\(\int \frac {x+x^7}{(-1+x^6)^{2/3} (-1+x^3+x^6)} \, dx\) [1054]

Optimal result

Integrand size = 25, antiderivative size = 79 \[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]

[Out]

-1/3*arctan(3^(1/2)*x/(-x+2*(x^6-1)^(1/3)))*3^(1/2)+1/3*ln(x+(x^6-1)^(1/3))-1/6*ln(x^2-x*(x^6-1)^(1/3)+(x^6-1)
^(2/3))

Rubi [C] (verified)

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

Time = 1.78 (sec) , antiderivative size = 601, normalized size of antiderivative = 7.61, number of steps used = 40, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1607, 6860, 281, 252, 251, 1576, 476, 441, 440, 525, 524} \[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=\frac {\left (5-\sqrt {5}\right ) \left (1-x^6\right )^{2/3} x^5 \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{25 \left (3-\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}-\frac {4 \left (1-x^6\right )^{2/3} x^5 \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \sqrt {5} \left (3-\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}+\frac {\left (5+\sqrt {5}\right ) \left (1-x^6\right )^{2/3} x^5 \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{25 \left (3+\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}+\frac {4 \left (1-x^6\right )^{2/3} x^5 \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{5 \sqrt {5} \left (3+\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}-\frac {\left (5-\sqrt {5}\right ) \left (1-x^6\right )^{2/3} x^2 \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \left (3-\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}+\frac {\left (1-x^6\right )^{2/3} x^2 \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{2 \sqrt {5} \left (x^6-1\right )^{2/3}}-\frac {\left (5+\sqrt {5}\right ) \left (1-x^6\right )^{2/3} x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {2}{3},\frac {4}{3},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{5 \left (3+\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}-\frac {\left (1-x^6\right )^{2/3} x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {2}{3},\frac {4}{3},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{2 \sqrt {5} \left (x^6-1\right )^{2/3}}+\frac {\left (1-x^6\right )^{2/3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^6\right )}{2 \left (x^6-1\right )^{2/3}} \]

[In]

Int[(x + x^7)/((-1 + x^6)^(2/3)*(-1 + x^3 + x^6)),x]

[Out]

(x^2*(1 - x^6)^(2/3)*AppellF1[1/3, 2/3, 1, 4/3, x^6, (2*x^6)/(3 - Sqrt[5])])/(2*Sqrt[5]*(-1 + x^6)^(2/3)) - ((
5 - Sqrt[5])*x^2*(1 - x^6)^(2/3)*AppellF1[1/3, 2/3, 1, 4/3, x^6, (2*x^6)/(3 - Sqrt[5])])/(5*(3 - Sqrt[5])*(-1
+ x^6)^(2/3)) - (x^2*(1 - x^6)^(2/3)*AppellF1[1/3, 1, 2/3, 4/3, (2*x^6)/(3 + Sqrt[5]), x^6])/(2*Sqrt[5]*(-1 +
x^6)^(2/3)) - ((5 + Sqrt[5])*x^2*(1 - x^6)^(2/3)*AppellF1[1/3, 1, 2/3, 4/3, (2*x^6)/(3 + Sqrt[5]), x^6])/(5*(3
+ Sqrt[5])*(-1 + x^6)^(2/3)) - (4*x^5*(1 - x^6)^(2/3)*AppellF1[5/6, 2/3, 1, 11/6, x^6, (2*x^6)/(3 - Sqrt[5])]
)/(5*Sqrt[5]*(3 - Sqrt[5])*(-1 + x^6)^(2/3)) + ((5 - Sqrt[5])*x^5*(1 - x^6)^(2/3)*AppellF1[5/6, 2/3, 1, 11/6,
x^6, (2*x^6)/(3 - Sqrt[5])])/(25*(3 - Sqrt[5])*(-1 + x^6)^(2/3)) + (4*x^5*(1 - x^6)^(2/3)*AppellF1[5/6, 2/3, 1
, 11/6, x^6, (2*x^6)/(3 + Sqrt[5])])/(5*Sqrt[5]*(3 + Sqrt[5])*(-1 + x^6)^(2/3)) + ((5 + Sqrt[5])*x^5*(1 - x^6)
^(2/3)*AppellF1[5/6, 2/3, 1, 11/6, x^6, (2*x^6)/(3 + Sqrt[5])])/(25*(3 + Sqrt[5])*(-1 + x^6)^(2/3)) + (x^2*(1
- x^6)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, x^6])/(2*(-1 + x^6)^(2/3))

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 440

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

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 476

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

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 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{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])

Rule 1576

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Dist[(f*x)^m
/x^m, Int[ExpandIntegrand[x^m*(a + c*x^(2*n))^p, (d/(d^2 - e^2*x^(2*n)) - e*(x^n/(d^2 - e^2*x^(2*n))))^(-q), x
], x], x] /; FreeQ[{a, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] &&  !IntegerQ[p] && ILtQ[q, 0]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 \frac {x \left (1+x^6\right )}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx \\ & = \int \left (\frac {x}{\left (-1+x^6\right )^{2/3}}+\frac {x \left (2-x^3\right )}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )}\right ) \, dx \\ & = \int \frac {x}{\left (-1+x^6\right )^{2/3}} \, dx+\int \frac {x \left (2-x^3\right )}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right )+\int \left (\frac {2 x}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )}-\frac {x^4}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )}\right ) \, dx \\ & = 2 \int \frac {x}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx+\frac {\left (1-x^6\right )^{2/3} \text {Subst}\left (\int \frac {1}{\left (1-x^3\right )^{2/3}} \, dx,x,x^2\right )}{2 \left (-1+x^6\right )^{2/3}}-\int \frac {x^4}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx \\ & = \frac {x^2 \left (1-x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^6\right )}{2 \left (-1+x^6\right )^{2/3}}+2 \int \left (-\frac {2 x}{\sqrt {5} \left (-1+\sqrt {5}-2 x^3\right ) \left (-1+x^6\right )^{2/3}}-\frac {2 x}{\sqrt {5} \left (1+\sqrt {5}+2 x^3\right ) \left (-1+x^6\right )^{2/3}}\right ) \, dx-\int \left (-\frac {\left (-1+\sqrt {5}\right ) x}{\sqrt {5} \left (-1+\sqrt {5}-2 x^3\right ) \left (-1+x^6\right )^{2/3}}+\frac {\left (1+\sqrt {5}\right ) x}{\sqrt {5} \left (1+\sqrt {5}+2 x^3\right ) \left (-1+x^6\right )^{2/3}}\right ) \, dx \\ & = \frac {x^2 \left (1-x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^6\right )}{2 \left (-1+x^6\right )^{2/3}}-\frac {4 \int \frac {x}{\left (-1+\sqrt {5}-2 x^3\right ) \left (-1+x^6\right )^{2/3}} \, dx}{\sqrt {5}}-\frac {4 \int \frac {x}{\left (1+\sqrt {5}+2 x^3\right ) \left (-1+x^6\right )^{2/3}} \, dx}{\sqrt {5}}+\frac {1}{5} \left (5-\sqrt {5}\right ) \int \frac {x}{\left (-1+\sqrt {5}-2 x^3\right ) \left (-1+x^6\right )^{2/3}} \, dx-\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {x}{\left (1+\sqrt {5}+2 x^3\right ) \left (-1+x^6\right )^{2/3}} \, dx \\ & = \frac {x^2 \left (1-x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^6\right )}{2 \left (-1+x^6\right )^{2/3}}-\frac {4 \int \left (\frac {\left (1+\sqrt {5}\right ) x}{2 \left (3+\sqrt {5}-2 x^6\right ) \left (-1+x^6\right )^{2/3}}+\frac {x^4}{\left (-1+x^6\right )^{2/3} \left (-3-\sqrt {5}+2 x^6\right )}\right ) \, dx}{\sqrt {5}}-\frac {4 \int \left (\frac {\left (1-\sqrt {5}\right ) x}{2 \left (-1+x^6\right )^{2/3} \left (-3+\sqrt {5}+2 x^6\right )}-\frac {x^4}{\left (-1+x^6\right )^{2/3} \left (-3+\sqrt {5}+2 x^6\right )}\right ) \, dx}{\sqrt {5}}+\frac {1}{5} \left (5-\sqrt {5}\right ) \int \left (\frac {\left (1-\sqrt {5}\right ) x}{2 \left (-1+x^6\right )^{2/3} \left (-3+\sqrt {5}+2 x^6\right )}-\frac {x^4}{\left (-1+x^6\right )^{2/3} \left (-3+\sqrt {5}+2 x^6\right )}\right ) \, dx-\frac {1}{5} \left (5+\sqrt {5}\right ) \int \left (\frac {\left (1+\sqrt {5}\right ) x}{2 \left (3+\sqrt {5}-2 x^6\right ) \left (-1+x^6\right )^{2/3}}+\frac {x^4}{\left (-1+x^6\right )^{2/3} \left (-3-\sqrt {5}+2 x^6\right )}\right ) \, dx \\ & = \frac {x^2 \left (1-x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^6\right )}{2 \left (-1+x^6\right )^{2/3}}-\frac {4 \int \frac {x^4}{\left (-1+x^6\right )^{2/3} \left (-3-\sqrt {5}+2 x^6\right )} \, dx}{\sqrt {5}}+\frac {4 \int \frac {x^4}{\left (-1+x^6\right )^{2/3} \left (-3+\sqrt {5}+2 x^6\right )} \, dx}{\sqrt {5}}+\frac {1}{5} \left (5-3 \sqrt {5}\right ) \int \frac {x}{\left (-1+x^6\right )^{2/3} \left (-3+\sqrt {5}+2 x^6\right )} \, dx+\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \int \frac {x}{\left (-1+x^6\right )^{2/3} \left (-3+\sqrt {5}+2 x^6\right )} \, dx+\frac {1}{5} \left (-5+\sqrt {5}\right ) \int \frac {x^4}{\left (-1+x^6\right )^{2/3} \left (-3+\sqrt {5}+2 x^6\right )} \, dx-\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {x^4}{\left (-1+x^6\right )^{2/3} \left (-3-\sqrt {5}+2 x^6\right )} \, dx-\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \int \frac {x}{\left (3+\sqrt {5}-2 x^6\right ) \left (-1+x^6\right )^{2/3}} \, dx-\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {x}{\left (3+\sqrt {5}-2 x^6\right ) \left (-1+x^6\right )^{2/3}} \, dx \\ & = \frac {x^2 \left (1-x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^6\right )}{2 \left (-1+x^6\right )^{2/3}}+\frac {1}{10} \left (5-3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^3\right )^{2/3} \left (-3+\sqrt {5}+2 x^3\right )} \, dx,x,x^2\right )+\frac {1}{5} \left (5-\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^3\right )^{2/3} \left (-3+\sqrt {5}+2 x^3\right )} \, dx,x,x^2\right )-\frac {1}{5} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\left (3+\sqrt {5}-2 x^3\right ) \left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right )-\frac {1}{10} \left (5+3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\left (3+\sqrt {5}-2 x^3\right ) \left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right )-\frac {\left (4 \left (1-x^6\right )^{2/3}\right ) \int \frac {x^4}{\left (1-x^6\right )^{2/3} \left (-3-\sqrt {5}+2 x^6\right )} \, dx}{\sqrt {5} \left (-1+x^6\right )^{2/3}}+\frac {\left (4 \left (1-x^6\right )^{2/3}\right ) \int \frac {x^4}{\left (1-x^6\right )^{2/3} \left (-3+\sqrt {5}+2 x^6\right )} \, dx}{\sqrt {5} \left (-1+x^6\right )^{2/3}}+\frac {\left (\left (-5+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}\right ) \int \frac {x^4}{\left (1-x^6\right )^{2/3} \left (-3+\sqrt {5}+2 x^6\right )} \, dx}{5 \left (-1+x^6\right )^{2/3}}-\frac {\left (\left (5+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}\right ) \int \frac {x^4}{\left (1-x^6\right )^{2/3} \left (-3-\sqrt {5}+2 x^6\right )} \, dx}{5 \left (-1+x^6\right )^{2/3}} \\ & = -\frac {4 x^5 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \sqrt {5} \left (3-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}+\frac {\left (5-\sqrt {5}\right ) x^5 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{25 \left (3-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}+\frac {4 x^5 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{5 \sqrt {5} \left (3+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}+\frac {\left (5+\sqrt {5}\right ) x^5 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{25 \left (3+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}+\frac {x^2 \left (1-x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^6\right )}{2 \left (-1+x^6\right )^{2/3}}+\frac {\left (\left (5-3 \sqrt {5}\right ) \left (1-x^6\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^3\right )^{2/3} \left (-3+\sqrt {5}+2 x^3\right )} \, dx,x,x^2\right )}{10 \left (-1+x^6\right )^{2/3}}+\frac {\left (\left (5-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^3\right )^{2/3} \left (-3+\sqrt {5}+2 x^3\right )} \, dx,x,x^2\right )}{5 \left (-1+x^6\right )^{2/3}}-\frac {\left (\left (5+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (3+\sqrt {5}-2 x^3\right ) \left (1-x^3\right )^{2/3}} \, dx,x,x^2\right )}{5 \left (-1+x^6\right )^{2/3}}-\frac {\left (\left (5+3 \sqrt {5}\right ) \left (1-x^6\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (3+\sqrt {5}-2 x^3\right ) \left (1-x^3\right )^{2/3}} \, dx,x,x^2\right )}{10 \left (-1+x^6\right )^{2/3}} \\ & = \frac {x^2 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{2 \sqrt {5} \left (-1+x^6\right )^{2/3}}-\frac {\left (5-\sqrt {5}\right ) x^2 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \left (3-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}-\frac {x^2 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {2}{3},\frac {4}{3},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{2 \sqrt {5} \left (-1+x^6\right )^{2/3}}-\frac {\left (5+\sqrt {5}\right ) x^2 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {2}{3},\frac {4}{3},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{5 \left (3+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}-\frac {4 x^5 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \sqrt {5} \left (3-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}+\frac {\left (5-\sqrt {5}\right ) x^5 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{25 \left (3-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}+\frac {4 x^5 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{5 \sqrt {5} \left (3+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}+\frac {\left (5+\sqrt {5}\right ) x^5 \left (1-x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{25 \left (3+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}+\frac {x^2 \left (1-x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^6\right )}{2 \left (-1+x^6\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]

[In]

Integrate[(x + x^7)/((-1 + x^6)^(2/3)*(-1 + x^3 + x^6)),x]

[Out]

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

Maple [A] (verified)

Time = 3.48 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {x +\left (x^{6}-1\right )^{\frac {1}{3}}}{x}\right )}{3}-\frac {\ln \left (\frac {x^{2}-x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{2}}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )}{3}\)	\(71\)
trager	\(\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-3 x \left (x^{6}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{x^{6}+x^{3}-1}\right )-\frac {\ln \left (\frac {-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}+x^{3}-1}\right )}{3}-\ln \left (\frac {-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}+x^{3}-1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\)	\(433\)

[In]

int((x^7+x)/(x^6-1)^(2/3)/(x^6+x^3-1),x,method=_RETURNVERBOSE)

[Out]

1/3*ln((x+(x^6-1)^(1/3))/x)-1/6*ln((x^2-x*(x^6-1)^(1/3)+(x^6-1)^(2/3))/x^2)-1/3*3^(1/2)*arctan(1/3*3^(1/2)*(x-
2*(x^6-1)^(1/3))/x)

Fricas [A] (verification not implemented)

none

Time = 1.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.29 \[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{6} - 1\right )}}{x^{6} - 8 \, x^{3} - 1}\right ) + \frac {1}{6} \, \log \left (\frac {x^{6} + x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} + x^{3} - 1}\right ) \]

[In]

integrate((x^7+x)/(x^6-1)^(2/3)/(x^6+x^3-1),x, algorithm="fricas")

[Out]

-1/3*sqrt(3)*arctan((4*sqrt(3)*(x^6 - 1)^(1/3)*x^2 + 2*sqrt(3)*(x^6 - 1)^(2/3)*x + sqrt(3)*(x^6 - 1))/(x^6 - 8
*x^3 - 1)) + 1/6*log((x^6 + x^3 + 3*(x^6 - 1)^(1/3)*x^2 + 3*(x^6 - 1)^(2/3)*x - 1)/(x^6 + x^3 - 1))

Sympy [F]

\[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=\int \frac {x \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}{\left (\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + x^{3} - 1\right )}\, dx \]

[In]

integrate((x**7+x)/(x**6-1)**(2/3)/(x**6+x**3-1),x)

[Out]

Integral(x*(x**2 + 1)*(x**4 - x**2 + 1)/(((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))**(2/3)*(x**6 + x**3 -
1)), x)

Maxima [F]

\[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=\int { \frac {x^{7} + x}{{\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}} \,d x } \]

[In]

integrate((x^7+x)/(x^6-1)^(2/3)/(x^6+x^3-1),x, algorithm="maxima")

[Out]

integrate((x^7 + x)/((x^6 + x^3 - 1)*(x^6 - 1)^(2/3)), x)

Giac [F]

\[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=\int { \frac {x^{7} + x}{{\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}} \,d x } \]

[In]

integrate((x^7+x)/(x^6-1)^(2/3)/(x^6+x^3-1),x, algorithm="giac")

[Out]

integrate((x^7 + x)/((x^6 + x^3 - 1)*(x^6 - 1)^(2/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=\int \frac {x^7+x}{{\left (x^6-1\right )}^{2/3}\,\left (x^6+x^3-1\right )} \,d x \]

[In]

int((x + x^7)/((x^6 - 1)^(2/3)*(x^3 + x^6 - 1)),x)

[Out]

int((x + x^7)/((x^6 - 1)^(2/3)*(x^3 + x^6 - 1)), x)