3.23.100 \(\int \frac {a+b x}{(1+x)^3 (1-x+x^2)^3} \, dx\) [2300]

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

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {823, 1869, 1874, 31, 648, 632, 210, 642} \begin {gather*} -\frac {(5 a+2 b) \text {ArcTan}\left (\frac {1-2 x}{\sqrt {3}}\right )}{9 \sqrt {3}}+\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) \end {gather*}

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[
(d + e*x)^FracPart[p]*((a + b*x + c*x^2)^FracPart[p]/(a*d + c*e*x^3)^FracPart[p]), Int[(f + g*x)*(a*d + c*e*x^
3)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[m, p] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0]

Rule 1869

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] +
Dist[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b
}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rubi steps

\begin {align*} \int \frac {a+b x}{(1+x)^3 \left (1-x+x^2\right )^3} \, dx &=\int \frac {a+b x}{\left (1+x^3\right )^3} \, dx\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}-\frac {1}{6} \int \frac {-5 a-4 b x}{\left (1+x^3\right )^2} \, dx\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}+\frac {x (5 a+4 b x)}{18 \left (1+x^3\right )}+\frac {1}{18} \int \frac {10 a+4 b x}{1+x^3} \, dx\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}+\frac {x (5 a+4 b x)}{18 \left (1+x^3\right )}+\frac {1}{54} \int \frac {20 a+4 b+(-10 a+4 b) x}{1-x+x^2} \, dx+\frac {1}{27} (5 a-2 b) \int \frac {1}{1+x} \, dx\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}+\frac {x (5 a+4 b x)}{18 \left (1+x^3\right )}+\frac {1}{27} (5 a-2 b) \log (1+x)+\frac {1}{54} (-5 a+2 b) \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{18} (5 a+2 b) \int \frac {1}{1-x+x^2} \, dx\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}+\frac {x (5 a+4 b x)}{18 \left (1+x^3\right )}+\frac {1}{27} (5 a-2 b) \log (1+x)-\frac {1}{54} (5 a-2 b) \log \left (1-x+x^2\right )+\frac {1}{9} (-5 a-2 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=\frac {x (a+b x)}{6 \left (1+x^3\right )^2}+\frac {x (5 a+4 b x)}{18 \left (1+x^3\right )}-\frac {(5 a+2 b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{27} (5 a-2 b) \log (1+x)-\frac {1}{54} (5 a-2 b) \log \left (1-x+x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 94, normalized size = 0.93 \begin {gather*} \frac {1}{54} \left (\frac {9 x (a+b x)}{\left (1+x^3\right )^2}+\frac {3 x (5 a+4 b x)}{1+x^3}+2 \sqrt {3} (5 a+2 b) \tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )+2 (5 a-2 b) \log (1+x)+(-5 a+2 b) \log \left (1-x+x^2\right )\right ) \end {gather*}

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

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Maple [A]
time = 0.06, size = 131, normalized size = 1.30

method	result	size
risch	\(\frac {\frac {2}{9} b \,x^{5}+\frac {5}{18} a \,x^{4}+\frac {7}{18} b \,x^{2}+\frac {4}{9} a x}{\left (1+x \right )^{2} \left (x^{2}-x +1\right )^{2}}+\frac {5 \ln \left (1+x \right ) a}{27}-\frac {2 \ln \left (1+x \right ) b}{27}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{2}+\left (5 a -2 b \right ) \textit {\_Z} +25 a^{2}+10 a b +4 b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (5 \textit {\_R} a +4 b^{2}\right ) x +\textit {\_R}^{2}+10 a b \right )\right )}{27}\)	\(111\)
default	\(-\frac {\frac {a}{27}-\frac {b}{27}}{2 \left (1+x \right )^{2}}-\frac {\frac {a}{9}-\frac {2 b}{27}}{1+x}+\left (\frac {5 a}{27}-\frac {2 b}{27}\right ) \ln \left (1+x \right )-\frac {\left (-3 a -4 b \right ) x^{3}+\left (a +\frac {13 b}{2}\right ) x^{2}+\left (-a -8 b \right ) x -\frac {7 a}{2}+\frac {5 b}{2}}{27 \left (x^{2}-x +1\right )^{2}}-\frac {\left (5 a -2 b \right ) \ln \left (x^{2}-x +1\right )}{54}-\frac {2 \left (-\frac {15 a}{2}-3 b \right ) \sqrt {3}\, \arctan \left (\frac {\left (-1+2 x \right ) \sqrt {3}}{3}\right )}{81}\)	\(131\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(1+x)^3/(x^2-x+1)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*(1/27*a-1/27*b)/(1+x)^2-(1/9*a-2/27*b)/(1+x)+(5/27*a-2/27*b)*ln(1+x)-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)^2-1/54*(5*a-2*b)*ln(x^2-x+1)-2/81*(-15/2*a-3*b)*3^(1/2)*arctan(1/3*(-1+2*x)*
3^(1/2))

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Maxima [A]
time = 0.48, size = 92, normalized size = 0.91 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(1+x)^3/(x^2-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)

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Fricas [A]
time = 3.54, size = 160, normalized size = 1.58 \begin {gather*} \frac {12 \, b x^{5} + 15 \, a x^{4} + 21 \, b x^{2} + 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 )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 24 \, a x - {\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 \, {\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 )}{54 \, {\left (x^{6} + 2 \, x^{3} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(12*b*x^5 + 15*a*x^4 + 21*b*x^2 + 2*sqrt(3)*((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)) + 24*a*x - ((5*a - 2*b)*x^6 + 2*(5*a - 2*b)*x^3 + 5*a - 2*b)*log(x^2 - x + 1) + 2*((5*a - 2
*b)*x^6 + 2*(5*a - 2*b)*x^3 + 5*a - 2*b)*log(x + 1))/(x^6 + 2*x^3 + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.36, size = 292, normalized size = 2.89 \begin {gather*} \frac {\left (5 a - 2 b\right ) \log {\left (x + \frac {25 a^{2} \cdot \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} \end {gather*}

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 + 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/2
7 + 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 + 3
6*x**3 + 18)

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Giac [A]
time = 1.32, size = 88, normalized size = 0.87 \begin {gather*} \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 ({\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(1+x)^3/(x^2-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)*log(x^2 - x + 1) + 1/27*(5*a - 2*b)*
log(abs(x + 1)) + 1/18*(4*b*x^5 + 5*a*x^4 + 7*b*x^2 + 8*a*x)/(x^3 + 1)^2

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Mupad [B]
time = 0.14, size = 114, normalized size = 1.13 \begin {gather*} \ln \left (x+1\right )\,\left (\frac {5\,a}{27}-\frac {2\,b}{27}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {b}{27}-\frac {5\,a}{54}+\frac {\sqrt {3}\,a\,5{}\mathrm {i}}{54}+\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{27}\right )-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {5\,a}{54}-\frac {b}{27}+\frac {\sqrt {3}\,a\,5{}\mathrm {i}}{54}+\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{27}\right )+\frac {\frac {2\,b\,x^5}{9}+\frac {5\,a\,x^4}{18}+\frac {7\,b\,x^2}{18}+\frac {4\,a\,x}{9}}{x^6+2\,x^3+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + (3^(1/2)*1i)/2 - 1/2)*(b/27 - (5*a)/54 + (3^(1/2)*a*5i)/54 + (3^(1/2)*b*1i)/27) - log(x - (3^(1/2)*1i)
/2 - 1/2)*((5*a)/54 - b/27 + (3^(1/2)*a*5i)/54 + (3^(1/2)*b*1i)/27) + log(x + 1)*((5*a)/27 - (2*b)/27) + ((4*a
*x)/9 + (5*a*x^4)/18 + (7*b*x^2)/18 + (2*b*x^5)/9)/(2*x^3 + x^6 + 1)

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