3.2298 \(\int \frac{a+b x}{(1+x)^2 \left (1-x+x^2\right )^2} \, dx\)

Optimal. Leaf size=79 \[ \frac{x (a+b x)}{3 \left (x^3+1\right )}-\frac{1}{18} (2 a-b) \log \left (x^2-x+1\right )+\frac{1}{9} (2 a-b) \log (x+1)-\frac{(2 a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

(x*(a + b*x))/(3*(1 + x^3)) - ((2*a + b)*ArcTan[(1 - 2*x)/Sqrt[3]])/(3*Sqrt[3])
+ ((2*a - b)*Log[1 + x])/9 - ((2*a - b)*Log[1 - x + x^2])/18

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Rubi [A]  time = 0.13984, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{x (a+b x)}{3 \left (x^3+1\right )}-\frac{1}{18} (2 a-b) \log \left (x^2-x+1\right )+\frac{1}{9} (2 a-b) \log (x+1)-\frac{(2 a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((1 + x)^2*(1 - x + x^2)^2),x]

[Out]

(x*(a + b*x))/(3*(1 + x^3)) - ((2*a + b)*ArcTan[(1 - 2*x)/Sqrt[3]])/(3*Sqrt[3])
+ ((2*a - b)*Log[1 + x])/9 - ((2*a - b)*Log[1 - x + x^2])/18

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Rubi in Sympy [A]  time = 25.3983, size = 70, normalized size = 0.89 \[ \frac{x \left (a + b x\right )}{3 \left (x^{3} + 1\right )} - \left (\frac{a}{9} - \frac{b}{18}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{2 a}{9} - \frac{b}{9}\right ) \log{\left (x + 1 \right )} + \frac{2 \sqrt{3} \left (a + \frac{b}{2}\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(1+x)**2/(x**2-x+1)**2,x)

[Out]

x*(a + b*x)/(3*(x**3 + 1)) - (a/9 - b/18)*log(x**2 - x + 1) + (2*a/9 - b/9)*log(
x + 1) + 2*sqrt(3)*(a + b/2)*atan(sqrt(3)*(2*x/3 - 1/3))/9

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Mathematica [A]  time = 0.0794367, size = 72, normalized size = 0.91 \[ \frac{1}{18} \left (\frac{6 x (a+b x)}{x^3+1}+(b-2 a) \log \left (x^2-x+1\right )+2 (2 a-b) \log (x+1)+2 \sqrt{3} (2 a+b) \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((1 + x)^2*(1 - x + x^2)^2),x]

[Out]

((6*x*(a + b*x))/(1 + x^3) + 2*Sqrt[3]*(2*a + b)*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*
(2*a - b)*Log[1 + x] + (-2*a + b)*Log[1 - x + x^2])/18

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Maple [A]  time = 0.017, size = 116, normalized size = 1.5 \[ -{\frac{ \left ( -a-2\,b \right ) x-a+b}{9\,{x}^{2}-9\,x+9}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) a}{9}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) b}{18}}+{\frac{2\,\sqrt{3}a}{9}\arctan \left ({\frac{ \left ( -1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{b\sqrt{3}}{9}\arctan \left ({\frac{ \left ( -1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{a}{9+9\,x}}+{\frac{b}{9+9\,x}}-{\frac{\ln \left ( 1+x \right ) b}{9}}+{\frac{2\,\ln \left ( 1+x \right ) a}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(1+x)^2/(x^2-x+1)^2,x)

[Out]

-1/9*((-a-2*b)*x-a+b)/(x^2-x+1)-1/9*ln(x^2-x+1)*a+1/18*ln(x^2-x+1)*b+2/9*3^(1/2)
*arctan(1/3*(-1+2*x)*3^(1/2))*a+1/9*3^(1/2)*arctan(1/3*(-1+2*x)*3^(1/2))*b-1/9/(
1+x)*a+1/9/(1+x)*b-1/9*ln(1+x)*b+2/9*ln(1+x)*a

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Maxima [A]  time = 0.766868, size = 96, normalized size = 1.22 \[ \frac{1}{9} \, \sqrt{3}{\left (2 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{18} \,{\left (2 \, a - b\right )} \log \left (x^{2} - x + 1\right ) + \frac{1}{9} \,{\left (2 \, a - b\right )} \log \left (x + 1\right ) + \frac{b x^{2} + a x}{3 \,{\left (x^{3} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((x^2 - x + 1)^2*(x + 1)^2),x, algorithm="maxima")

[Out]

1/9*sqrt(3)*(2*a + b)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/18*(2*a - b)*log(x^2 - x
+ 1) + 1/9*(2*a - b)*log(x + 1) + 1/3*(b*x^2 + a*x)/(x^3 + 1)

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Fricas [A]  time = 0.271972, size = 151, normalized size = 1.91 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (2 \, a - b\right )} x^{3} + 2 \, a - b\right )} \log \left (x^{2} - x + 1\right ) - 2 \, \sqrt{3}{\left ({\left (2 \, a - b\right )} x^{3} + 2 \, a - b\right )} \log \left (x + 1\right ) - 6 \,{\left ({\left (2 \, a + b\right )} x^{3} + 2 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 6 \, \sqrt{3}{\left (b x^{2} + a x\right )}\right )}}{54 \,{\left (x^{3} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((x^2 - x + 1)^2*(x + 1)^2),x, algorithm="fricas")

[Out]

-1/54*sqrt(3)*(sqrt(3)*((2*a - b)*x^3 + 2*a - b)*log(x^2 - x + 1) - 2*sqrt(3)*((
2*a - b)*x^3 + 2*a - b)*log(x + 1) - 6*((2*a + b)*x^3 + 2*a + b)*arctan(1/3*sqrt
(3)*(2*x - 1)) - 6*sqrt(3)*(b*x^2 + a*x))/(x^3 + 1)

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Sympy [A]  time = 2.40309, size = 238, normalized size = 3.01 \[ \frac{\left (2 a - b\right ) \log{\left (x + \frac{4 a^{2} \left (2 a - b\right ) + 4 a b^{2} + b \left (2 a - b\right )^{2}}{8 a^{3} + b^{3}} \right )}}{9} + \left (- \frac{a}{9} + \frac{b}{18} - \frac{\sqrt{3} i \left (2 a + b\right )}{18}\right ) \log{\left (x + \frac{36 a^{2} \left (- \frac{a}{9} + \frac{b}{18} - \frac{\sqrt{3} i \left (2 a + b\right )}{18}\right ) + 4 a b^{2} + 81 b \left (- \frac{a}{9} + \frac{b}{18} - \frac{\sqrt{3} i \left (2 a + b\right )}{18}\right )^{2}}{8 a^{3} + b^{3}} \right )} + \left (- \frac{a}{9} + \frac{b}{18} + \frac{\sqrt{3} i \left (2 a + b\right )}{18}\right ) \log{\left (x + \frac{36 a^{2} \left (- \frac{a}{9} + \frac{b}{18} + \frac{\sqrt{3} i \left (2 a + b\right )}{18}\right ) + 4 a b^{2} + 81 b \left (- \frac{a}{9} + \frac{b}{18} + \frac{\sqrt{3} i \left (2 a + b\right )}{18}\right )^{2}}{8 a^{3} + b^{3}} \right )} + \frac{a x + b x^{2}}{3 x^{3} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(1+x)**2/(x**2-x+1)**2,x)

[Out]

(2*a - b)*log(x + (4*a**2*(2*a - b) + 4*a*b**2 + b*(2*a - b)**2)/(8*a**3 + b**3)
)/9 + (-a/9 + b/18 - sqrt(3)*I*(2*a + b)/18)*log(x + (36*a**2*(-a/9 + b/18 - sqr
t(3)*I*(2*a + b)/18) + 4*a*b**2 + 81*b*(-a/9 + b/18 - sqrt(3)*I*(2*a + b)/18)**2
)/(8*a**3 + b**3)) + (-a/9 + b/18 + sqrt(3)*I*(2*a + b)/18)*log(x + (36*a**2*(-a
/9 + b/18 + sqrt(3)*I*(2*a + b)/18) + 4*a*b**2 + 81*b*(-a/9 + b/18 + sqrt(3)*I*(
2*a + b)/18)**2)/(8*a**3 + b**3)) + (a*x + b*x**2)/(3*x**3 + 3)

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GIAC/XCAS [A]  time = 0.267026, size = 136, normalized size = 1.72 \[ \frac{1}{9} \, \sqrt{3}{\left (2 \, a + b\right )} \arctan \left (-\sqrt{3}{\left (\frac{2}{x + 1} - 1\right )}\right ) - \frac{1}{18} \,{\left (2 \, a - b\right )}{\rm ln}\left (-\frac{3}{x + 1} + \frac{3}{{\left (x + 1\right )}^{2}} + 1\right ) - \frac{a}{9 \,{\left (x + 1\right )}} + \frac{b}{9 \,{\left (x + 1\right )}} - \frac{b + \frac{a - b}{x + 1}}{9 \,{\left (\frac{3}{x + 1} - \frac{3}{{\left (x + 1\right )}^{2}} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((x^2 - x + 1)^2*(x + 1)^2),x, algorithm="giac")

[Out]

1/9*sqrt(3)*(2*a + b)*arctan(-sqrt(3)*(2/(x + 1) - 1)) - 1/18*(2*a - b)*ln(-3/(x
+ 1) + 3/(x + 1)^2 + 1) - 1/9*a/(x + 1) + 1/9*b/(x + 1) - 1/9*(b + (a - b)/(x +
1))/(3/(x + 1) - 3/(x + 1)^2 - 1)