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

Optimal. Leaf size=63 \[ \frac{121 (19-7 x)}{184 \left (2 x^2-x+3\right )}+\frac{55}{8} \log \left (2 x^2-x+3\right )+\frac{25 x}{4}+\frac{1859 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{92 \sqrt{23}} \]

[Out]

(25*x)/4 + (121*(19 - 7*x))/(184*(3 - x + 2*x^2)) + (1859*ArcTan[(1 - 4*x)/Sqrt[
23]])/(92*Sqrt[23]) + (55*Log[3 - x + 2*x^2])/8

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Rubi [A]  time = 0.110354, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{121 (19-7 x)}{184 \left (2 x^2-x+3\right )}+\frac{55}{8} \log \left (2 x^2-x+3\right )+\frac{25 x}{4}+\frac{1859 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{92 \sqrt{23}} \]

Antiderivative was successfully verified.

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

[Out]

(25*x)/4 + (121*(19 - 7*x))/(184*(3 - x + 2*x^2)) + (1859*ArcTan[(1 - 4*x)/Sqrt[
23]])/(92*Sqrt[23]) + (55*Log[3 - x + 2*x^2])/8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{50 x^{3}}{23} - \frac{\left (- 4 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )^{2}}{23 \left (2 x^{2} - x + 3\right )} + \frac{55 \log{\left (2 x^{2} - x + 3 \right )}}{8} - \frac{1859 \sqrt{23} \operatorname{atan}{\left (\sqrt{23} \left (\frac{4 x}{23} - \frac{1}{23}\right ) \right )}}{2116} - \frac{\int \left (- \frac{279}{2}\right )\, dx}{23} - \frac{145 \int x\, dx}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-50*x**3/23 - (-4*x + 1)*(5*x**2 + 3*x + 2)**2/(23*(2*x**2 - x + 3)) + 55*log(2*
x**2 - x + 3)/8 - 1859*sqrt(23)*atan(sqrt(23)*(4*x/23 - 1/23))/2116 - Integral(-
279/2, x)/23 - 145*Integral(x, x)/23

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Mathematica [A]  time = 0.0592621, size = 63, normalized size = 1. \[ -\frac{121 (7 x-19)}{184 \left (2 x^2-x+3\right )}+\frac{55}{8} \log \left (2 x^2-x+3\right )+\frac{25 x}{4}-\frac{1859 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{92 \sqrt{23}} \]

Antiderivative was successfully verified.

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

[Out]

(25*x)/4 - (121*(-19 + 7*x))/(184*(3 - x + 2*x^2)) - (1859*ArcTan[(-1 + 4*x)/Sqr
t[23]])/(92*Sqrt[23]) + (55*Log[3 - x + 2*x^2])/8

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Maple [A]  time = 0.009, size = 51, normalized size = 0.8 \[{\frac{25\,x}{4}}+{\frac{11}{4} \left ( -{\frac{77\,x}{92}}+{\frac{209}{92}} \right ) \left ({x}^{2}-{\frac{x}{2}}+{\frac{3}{2}} \right ) ^{-1}}+{\frac{55\,\ln \left ( 4\,{x}^{2}-2\,x+6 \right ) }{8}}-{\frac{1859\,\sqrt{23}}{2116}\arctan \left ({\frac{ \left ( 8\,x-2 \right ) \sqrt{23}}{46}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/4*x+11/4*(-77/92*x+209/92)/(x^2-1/2*x+3/2)+55/8*ln(4*x^2-2*x+6)-1859/2116*23^
(1/2)*arctan(1/46*(8*x-2)*23^(1/2))

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Maxima [A]  time = 0.769517, size = 70, normalized size = 1.11 \[ -\frac{1859}{2116} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{25}{4} \, x - \frac{121 \,{\left (7 \, x - 19\right )}}{184 \,{\left (2 \, x^{2} - x + 3\right )}} + \frac{55}{8} \, \log \left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1859/2116*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 25/4*x - 121/184*(7*x - 19
)/(2*x^2 - x + 3) + 55/8*log(2*x^2 - x + 3)

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Fricas [A]  time = 0.260563, size = 116, normalized size = 1.84 \[ \frac{\sqrt{23}{\left (1265 \, \sqrt{23}{\left (2 \, x^{2} - x + 3\right )} \log \left (2 \, x^{2} - x + 3\right ) - 3718 \,{\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \sqrt{23}{\left (2300 \, x^{3} - 1150 \, x^{2} + 2603 \, x + 2299\right )}\right )}}{4232 \,{\left (2 \, x^{2} - x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4232*sqrt(23)*(1265*sqrt(23)*(2*x^2 - x + 3)*log(2*x^2 - x + 3) - 3718*(2*x^2
- x + 3)*arctan(1/23*sqrt(23)*(4*x - 1)) + sqrt(23)*(2300*x^3 - 1150*x^2 + 2603*
x + 2299))/(2*x^2 - x + 3)

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Sympy [A]  time = 0.208861, size = 61, normalized size = 0.97 \[ \frac{25 x}{4} - \frac{847 x - 2299}{368 x^{2} - 184 x + 552} + \frac{55 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{8} - \frac{1859 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{2116} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*x/4 - (847*x - 2299)/(368*x**2 - 184*x + 552) + 55*log(x**2 - x/2 + 3/2)/8 -
1859*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/2116

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GIAC/XCAS [A]  time = 0.265023, size = 70, normalized size = 1.11 \[ -\frac{1859}{2116} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{25}{4} \, x - \frac{121 \,{\left (7 \, x - 19\right )}}{184 \,{\left (2 \, x^{2} - x + 3\right )}} + \frac{55}{8} \,{\rm ln}\left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1859/2116*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 25/4*x - 121/184*(7*x - 19
)/(2*x^2 - x + 3) + 55/8*ln(2*x^2 - x + 3)