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

Optimal. Leaf size=56 \[ \frac{25 x^3}{6}+\frac{85 x^2}{8}-\frac{363}{32} \log \left (2 x^2-x+3\right )+\frac{51 x}{8}+\frac{847 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16 \sqrt{23}} \]

[Out]

(51*x)/8 + (85*x^2)/8 + (25*x^3)/6 + (847*ArcTan[(1 - 4*x)/Sqrt[23]])/(16*Sqrt[2
3]) - (363*Log[3 - x + 2*x^2])/32

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Rubi [A]  time = 0.0878965, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{25 x^3}{6}+\frac{85 x^2}{8}-\frac{363}{32} \log \left (2 x^2-x+3\right )+\frac{51 x}{8}+\frac{847 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16 \sqrt{23}} \]

Antiderivative was successfully verified.

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

[Out]

(51*x)/8 + (85*x^2)/8 + (25*x^3)/6 + (847*ArcTan[(1 - 4*x)/Sqrt[23]])/(16*Sqrt[2
3]) - (363*Log[3 - x + 2*x^2])/32

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Rubi in Sympy [A]  time = 27.1914, size = 58, normalized size = 1.04 \[ - \frac{x}{6} + \left (\frac{5 x}{6} + \frac{13}{8}\right ) \left (5 x^{2} + 3 x + 2\right ) - \frac{363 \log{\left (2 x^{2} - x + 3 \right )}}{32} - \frac{847 \sqrt{23} \operatorname{atan}{\left (\sqrt{23} \left (\frac{4 x}{23} - \frac{1}{23}\right ) \right )}}{368} \]

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),x)

[Out]

-x/6 + (5*x/6 + 13/8)*(5*x**2 + 3*x + 2) - 363*log(2*x**2 - x + 3)/32 - 847*sqrt
(23)*atan(sqrt(23)*(4*x/23 - 1/23))/368

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Mathematica [A]  time = 0.0317148, size = 52, normalized size = 0.93 \[ \frac{1}{24} x \left (100 x^2+255 x+153\right )-\frac{363}{32} \log \left (2 x^2-x+3\right )-\frac{847 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{16 \sqrt{23}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(153 + 255*x + 100*x^2))/24 - (847*ArcTan[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[23])
- (363*Log[3 - x + 2*x^2])/32

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Maple [A]  time = 0.005, size = 44, normalized size = 0.8 \[{\frac{25\,{x}^{3}}{6}}+{\frac{85\,{x}^{2}}{8}}+{\frac{51\,x}{8}}-{\frac{363\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{32}}-{\frac{847\,\sqrt{23}}{368}\arctan \left ({\frac{ \left ( 4\,x-1 \right ) \sqrt{23}}{23}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/6*x^3+85/8*x^2+51/8*x-363/32*ln(2*x^2-x+3)-847/368*23^(1/2)*arctan(1/23*(4*x-
1)*23^(1/2))

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Maxima [A]  time = 0.766298, size = 58, normalized size = 1.04 \[ \frac{25}{6} \, x^{3} + \frac{85}{8} \, x^{2} - \frac{847}{368} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{51}{8} \, x - \frac{363}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/6*x^3 + 85/8*x^2 - 847/368*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 51/8*x
- 363/32*log(2*x^2 - x + 3)

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Fricas [A]  time = 0.270894, size = 73, normalized size = 1.3 \[ \frac{1}{2208} \, \sqrt{23}{\left (4 \, \sqrt{23}{\left (100 \, x^{3} + 255 \, x^{2} + 153 \, x\right )} - 1089 \, \sqrt{23} \log \left (2 \, x^{2} - x + 3\right ) - 5082 \, \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2208*sqrt(23)*(4*sqrt(23)*(100*x^3 + 255*x^2 + 153*x) - 1089*sqrt(23)*log(2*x^
2 - x + 3) - 5082*arctan(1/23*sqrt(23)*(4*x - 1)))

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Sympy [A]  time = 0.153796, size = 60, normalized size = 1.07 \[ \frac{25 x^{3}}{6} + \frac{85 x^{2}}{8} + \frac{51 x}{8} - \frac{363 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{32} - \frac{847 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{368} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*x**3/6 + 85*x**2/8 + 51*x/8 - 363*log(x**2 - x/2 + 3/2)/32 - 847*sqrt(23)*ata
n(4*sqrt(23)*x/23 - sqrt(23)/23)/368

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GIAC/XCAS [A]  time = 0.264858, size = 58, normalized size = 1.04 \[ \frac{25}{6} \, x^{3} + \frac{85}{8} \, x^{2} - \frac{847}{368} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{51}{8} \, x - \frac{363}{32} \,{\rm ln}\left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/6*x^3 + 85/8*x^2 - 847/368*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 51/8*x
- 363/32*ln(2*x^2 - x + 3)