3.2299 \(\int \frac{a+b x}{(1+x)^3 \left (1-x+x^2\right )^3} \, dx\)

Optimal. Leaf size=101 \[ \frac{x (a+b x)}{6 \left (x^3+1\right )^2}+\frac{x (5 a+4 b x)}{18 \left (x^3+1\right )}-\frac{1}{54} (5 a-2 b) \log \left (x^2-x+1\right )+\frac{1}{27} (5 a-2 b) \log (x+1)-\frac{(5 a+2 b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{9 \sqrt{3}} \]

[Out]

(x*(a + b*x))/(6*(1 + x^3)^2) + (x*(5*a + 4*b*x))/(18*(1 + x^3)) - ((5*a + 2*b)*
ArcTan[(1 - 2*x)/Sqrt[3]])/(9*Sqrt[3]) + ((5*a - 2*b)*Log[1 + x])/27 - ((5*a - 2
*b)*Log[1 - x + x^2])/54

_______________________________________________________________________________________

Rubi [A]  time = 0.183888, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{x (a+b x)}{6 \left (x^3+1\right )^2}+\frac{x (5 a+4 b x)}{18 \left (x^3+1\right )}-\frac{1}{54} (5 a-2 b) \log \left (x^2-x+1\right )+\frac{1}{27} (5 a-2 b) \log (x+1)-\frac{(5 a+2 b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x))/(6*(1 + x^3)^2) + (x*(5*a + 4*b*x))/(18*(1 + x^3)) - ((5*a + 2*b)*
ArcTan[(1 - 2*x)/Sqrt[3]])/(9*Sqrt[3]) + ((5*a - 2*b)*Log[1 + x])/27 - ((5*a - 2
*b)*Log[1 - x + x^2])/54

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.6193, size = 92, normalized size = 0.91 \[ \frac{x \left (a + b x\right )}{6 \left (x^{3} + 1\right )^{2}} + \frac{x \left (5 a + 4 b x\right )}{18 \left (x^{3} + 1\right )} - \left (\frac{5 a}{54} - \frac{b}{27}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{5 a}{27} - \frac{2 b}{27}\right ) \log{\left (x + 1 \right )} + \frac{\sqrt{3} \left (5 a + 2 b\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(1+x)**3/(x**2-x+1)**3,x)

[Out]

x*(a + b*x)/(6*(x**3 + 1)**2) + x*(5*a + 4*b*x)/(18*(x**3 + 1)) - (5*a/54 - b/27
)*log(x**2 - x + 1) + (5*a/27 - 2*b/27)*log(x + 1) + sqrt(3)*(5*a + 2*b)*atan(sq
rt(3)*(2*x/3 - 1/3))/27

_______________________________________________________________________________________

Mathematica [A]  time = 0.115268, size = 94, normalized size = 0.93 \[ \frac{1}{54} \left (\frac{9 x (a+b x)}{\left (x^3+1\right )^2}+\frac{3 x (5 a+4 b x)}{x^3+1}+(2 b-5 a) \log \left (x^2-x+1\right )+2 (5 a-2 b) \log (x+1)+2 \sqrt{3} (5 a+2 b) \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((9*x*(a + b*x))/(1 + x^3)^2 + (3*x*(5*a + 4*b*x))/(1 + x^3) + 2*Sqrt[3]*(5*a +
2*b)*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*(5*a - 2*b)*Log[1 + x] + (-5*a + 2*b)*Log[1
- x + x^2])/54

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 154, normalized size = 1.5 \[ -{\frac{1}{27\, \left ({x}^{2}-x+1 \right ) ^{2}} \left ( \left ( -3\,a-4\,b \right ){x}^{3}+ \left ( a+{\frac{13\,b}{2}} \right ){x}^{2}+ \left ( -a-8\,b \right ) x-{\frac{7\,a}{2}}+{\frac{5\,b}{2}} \right ) }-{\frac{5\,\ln \left ({x}^{2}-x+1 \right ) a}{54}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) b}{27}}+{\frac{5\,\sqrt{3}a}{27}\arctan \left ({\frac{ \left ( -1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,b\sqrt{3}}{27}\arctan \left ({\frac{ \left ( -1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{a}{54\, \left ( 1+x \right ) ^{2}}}+{\frac{b}{54\, \left ( 1+x \right ) ^{2}}}-{\frac{2\,\ln \left ( 1+x \right ) b}{27}}+{\frac{5\,\ln \left ( 1+x \right ) a}{27}}+{\frac{2\,b}{27+27\,x}}-{\frac{a}{9+9\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*((-3*a-4*b)*x^3+(a+13/2*b)*x^2+(-a-8*b)*x-7/2*a+5/2*b)/(x^2-x+1)^2-5/54*ln
(x^2-x+1)*a+1/27*ln(x^2-x+1)*b+5/27*3^(1/2)*arctan(1/3*(-1+2*x)*3^(1/2))*a+2/27*
3^(1/2)*arctan(1/3*(-1+2*x)*3^(1/2))*b-1/54/(1+x)^2*a+1/54/(1+x)^2*b-2/27*ln(1+x
)*b+5/27*ln(1+x)*a+2/27/(1+x)*b-1/9/(1+x)*a

_______________________________________________________________________________________

Maxima [A]  time = 0.767708, size = 124, normalized size = 1.23 \[ \frac{1}{27} \, \sqrt{3}{\left (5 \, a + 2 \, b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{54} \,{\left (5 \, a - 2 \, b\right )} \log \left (x^{2} - x + 1\right ) + \frac{1}{27} \,{\left (5 \, a - 2 \, b\right )} \log \left (x + 1\right ) + \frac{4 \, b x^{5} + 5 \, a x^{4} + 7 \, b x^{2} + 8 \, a x}{18 \,{\left (x^{6} + 2 \, x^{3} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27*sqrt(3)*(5*a + 2*b)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/54*(5*a - 2*b)*log(x^
2 - x + 1) + 1/27*(5*a - 2*b)*log(x + 1) + 1/18*(4*b*x^5 + 5*a*x^4 + 7*b*x^2 + 8
*a*x)/(x^6 + 2*x^3 + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.273471, size = 231, normalized size = 2.29 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (5 \, a - 2 \, b\right )} x^{6} + 2 \,{\left (5 \, a - 2 \, b\right )} x^{3} + 5 \, a - 2 \, b\right )} \log \left (x^{2} - x + 1\right ) - 2 \, \sqrt{3}{\left ({\left (5 \, a - 2 \, b\right )} x^{6} + 2 \,{\left (5 \, a - 2 \, b\right )} x^{3} + 5 \, a - 2 \, b\right )} \log \left (x + 1\right ) - 6 \,{\left ({\left (5 \, a + 2 \, b\right )} x^{6} + 2 \,{\left (5 \, a + 2 \, b\right )} x^{3} + 5 \, a + 2 \, b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 3 \, \sqrt{3}{\left (4 \, b x^{5} + 5 \, a x^{4} + 7 \, b x^{2} + 8 \, a x\right )}\right )}}{162 \,{\left (x^{6} + 2 \, x^{3} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(sqrt(3)*((5*a - 2*b)*x^6 + 2*(5*a - 2*b)*x^3 + 5*a - 2*b)*log(x^
2 - x + 1) - 2*sqrt(3)*((5*a - 2*b)*x^6 + 2*(5*a - 2*b)*x^3 + 5*a - 2*b)*log(x +
1) - 6*((5*a + 2*b)*x^6 + 2*(5*a + 2*b)*x^3 + 5*a + 2*b)*arctan(1/3*sqrt(3)*(2*
x - 1)) - 3*sqrt(3)*(4*b*x^5 + 5*a*x^4 + 7*b*x^2 + 8*a*x))/(x^6 + 2*x^3 + 1)

_______________________________________________________________________________________

Sympy [A]  time = 2.76651, size = 292, normalized size = 2.89 \[ \frac{\left (5 a - 2 b\right ) \log{\left (x + \frac{25 a^{2} \left (5 a - 2 b\right ) + 40 a b^{2} + 2 b \left (5 a - 2 b\right )^{2}}{125 a^{3} + 8 b^{3}} \right )}}{27} + \left (- \frac{5 a}{54} + \frac{b}{27} - \frac{\sqrt{3} i \left (5 a + 2 b\right )}{54}\right ) \log{\left (x + \frac{675 a^{2} \left (- \frac{5 a}{54} + \frac{b}{27} - \frac{\sqrt{3} i \left (5 a + 2 b\right )}{54}\right ) + 40 a b^{2} + 1458 b \left (- \frac{5 a}{54} + \frac{b}{27} - \frac{\sqrt{3} i \left (5 a + 2 b\right )}{54}\right )^{2}}{125 a^{3} + 8 b^{3}} \right )} + \left (- \frac{5 a}{54} + \frac{b}{27} + \frac{\sqrt{3} i \left (5 a + 2 b\right )}{54}\right ) \log{\left (x + \frac{675 a^{2} \left (- \frac{5 a}{54} + \frac{b}{27} + \frac{\sqrt{3} i \left (5 a + 2 b\right )}{54}\right ) + 40 a b^{2} + 1458 b \left (- \frac{5 a}{54} + \frac{b}{27} + \frac{\sqrt{3} i \left (5 a + 2 b\right )}{54}\right )^{2}}{125 a^{3} + 8 b^{3}} \right )} + \frac{5 a x^{4} + 8 a x + 4 b x^{5} + 7 b x^{2}}{18 x^{6} + 36 x^{3} + 18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(1+x)**3/(x**2-x+1)**3,x)

[Out]

(5*a - 2*b)*log(x + (25*a**2*(5*a - 2*b) + 40*a*b**2 + 2*b*(5*a - 2*b)**2)/(125*
a**3 + 8*b**3))/27 + (-5*a/54 + b/27 - sqrt(3)*I*(5*a + 2*b)/54)*log(x + (675*a*
*2*(-5*a/54 + b/27 - sqrt(3)*I*(5*a + 2*b)/54) + 40*a*b**2 + 1458*b*(-5*a/54 + b
/27 - sqrt(3)*I*(5*a + 2*b)/54)**2)/(125*a**3 + 8*b**3)) + (-5*a/54 + b/27 + sqr
t(3)*I*(5*a + 2*b)/54)*log(x + (675*a**2*(-5*a/54 + b/27 + sqrt(3)*I*(5*a + 2*b)
/54) + 40*a*b**2 + 1458*b*(-5*a/54 + b/27 + sqrt(3)*I*(5*a + 2*b)/54)**2)/(125*a
**3 + 8*b**3)) + (5*a*x**4 + 8*a*x + 4*b*x**5 + 7*b*x**2)/(18*x**6 + 36*x**3 + 1
8)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.340801, size = 119, normalized size = 1.18 \[ \frac{1}{27} \, \sqrt{3}{\left (5 \, a + 2 \, b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{54} \,{\left (5 \, a - 2 \, b\right )}{\rm ln}\left (x^{2} - x + 1\right ) + \frac{1}{27} \,{\left (5 \, a - 2 \, b\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{4 \, b x^{5} + 5 \, a x^{4} + 7 \, b x^{2} + 8 \, a x}{18 \,{\left (x^{3} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27*sqrt(3)*(5*a + 2*b)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/54*(5*a - 2*b)*ln(x^2
- x + 1) + 1/27*(5*a - 2*b)*ln(abs(x + 1)) + 1/18*(4*b*x^5 + 5*a*x^4 + 7*b*x^2
+ 8*a*x)/(x^3 + 1)^2