\(\int \frac {1+x}{(1+3 x+x^2) \sqrt [3]{1-x^3}} \, dx\) [2627]

Optimal result

Integrand size = 25, antiderivative size = 231 \[ \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {5 \sqrt {3} \sqrt [3]{1-x^3}}{2 \sqrt [3]{2} 5^{2/3}-2 \sqrt [3]{2} 5^{2/3} x+5 \sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{2} 5^{2/3}+\sqrt [3]{2} 5^{2/3} x+5 \sqrt [3]{1-x^3}\right )}{\sqrt [3]{2} 5^{2/3}}-\frac {\log \left (2^{2/3} \sqrt [3]{5}-2\ 2^{2/3} \sqrt [3]{5} x+2^{2/3} \sqrt [3]{5} x^2+\left (\sqrt [3]{2} 5^{2/3}-\sqrt [3]{2} 5^{2/3} x\right ) \sqrt [3]{1-x^3}+5 \left (1-x^3\right )^{2/3}\right )}{2 \sqrt [3]{2} 5^{2/3}} \]

[Out]

1/10*3^(1/2)*arctan(5*3^(1/2)*(-x^3+1)^(1/3)/(2*2^(1/3)*5^(2/3)-2*2^(1/3)*5^(2/3)*x+5*(-x^3+1)^(1/3)))*2^(2/3)
*5^(1/3)+1/10*ln(-2^(1/3)*5^(2/3)+2^(1/3)*5^(2/3)*x+5*(-x^3+1)^(1/3))*2^(2/3)*5^(1/3)-1/20*ln(2^(2/3)*5^(1/3)-
2*2^(2/3)*5^(1/3)*x+2^(2/3)*5^(1/3)*x^2+(2^(1/3)*5^(2/3)-2^(1/3)*5^(2/3)*x)*(-x^3+1)^(1/3)+5*(-x^3+1)^(2/3))*2
^(2/3)*5^(1/3)

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.90 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.73, number of steps used = 20, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6860, 2181, 384, 524, 455, 57, 631, 210, 31} \[ \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=-\frac {\left (5-\sqrt {5}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {8 x^3}{\left (3-\sqrt {5}\right )^3}\right )}{5 \left (3-\sqrt {5}\right )^2}-\frac {\left (5+\sqrt {5}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {8 x^3}{\left (3+\sqrt {5}\right )^3}\right )}{5 \left (3+\sqrt {5}\right )^2}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}+\frac {\arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{1-x^3}}{\sqrt [3]{5+2 \sqrt {5}}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}+\frac {\arctan \left (\frac {10^{2/3} \sqrt [3]{5+2 \sqrt {5}} \sqrt [3]{1-x^3}+5}{5 \sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}-\frac {\log \left (x^3-4 \sqrt {5}+9\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (x^3+4 \sqrt {5}+9\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (8 x^3+8 \left (9-4 \sqrt {5}\right )\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (8 x^3+\left (3+\sqrt {5}\right )^3\right )}{6 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (\sqrt [3]{2 \left (5-2 \sqrt {5}\right )}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (\sqrt [3]{2 \left (5+2 \sqrt {5}\right )}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x\right )}{2 \sqrt [3]{2} 5^{2/3}} \]

[In]

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

[Out]

-1/5*((5 - Sqrt[5])*x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, (-8*x^3)/(3 - Sqrt[5])^3])/(3 - Sqrt[5])^2 - ((5 + Sqr
t[5])*x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, (-8*x^3)/(3 + Sqrt[5])^3])/(5*(3 + Sqrt[5])^2) - ArcTan[(1 - (2*(2*(
5 - 2*Sqrt[5]))^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3]*5^(2/3)) - ArcTan[(1 - (2*(2*(5 + 2*Sqrt[5
]))^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3]*5^(2/3)) + ArcTan[(1 + (2^(2/3)*(1 - x^3)^(1/3))/(5 +
2*Sqrt[5])^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3]*5^(2/3)) + ArcTan[(5 + 10^(2/3)*(5 + 2*Sqrt[5])^(1/3)*(1 - x^3)^(1
/3))/(5*Sqrt[3])]/(2^(1/3)*Sqrt[3]*5^(2/3)) - Log[9 - 4*Sqrt[5] + x^3]/(6*2^(1/3)*5^(2/3)) - Log[9 + 4*Sqrt[5]
+ x^3]/(6*2^(1/3)*5^(2/3)) - Log[8*(9 - 4*Sqrt[5]) + 8*x^3]/(6*2^(1/3)*5^(2/3)) - Log[(3 + Sqrt[5])^3 + 8*x^3
]/(6*2^(1/3)*5^(2/3)) + Log[(2*(5 - 2*Sqrt[5]))^(1/3) - (1 - x^3)^(1/3)]/(2*2^(1/3)*5^(2/3)) + Log[(2*(5 + 2*S
qrt[5]))^(1/3) - (1 - x^3)^(1/3)]/(2*2^(1/3)*5^(2/3)) + Log[-((2*(5 - 2*Sqrt[5]))^(1/3)*x) - (1 - x^3)^(1/3)]/
(2*2^(1/3)*5^(2/3)) + Log[-((2*(5 + 2*Sqrt[5]))^(1/3)*x) - (1 - x^3)^(1/3)]/(2*2^(1/3)*5^(2/3))

Rule 31

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

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

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 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2181

Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c^3 + d^3*x
^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ
[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1-\frac {1}{\sqrt {5}}}{\left (3-\sqrt {5}+2 x\right ) \sqrt [3]{1-x^3}}+\frac {1+\frac {1}{\sqrt {5}}}{\left (3+\sqrt {5}+2 x\right ) \sqrt [3]{1-x^3}}\right ) \, dx \\ & = \frac {1}{5} \left (5-\sqrt {5}\right ) \int \frac {1}{\left (3-\sqrt {5}+2 x\right ) \sqrt [3]{1-x^3}} \, dx+\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x\right ) \sqrt [3]{1-x^3}} \, dx \\ & = \frac {1}{5} \left (5-\sqrt {5}\right ) \int \left (-\frac {2 \left (-7+3 \sqrt {5}\right )}{\sqrt [3]{1-x^3} \left (\left (3-\sqrt {5}\right )^3+8 x^3\right )}+\frac {2 \left (-3+\sqrt {5}\right ) x}{\sqrt [3]{1-x^3} \left (\left (3-\sqrt {5}\right )^3+8 x^3\right )}+\frac {4 x^2}{\sqrt [3]{1-x^3} \left (\left (3-\sqrt {5}\right )^3+8 x^3\right )}\right ) \, dx+\frac {1}{5} \left (5+\sqrt {5}\right ) \int \left (\frac {\left (3+\sqrt {5}\right )^2}{\sqrt [3]{1-x^3} \left (\left (3+\sqrt {5}\right )^3+8 x^3\right )}-\frac {2 \left (3+\sqrt {5}\right ) x}{\sqrt [3]{1-x^3} \left (\left (3+\sqrt {5}\right )^3+8 x^3\right )}+\frac {4 x^2}{\sqrt [3]{1-x^3} \left (\left (3+\sqrt {5}\right )^3+8 x^3\right )}\right ) \, dx \\ & = \frac {1}{5} \left (4 \left (25-11 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt [3]{1-x^3} \left (\left (3-\sqrt {5}\right )^3+8 x^3\right )} \, dx-\frac {1}{5} \left (8 \left (5-2 \sqrt {5}\right )\right ) \int \frac {x}{\sqrt [3]{1-x^3} \left (\left (3-\sqrt {5}\right )^3+8 x^3\right )} \, dx+\frac {1}{5} \left (4 \left (5-\sqrt {5}\right )\right ) \int \frac {x^2}{\sqrt [3]{1-x^3} \left (\left (3-\sqrt {5}\right )^3+8 x^3\right )} \, dx+\frac {1}{5} \left (4 \left (5+\sqrt {5}\right )\right ) \int \frac {x^2}{\sqrt [3]{1-x^3} \left (\left (3+\sqrt {5}\right )^3+8 x^3\right )} \, dx-\frac {1}{5} \left (8 \left (5+2 \sqrt {5}\right )\right ) \int \frac {x}{\sqrt [3]{1-x^3} \left (\left (3+\sqrt {5}\right )^3+8 x^3\right )} \, dx+\frac {1}{5} \left (4 \left (25+11 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt [3]{1-x^3} \left (\left (3+\sqrt {5}\right )^3+8 x^3\right )} \, dx \\ & = -\frac {4 \left (5-2 \sqrt {5}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {8 x^3}{\left (3-\sqrt {5}\right )^3}\right )}{5 \left (3-\sqrt {5}\right )^3}-\frac {4 \left (5+2 \sqrt {5}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {8 x^3}{\left (3+\sqrt {5}\right )^3}\right )}{5 \left (3+\sqrt {5}\right )^3}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}-\frac {\log \left (8 \left (9-4 \sqrt {5}\right )+8 x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (\left (3+\sqrt {5}\right )^3+8 x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {1}{15} \left (4 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} \left (\left (3-\sqrt {5}\right )^3+8 x\right )} \, dx,x,x^3\right )+\frac {1}{15} \left (4 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} \left (\left (3+\sqrt {5}\right )^3+8 x\right )} \, dx,x,x^3\right ) \\ & = -\frac {4 \left (5-2 \sqrt {5}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {8 x^3}{\left (3-\sqrt {5}\right )^3}\right )}{5 \left (3-\sqrt {5}\right )^3}-\frac {4 \left (5+2 \sqrt {5}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {8 x^3}{\left (3+\sqrt {5}\right )^3}\right )}{5 \left (3+\sqrt {5}\right )^3}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}-\frac {\log \left (9-4 \sqrt {5}+x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (9+4 \sqrt {5}+x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (8 \left (9-4 \sqrt {5}\right )+8 x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (\left (3+\sqrt {5}\right )^3+8 x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2 \left (5-2 \sqrt {5}\right )}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2 \left (5+2 \sqrt {5}\right )}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {1}{20} \left (5-\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \left (5-2 \sqrt {5}\right )\right )^{2/3}+\sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac {1}{20} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \left (5+2 \sqrt {5}\right )\right )^{2/3}+\sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right ) \\ & = -\frac {4 \left (5-2 \sqrt {5}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {8 x^3}{\left (3-\sqrt {5}\right )^3}\right )}{5 \left (3-\sqrt {5}\right )^3}-\frac {4 \left (5+2 \sqrt {5}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {8 x^3}{\left (3+\sqrt {5}\right )^3}\right )}{5 \left (3+\sqrt {5}\right )^3}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}-\frac {\log \left (9-4 \sqrt {5}+x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (9+4 \sqrt {5}+x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (8 \left (9-4 \sqrt {5}\right )+8 x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (\left (3+\sqrt {5}\right )^3+8 x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (\sqrt [3]{2 \left (5-2 \sqrt {5}\right )}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (\sqrt [3]{2 \left (5+2 \sqrt {5}\right )}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{1-x^3}}{\sqrt [3]{5+2 \sqrt {5}}}\right )}{\sqrt [3]{2} 5^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{\left (-1-\frac {2}{\sqrt {5}}\right ) \left (-1+x^3\right )}\right )}{\sqrt [3]{2} 5^{2/3}} \\ & = -\frac {4 \left (5-2 \sqrt {5}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {8 x^3}{\left (3-\sqrt {5}\right )^3}\right )}{5 \left (3-\sqrt {5}\right )^3}-\frac {4 \left (5+2 \sqrt {5}\right ) x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^3,-\frac {8 x^3}{\left (3+\sqrt {5}\right )^3}\right )}{5 \left (3+\sqrt {5}\right )^3}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}+\frac {\arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{1-x^3}}{\sqrt [3]{5+2 \sqrt {5}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}+\frac {\arctan \left (\frac {5+10^{2/3} \sqrt [3]{5+2 \sqrt {5}} \sqrt [3]{1-x^3}}{5 \sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} 5^{2/3}}-\frac {\log \left (9-4 \sqrt {5}+x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (9+4 \sqrt {5}+x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (8 \left (9-4 \sqrt {5}\right )+8 x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}-\frac {\log \left (\left (3+\sqrt {5}\right )^3+8 x^3\right )}{6 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (\sqrt [3]{2 \left (5-2 \sqrt {5}\right )}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (\sqrt [3]{2 \left (5+2 \sqrt {5}\right )}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{2 \left (5-2 \sqrt {5}\right )} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{2 \left (5+2 \sqrt {5}\right )} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2} 5^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.86 \[ \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {5 \sqrt {3} \sqrt [3]{1-x^3}}{2 \sqrt [3]{2} 5^{2/3}-2 \sqrt [3]{2} 5^{2/3} x+5 \sqrt [3]{1-x^3}}\right )+2 \log \left (-\sqrt [3]{2} 5^{2/3}+\sqrt [3]{2} 5^{2/3} x+5 \sqrt [3]{1-x^3}\right )-\log \left (2^{2/3} \sqrt [3]{5}-2\ 2^{2/3} \sqrt [3]{5} x+2^{2/3} \sqrt [3]{5} x^2-5^{2/3} (-1+x) \sqrt [3]{2-2 x^3}+5 \left (1-x^3\right )^{2/3}\right )}{2 \sqrt [3]{2} 5^{2/3}} \]

[In]

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

[Out]

(2*Sqrt[3]*ArcTan[(5*Sqrt[3]*(1 - x^3)^(1/3))/(2*2^(1/3)*5^(2/3) - 2*2^(1/3)*5^(2/3)*x + 5*(1 - x^3)^(1/3))] +
2*Log[-(2^(1/3)*5^(2/3)) + 2^(1/3)*5^(2/3)*x + 5*(1 - x^3)^(1/3)] - Log[2^(2/3)*5^(1/3) - 2*2^(2/3)*5^(1/3)*x
+ 2^(2/3)*5^(1/3)*x^2 - 5^(2/3)*(-1 + x)*(2 - 2*x^3)^(1/3) + 5*(1 - x^3)^(2/3)])/(2*2^(1/3)*5^(2/3))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 5.08 (sec) , antiderivative size = 778, normalized size of antiderivative = 3.37

method	result	size
trager	\(\text {Expression too large to display}\)	\(778\)

[In]

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

[Out]

1/10*RootOf(_Z^3-20)*ln((25*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2)*RootOf(_Z^
3-20)^2-50*RootOf(RootOf(_Z^3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2)^2*RootOf(_Z^3-20)^2*x-15*RootOf(RootOf(_Z^
3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2)*RootOf(_Z^3-20)^3*x+110*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-20)^2+10*_Z*
RootOf(_Z^3-20)+100*_Z^2)*RootOf(_Z^3-20)*x+5*(-x^3+1)^(1/3)*RootOf(_Z^3-20)^2*x-110*(-x^3+1)^(1/3)*RootOf(Roo
tOf(_Z^3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2)*RootOf(_Z^3-20)-5*(-x^3+1)^(1/3)*RootOf(_Z^3-20)^2+130*RootOf(R
ootOf(_Z^3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2)*x^2+39*RootOf(_Z^3-20)*x^2-60*(-x^3+1)^(2/3)-10*RootOf(RootOf
(_Z^3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2)*x-3*RootOf(_Z^3-20)*x+130*RootOf(RootOf(_Z^3-20)^2+10*_Z*RootOf(_Z
^3-20)+100*_Z^2)+39*RootOf(_Z^3-20))/(x^2+3*x+1))+RootOf(RootOf(_Z^3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2)*ln(
-(25*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2)*RootOf(_Z^3-20)^2-150*RootOf(Root
Of(_Z^3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2)^2*RootOf(_Z^3-20)^2*x-5*RootOf(RootOf(_Z^3-20)^2+10*_Z*RootOf(_Z
^3-20)+100*_Z^2)*RootOf(_Z^3-20)^3*x-60*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2
)*RootOf(_Z^3-20)*x+5*(-x^3+1)^(1/3)*RootOf(_Z^3-20)^2*x+60*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-20)^2+10*_Z*Root
Of(_Z^3-20)+100*_Z^2)*RootOf(_Z^3-20)-5*(-x^3+1)^(1/3)*RootOf(_Z^3-20)^2-390*RootOf(RootOf(_Z^3-20)^2+10*_Z*Ro
otOf(_Z^3-20)+100*_Z^2)*x^2-13*RootOf(_Z^3-20)*x^2+110*(-x^3+1)^(2/3)-270*RootOf(RootOf(_Z^3-20)^2+10*_Z*RootO
f(_Z^3-20)+100*_Z^2)*x-9*RootOf(_Z^3-20)*x-390*RootOf(RootOf(_Z^3-20)^2+10*_Z*RootOf(_Z^3-20)+100*_Z^2)-13*Roo
tOf(_Z^3-20))/(x^2+3*x+1))

Fricas [A] (verification not implemented)

none

Time = 5.58 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.40 \[ \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\frac {1}{30} \cdot 50^{\frac {1}{6}} \sqrt {3} \sqrt {2} \arctan \left (\frac {50^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 50^{\frac {2}{3}} \sqrt {2} {\left (3 \, x^{4} + 8 \, x^{3} + 3 \, x^{2} + 8 \, x + 3\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 50^{\frac {1}{3}} \sqrt {2} {\left (41 \, x^{6} - 11 \, x^{5} + 50 \, x^{4} - 35 \, x^{3} + 50 \, x^{2} - 11 \, x + 41\right )} - 20 \, \sqrt {2} {\left (11 \, x^{5} - 15 \, x^{4} + 15 \, x^{3} - 15 \, x^{2} + 15 \, x - 11\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{30 \, {\left (19 \, x^{6} - 69 \, x^{5} + 30 \, x^{4} - 85 \, x^{3} + 30 \, x^{2} - 69 \, x + 19\right )}}\right ) - \frac {1}{300} \cdot 50^{\frac {2}{3}} \log \left (\frac {50^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (3 \, x^{2} - x + 3\right )} + 50^{\frac {1}{3}} {\left (11 \, x^{4} - 4 \, x^{3} + 11 \, x^{2} - 4 \, x + 11\right )} - 20 \, {\left (2 \, x^{3} - x^{2} + x - 2\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{4} + 6 \, x^{3} + 11 \, x^{2} + 6 \, x + 1}\right ) + \frac {1}{150} \cdot 50^{\frac {2}{3}} \log \left (\frac {50^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 1\right )} - 10 \cdot 50^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 50 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + 3 \, x + 1}\right ) \]

[In]

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

[Out]

1/30*50^(1/6)*sqrt(3)*sqrt(2)*arctan(1/30*50^(1/6)*sqrt(3)*(2*50^(2/3)*sqrt(2)*(3*x^4 + 8*x^3 + 3*x^2 + 8*x +
3)*(-x^3 + 1)^(2/3) + 50^(1/3)*sqrt(2)*(41*x^6 - 11*x^5 + 50*x^4 - 35*x^3 + 50*x^2 - 11*x + 41) - 20*sqrt(2)*(
11*x^5 - 15*x^4 + 15*x^3 - 15*x^2 + 15*x - 11)*(-x^3 + 1)^(1/3))/(19*x^6 - 69*x^5 + 30*x^4 - 85*x^3 + 30*x^2 -
69*x + 19)) - 1/300*50^(2/3)*log((50^(2/3)*(-x^3 + 1)^(2/3)*(3*x^2 - x + 3) + 50^(1/3)*(11*x^4 - 4*x^3 + 11*x
^2 - 4*x + 11) - 20*(2*x^3 - x^2 + x - 2)*(-x^3 + 1)^(1/3))/(x^4 + 6*x^3 + 11*x^2 + 6*x + 1)) + 1/150*50^(2/3)
*log((50^(2/3)*(x^2 + 3*x + 1) - 10*50^(1/3)*(-x^3 + 1)^(1/3)*(x - 1) - 50*(-x^3 + 1)^(2/3))/(x^2 + 3*x + 1))

Sympy [F]

\[ \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int \frac {x + 1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 3 x + 1\right )}\, dx \]

[In]

integrate((1+x)/(x**2+3*x+1)/(-x**3+1)**(1/3),x)

[Out]

Integral((x + 1)/((-(x - 1)*(x**2 + x + 1))**(1/3)*(x**2 + 3*x + 1)), x)

Maxima [F]

\[ \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int { \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 3 \, x + 1\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int { \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 3 \, x + 1\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int \frac {x+1}{{\left (1-x^3\right )}^{1/3}\,\left (x^2+3\,x+1\right )} \,d x \]

[In]

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

[Out]

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