3.61 \(\int \frac{3+12 x+20 x^2}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx\)

Optimal. Leaf size=59 \[ \frac{\tan ^{-1}\left (\frac{800 x^3-40 x^2+30 x+57}{6 \sqrt{11}}\right )}{2 \sqrt{11}}-\frac{\tan ^{-1}\left (\frac{7-40 x}{5 \sqrt{11}}\right )}{2 \sqrt{11}} \]

[Out]

-ArcTan[(7 - 40*x)/(5*Sqrt[11])]/(2*Sqrt[11]) + ArcTan[(57 + 30*x - 40*x^2 + 800
*x^3)/(6*Sqrt[11])]/(2*Sqrt[11])

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Rubi [A]  time = 0.0595764, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03 \[ \frac{\tan ^{-1}\left (\frac{800 x^3-40 x^2+30 x+57}{6 \sqrt{11}}\right )}{2 \sqrt{11}}-\frac{\tan ^{-1}\left (\frac{7-40 x}{5 \sqrt{11}}\right )}{2 \sqrt{11}} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 12*x + 20*x^2)/(9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4),x]

[Out]

-ArcTan[(7 - 40*x)/(5*Sqrt[11])]/(2*Sqrt[11]) + ArcTan[(57 + 30*x - 40*x^2 + 800
*x^3)/(6*Sqrt[11])]/(2*Sqrt[11])

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Rubi in Sympy [A]  time = 7.63274, size = 56, normalized size = 0.95 \[ \frac{\sqrt{11} \operatorname{atan}{\left (\sqrt{11} \left (\frac{8 x}{11} - \frac{7}{55}\right ) \right )}}{22} + \frac{\sqrt{11} \operatorname{atan}{\left (\sqrt{11} \left (\frac{400 x^{3}}{33} - \frac{20 x^{2}}{33} + \frac{5 x}{11} + \frac{19}{22}\right ) \right )}}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((20*x**2+12*x+3)/(320*x**4+80*x**3-12*x**2+24*x+9),x)

[Out]

sqrt(11)*atan(sqrt(11)*(8*x/11 - 7/55))/22 + sqrt(11)*atan(sqrt(11)*(400*x**3/33
- 20*x**2/33 + 5*x/11 + 19/22))/22

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Mathematica [C]  time = 0.0273115, size = 86, normalized size = 1.46 \[ \frac{1}{8} \text{RootSum}\left [320 \text{$\#$1}^4+80 \text{$\#$1}^3-12 \text{$\#$1}^2+24 \text{$\#$1}+9\&,\frac{20 \text{$\#$1}^2 \log (x-\text{$\#$1})+12 \text{$\#$1} \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{160 \text{$\#$1}^3+30 \text{$\#$1}^2-3 \text{$\#$1}+3}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 12*x + 20*x^2)/(9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4),x]

[Out]

RootSum[9 + 24*#1 - 12*#1^2 + 80*#1^3 + 320*#1^4 & , (3*Log[x - #1] + 12*Log[x -
#1]*#1 + 20*Log[x - #1]*#1^2)/(3 - 3*#1 + 30*#1^2 + 160*#1^3) & ]/8

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Maple [A]  time = 0.043, size = 52, normalized size = 0.9 \[{\frac{\sqrt{11}}{22}\arctan \left ({\frac{ \left ( 40\,x-7 \right ) \sqrt{11}}{55}} \right ) }+{\frac{\sqrt{11}}{22}\arctan \left ( -{\frac{20\,\sqrt{11}{x}^{2}}{33}}+{\frac{5\,\sqrt{11}x}{11}}+{\frac{19\,\sqrt{11}}{22}}+{\frac{400\,\sqrt{11}{x}^{3}}{33}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((20*x^2+12*x+3)/(320*x^4+80*x^3-12*x^2+24*x+9),x)

[Out]

1/22*11^(1/2)*arctan(1/55*(40*x-7)*11^(1/2))+1/22*11^(1/2)*arctan(-20/33*11^(1/2
)*x^2+5/11*11^(1/2)*x+19/22*11^(1/2)+400/33*11^(1/2)*x^3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{20 \, x^{2} + 12 \, x + 3}{320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((20*x^2 + 12*x + 3)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9),x, algorithm="maxima")

[Out]

integrate((20*x^2 + 12*x + 3)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9), x)

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Fricas [A]  time = 0.195492, size = 51, normalized size = 0.86 \[ \frac{1}{22} \, \sqrt{11}{\left (\arctan \left (\frac{1}{66} \, \sqrt{11}{\left (800 \, x^{3} - 40 \, x^{2} + 30 \, x + 57\right )}\right ) + \arctan \left (\frac{1}{55} \, \sqrt{11}{\left (40 \, x - 7\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((20*x^2 + 12*x + 3)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9),x, algorithm="fricas")

[Out]

1/22*sqrt(11)*(arctan(1/66*sqrt(11)*(800*x^3 - 40*x^2 + 30*x + 57)) + arctan(1/5
5*sqrt(11)*(40*x - 7)))

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Sympy [A]  time = 0.212998, size = 73, normalized size = 1.24 \[ \frac{\sqrt{11} \left (2 \operatorname{atan}{\left (\frac{8 \sqrt{11} x}{11} - \frac{7 \sqrt{11}}{55} \right )} + 2 \operatorname{atan}{\left (\frac{400 \sqrt{11} x^{3}}{33} - \frac{20 \sqrt{11} x^{2}}{33} + \frac{5 \sqrt{11} x}{11} + \frac{19 \sqrt{11}}{22} \right )}\right )}{44} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((20*x**2+12*x+3)/(320*x**4+80*x**3-12*x**2+24*x+9),x)

[Out]

sqrt(11)*(2*atan(8*sqrt(11)*x/11 - 7*sqrt(11)/55) + 2*atan(400*sqrt(11)*x**3/33
- 20*sqrt(11)*x**2/33 + 5*sqrt(11)*x/11 + 19*sqrt(11)/22))/44

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{20 \, x^{2} + 12 \, x + 3}{320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((20*x^2 + 12*x + 3)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9),x, algorithm="giac")

[Out]

integrate((20*x^2 + 12*x + 3)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9), x)