3.171 \(\int \frac{1+16 x}{(5+x)^2 (-3+2 x) \left (1+x+x^2\right )} \, dx\)

Optimal. Leaf size=60 \[ -\frac{481 \log \left (x^2+x+1\right )}{5586}-\frac{79}{273 (x+5)}+\frac{200 \log (3-2 x)}{3211}+\frac{2731 \log (x+5)}{24843}+\frac{451 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2793 \sqrt{3}} \]

[Out]

-79/(273*(5 + x)) + (451*ArcTan[(1 + 2*x)/Sqrt[3]])/(2793*Sqrt[3]) + (200*Log[3
- 2*x])/3211 + (2731*Log[5 + x])/24843 - (481*Log[1 + x + x^2])/5586

_______________________________________________________________________________________

Rubi [A]  time = 0.401882, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{481 \log \left (x^2+x+1\right )}{5586}-\frac{79}{273 (x+5)}+\frac{200 \log (3-2 x)}{3211}+\frac{2731 \log (x+5)}{24843}+\frac{451 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2793 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 16*x)/((5 + x)^2*(-3 + 2*x)*(1 + x + x^2)),x]

[Out]

-79/(273*(5 + x)) + (451*ArcTan[(1 + 2*x)/Sqrt[3]])/(2793*Sqrt[3]) + (200*Log[3
- 2*x])/3211 + (2731*Log[5 + x])/24843 - (481*Log[1 + x + x^2])/5586

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 65.3073, size = 60, normalized size = 1. \[ \frac{200 \log{\left (- 2 x + 3 \right )}}{3211} + \frac{2731 \log{\left (x + 5 \right )}}{24843} - \frac{481 \log{\left (x^{2} + x + 1 \right )}}{5586} + \frac{451 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{8379} - \frac{79}{273 \left (x + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+16*x)/(5+x)**2/(-3+2*x)/(x**2+x+1),x)

[Out]

200*log(-2*x + 3)/3211 + 2731*log(x + 5)/24843 - 481*log(x**2 + x + 1)/5586 + 45
1*sqrt(3)*atan(sqrt(3)*(2*x/3 + 1/3))/8379 - 79/(273*(x + 5))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0878075, size = 54, normalized size = 0.9 \[ \frac{-243867 \log \left (x^2+x+1\right )-\frac{819546}{x+5}+176400 \log (3-2 x)+311334 \log (x+5)+152438 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2832102} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 16*x)/((5 + x)^2*(-3 + 2*x)*(1 + x + x^2)),x]

[Out]

(-819546/(5 + x) + 152438*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + 176400*Log[3 - 2*x
] + 311334*Log[5 + x] - 243867*Log[1 + x + x^2])/2832102

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 48, normalized size = 0.8 \[ -{\frac{481\,\ln \left ({x}^{2}+x+1 \right ) }{5586}}+{\frac{451\,\sqrt{3}}{8379}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{79}{1365+273\,x}}+{\frac{2731\,\ln \left ( 5+x \right ) }{24843}}+{\frac{200\,\ln \left ( -3+2\,x \right ) }{3211}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+16*x)/(5+x)^2/(-3+2*x)/(x^2+x+1),x)

[Out]

-481/5586*ln(x^2+x+1)+451/8379*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-79/273/(5+x)+
2731/24843*ln(5+x)+200/3211*ln(-3+2*x)

_______________________________________________________________________________________

Maxima [A]  time = 1.52683, size = 63, normalized size = 1.05 \[ \frac{451}{8379} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{79}{273 \,{\left (x + 5\right )}} - \frac{481}{5586} \, \log \left (x^{2} + x + 1\right ) + \frac{200}{3211} \, \log \left (2 \, x - 3\right ) + \frac{2731}{24843} \, \log \left (x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x + 1)/((x^2 + x + 1)*(2*x - 3)*(x + 5)^2),x, algorithm="maxima")

[Out]

451/8379*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 79/273/(x + 5) - 481/5586*log(x
^2 + x + 1) + 200/3211*log(2*x - 3) + 2731/24843*log(x + 5)

_______________________________________________________________________________________

Fricas [A]  time = 0.204922, size = 99, normalized size = 1.65 \[ -\frac{\sqrt{3}{\left (81289 \, \sqrt{3}{\left (x + 5\right )} \log \left (x^{2} + x + 1\right ) - 58800 \, \sqrt{3}{\left (x + 5\right )} \log \left (2 \, x - 3\right ) - 103778 \, \sqrt{3}{\left (x + 5\right )} \log \left (x + 5\right ) - 152438 \,{\left (x + 5\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 273182 \, \sqrt{3}\right )}}{2832102 \,{\left (x + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x + 1)/((x^2 + x + 1)*(2*x - 3)*(x + 5)^2),x, algorithm="fricas")

[Out]

-1/2832102*sqrt(3)*(81289*sqrt(3)*(x + 5)*log(x^2 + x + 1) - 58800*sqrt(3)*(x +
5)*log(2*x - 3) - 103778*sqrt(3)*(x + 5)*log(x + 5) - 152438*(x + 5)*arctan(1/3*
sqrt(3)*(2*x + 1)) + 273182*sqrt(3))/(x + 5)

_______________________________________________________________________________________

Sympy [A]  time = 0.364901, size = 63, normalized size = 1.05 \[ \frac{200 \log{\left (x - \frac{3}{2} \right )}}{3211} + \frac{2731 \log{\left (x + 5 \right )}}{24843} - \frac{481 \log{\left (x^{2} + x + 1 \right )}}{5586} + \frac{451 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{8379} - \frac{79}{273 x + 1365} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+16*x)/(5+x)**2/(-3+2*x)/(x**2+x+1),x)

[Out]

200*log(x - 3/2)/3211 + 2731*log(x + 5)/24843 - 481*log(x**2 + x + 1)/5586 + 451
*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/8379 - 79/(273*x + 1365)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222988, size = 81, normalized size = 1.35 \[ \frac{451}{8379} \, \sqrt{3} \arctan \left (-\sqrt{3}{\left (\frac{14}{x + 5} - 3\right )}\right ) - \frac{79}{273 \,{\left (x + 5\right )}} - \frac{481}{5586} \,{\rm ln}\left (-\frac{9}{x + 5} + \frac{21}{{\left (x + 5\right )}^{2}} + 1\right ) + \frac{200}{3211} \,{\rm ln}\left ({\left | -\frac{13}{x + 5} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((16*x + 1)/((x^2 + x + 1)*(2*x - 3)*(x + 5)^2),x, algorithm="giac")

[Out]

451/8379*sqrt(3)*arctan(-sqrt(3)*(14/(x + 5) - 3)) - 79/273/(x + 5) - 481/5586*l
n(-9/(x + 5) + 21/(x + 5)^2 + 1) + 200/3211*ln(abs(-13/(x + 5) + 2))