3.62 \(\int -\frac{84+576 x+400 x^2-2560 x^3}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx\)

Optimal. Leaf size=78 \[ -2 \sqrt{11} \tan ^{-1}\left (\frac{800 x^3-40 x^2+30 x+57}{6 \sqrt{11}}\right )+2 \log \left (320 x^4+80 x^3-12 x^2+24 x+9\right )+2 \sqrt{11} \tan ^{-1}\left (\frac{7-40 x}{5 \sqrt{11}}\right ) \]

[Out]

2*Sqrt[11]*ArcTan[(7 - 40*x)/(5*Sqrt[11])] - 2*Sqrt[11]*ArcTan[(57 + 30*x - 40*x
^2 + 800*x^3)/(6*Sqrt[11])] + 2*Log[9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4]

_______________________________________________________________________________________

Rubi [A]  time = 0.124319, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ -2 \sqrt{11} \tan ^{-1}\left (\frac{800 x^3-40 x^2+30 x+57}{6 \sqrt{11}}\right )+2 \log \left (320 x^4+80 x^3-12 x^2+24 x+9\right )+2 \sqrt{11} \tan ^{-1}\left (\frac{7-40 x}{5 \sqrt{11}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[-((84 + 576*x + 400*x^2 - 2560*x^3)/(9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4)),x]

[Out]

2*Sqrt[11]*ArcTan[(7 - 40*x)/(5*Sqrt[11])] - 2*Sqrt[11]*ArcTan[(57 + 30*x - 40*x
^2 + 800*x^3)/(6*Sqrt[11])] + 2*Log[9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.7712, size = 80, normalized size = 1.03 \[ 2 \log{\left (320 x^{4} + 80 x^{3} - 12 x^{2} + 24 x + 9 \right )} - 2 \sqrt{11} \operatorname{atan}{\left (\sqrt{11} \left (\frac{8 x}{11} - \frac{7}{55}\right ) \right )} - 2 \sqrt{11} \operatorname{atan}{\left (\sqrt{11} \left (\frac{400 x^{3}}{33} - \frac{20 x^{2}}{33} + \frac{5 x}{11} + \frac{19}{22}\right ) \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2560*x**3-400*x**2-576*x-84)/(320*x**4+80*x**3-12*x**2+24*x+9),x)

[Out]

2*log(320*x**4 + 80*x**3 - 12*x**2 + 24*x + 9) - 2*sqrt(11)*atan(sqrt(11)*(8*x/1
1 - 7/55)) - 2*sqrt(11)*atan(sqrt(11)*(400*x**3/33 - 20*x**2/33 + 5*x/11 + 19/22
))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0294842, size = 99, normalized size = 1.27 \[ \frac{1}{2} \text{RootSum}\left [320 \text{$\#$1}^4+80 \text{$\#$1}^3-12 \text{$\#$1}^2+24 \text{$\#$1}+9\&,\frac{640 \text{$\#$1}^3 \log (x-\text{$\#$1})-100 \text{$\#$1}^2 \log (x-\text{$\#$1})-144 \text{$\#$1} \log (x-\text{$\#$1})-21 \log (x-\text{$\#$1})}{160 \text{$\#$1}^3+30 \text{$\#$1}^2-3 \text{$\#$1}+3}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[-((84 + 576*x + 400*x^2 - 2560*x^3)/(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 & , (-21*Log[x - #1] - 144*Log[
x - #1]*#1 - 100*Log[x - #1]*#1^2 + 640*Log[x - #1]*#1^3)/(3 - 3*#1 + 30*#1^2 +
160*#1^3) & ]/2

_______________________________________________________________________________________

Maple [A]  time = 0.033, size = 75, normalized size = 1. \[ 2\,\ln \left ( 6400\,{x}^{4}+1600\,{x}^{3}-240\,{x}^{2}+480\,x+180 \right ) -2\,\sqrt{11}\arctan \left ( -{\frac{20\,\sqrt{11}{x}^{2}}{33}}+{\frac{5\,\sqrt{11}x}{11}}+{\frac{19\,\sqrt{11}}{22}}+{\frac{400\,\sqrt{11}{x}^{3}}{33}} \right ) -2\,\sqrt{11}\arctan \left ({\frac{ \left ( 40\,x-7 \right ) \sqrt{11}}{55}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2560*x^3-400*x^2-576*x-84)/(320*x^4+80*x^3-12*x^2+24*x+9),x)

[Out]

2*ln(6400*x^4+1600*x^3-240*x^2+480*x+180)-2*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)-2*11^(1/2)*arctan(1/55*(40*x
-7)*11^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ 4 \, \int \frac{640 \, x^{3} - 100 \, x^{2} - 144 \, x - 21}{320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4*(640*x^3 - 100*x^2 - 144*x - 21)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9),x, algorithm="maxima")

[Out]

4*integrate((640*x^3 - 100*x^2 - 144*x - 21)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x +
9), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.198484, size = 89, normalized size = 1.14 \[ -2 \, \sqrt{11} \arctan \left (\frac{1}{66} \, \sqrt{11}{\left (800 \, x^{3} - 40 \, x^{2} + 30 \, x + 57\right )}\right ) - 2 \, \sqrt{11} \arctan \left (\frac{1}{55} \, \sqrt{11}{\left (40 \, x - 7\right )}\right ) + 2 \, \log \left (320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4*(640*x^3 - 100*x^2 - 144*x - 21)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9),x, algorithm="fricas")

[Out]

-2*sqrt(11)*arctan(1/66*sqrt(11)*(800*x^3 - 40*x^2 + 30*x + 57)) - 2*sqrt(11)*ar
ctan(1/55*sqrt(11)*(40*x - 7)) + 2*log(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9)

_______________________________________________________________________________________

Sympy [A]  time = 0.218207, size = 100, normalized size = 1.28 \[ \sqrt{11} \left (- 2 \operatorname{atan}{\left (\frac{8 \sqrt{11} x}{11} - \frac{7 \sqrt{11}}{55} \right )} - 2 \operatorname{atan}{\left (\frac{400 \sqrt{11} x^{3}}{33} - \frac{20 \sqrt{11} x^{2}}{33} + \frac{5 \sqrt{11} x}{11} + \frac{19 \sqrt{11}}{22} \right )}\right ) + 2 \log{\left (x^{4} + \frac{x^{3}}{4} - \frac{3 x^{2}}{80} + \frac{3 x}{40} + \frac{9}{320} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2560*x**3-400*x**2-576*x-84)/(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)) + 2*log(x**4 + x**3/
4 - 3*x**2/80 + 3*x/40 + 9/320)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{4 \,{\left (640 \, x^{3} - 100 \, x^{2} - 144 \, x - 21\right )}}{320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4*(640*x^3 - 100*x^2 - 144*x - 21)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9),x, algorithm="giac")

[Out]

integrate(4*(640*x^3 - 100*x^2 - 144*x - 21)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x +
9), x)