[
23])

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Rubi in Sympy [A]  time = 9.56137, size = 37, normalized size = 0.86 \[ - \frac{33 x + 55}{46 \left (2 x^{2} - x + 3\right )} + \frac{82 \sqrt{23} \operatorname{atan}{\left (\sqrt{23} \left (\frac{4 x}{23} - \frac{1}{23}\right ) \right )}}{529} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(33*x + 55)/(46*(2*x**2 - x + 3)) + 82*sqrt(23)*atan(sqrt(23)*(4*x/23 - 1/23))/
529

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Mathematica [A]  time = 0.0302141, size = 43, normalized size = 1. \[ \frac{82 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{23 \sqrt{23}}-\frac{11 (3 x+5)}{46 \left (2 x^2-x+3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-11*(5 + 3*x))/(46*(3 - x + 2*x^2)) + (82*ArcTan[(-1 + 4*x)/Sqrt[23]])/(23*Sqrt
[23])

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Maple [A]  time = 0.007, size = 34, normalized size = 0.8 \[{1 \left ( -{\frac{33\,x}{92}}-{\frac{55}{92}} \right ) \left ({x}^{2}-{\frac{x}{2}}+{\frac{3}{2}} \right ) ^{-1}}+{\frac{82\,\sqrt{23}}{529}\arctan \left ({\frac{ \left ( 8\,x-2 \right ) \sqrt{23}}{46}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-33/92*x-55/92)/(x^2-1/2*x+3/2)+82/529*23^(1/2)*arctan(1/46*(8*x-2)*23^(1/2))

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Maxima [A]  time = 0.765322, size = 49, normalized size = 1.14 \[ \frac{82}{529} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{11 \,{\left (3 \, x + 5\right )}}{46 \,{\left (2 \, x^{2} - x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

82/529*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) - 11/46*(3*x + 5)/(2*x^2 - x + 3
)

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Fricas [A]  time = 0.258337, size = 69, normalized size = 1.6 \[ \frac{\sqrt{23}{\left (164 \,{\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - 11 \, \sqrt{23}{\left (3 \, x + 5\right )}\right )}}{1058 \,{\left (2 \, x^{2} - x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1058*sqrt(23)*(164*(2*x^2 - x + 3)*arctan(1/23*sqrt(23)*(4*x - 1)) - 11*sqrt(2
3)*(3*x + 5))/(2*x^2 - x + 3)

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Sympy [A]  time = 0.170158, size = 41, normalized size = 0.95 \[ - \frac{33 x + 55}{92 x^{2} - 46 x + 138} + \frac{82 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{529} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(33*x + 55)/(92*x**2 - 46*x + 138) + 82*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23
)/23)/529

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GIAC/XCAS [A]  time = 0.26524, size = 49, normalized size = 1.14 \[ \frac{82}{529} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{11 \,{\left (3 \, x + 5\right )}}{46 \,{\left (2 \, x^{2} - x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

82/529*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) - 11/46*(3*x + 5)/(2*x^2 - x + 3
)