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

Optimal. Leaf size=79 \[ \frac {1}{3} \log \left (\sqrt [3]{x^6-1}+x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-1}-x}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (-\sqrt [3]{x^6-1} x+\left (x^6-1\right )^{2/3}+x^2\right ) \]

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Rubi [C]  time = 1.74, antiderivative size = 601, normalized size of antiderivative = 7.61, number of steps used = 40, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1593, 6728, 275, 246, 245, 1562, 465, 430, 429, 511, 510} \begin {gather*} \frac {\left (5-\sqrt {5}\right ) \left (1-x^6\right )^{2/3} x^5 F_1\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 F_1\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 F_1\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 F_1\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 F_1\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 F_1\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 F_1\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 F_1\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 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};x^6\right )}{2 \left (x^6-1\right )^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[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 245

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 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[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 275

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 429

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 430

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 465

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 510

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)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), 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 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[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 1562

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 1593

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 6728

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*} \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx &=\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} \operatorname {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} \operatorname {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} \, _2F_1\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} \, _2F_1\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} \, _2F_1\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} \, _2F_1\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} \, _2F_1\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} F_1\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} F_1\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} F_1\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} F_1\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} \, _2F_1\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};x^6\right )}{2 \left (-1+x^6\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.74, size = 79, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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

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fricas [A]  time = 2.29, size = 102, normalized size = 1.29 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7} + x}{{\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 1.42, size = 436, normalized size = 5.52

method	result	size
trager	\(\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{6}-1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}+x^{3}-1}\right )-\frac {\ln \left (\frac {-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}+18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x^{3}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}+x^{3}-1}\right )}{3}-\ln \left (\frac {-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}+18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x^{3}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}+x^{3}-1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\)	\(436\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(9*_Z^2+3*_Z+1)*ln(-(-6*RootOf(9*_Z^2+3*_Z+1)*x^6-x^6-18*RootOf(9*_Z^2+3*_Z+1)^2*x^3+9*RootOf(9*_Z^2+3*_
Z+1)*(x^6-1)^(2/3)*x-9*RootOf(9*_Z^2+3*_Z+1)*(x^6-1)^(1/3)*x^2-3*RootOf(9*_Z^2+3*_Z+1)*x^3+3*x*(x^6-1)^(2/3)-3
*x^2*(x^6-1)^(1/3)+6*RootOf(9*_Z^2+3*_Z+1)+1)/(x^6+x^3-1))-1/3*ln((-6*RootOf(9*_Z^2+3*_Z+1)*x^6-x^6+18*RootOf(
9*_Z^2+3*_Z+1)^2*x^3+9*RootOf(9*_Z^2+3*_Z+1)*(x^6-1)^(2/3)*x-9*RootOf(9*_Z^2+3*_Z+1)*(x^6-1)^(1/3)*x^2+9*RootO
f(9*_Z^2+3*_Z+1)*x^3+x^3+6*RootOf(9*_Z^2+3*_Z+1)+1)/(x^6+x^3-1))-ln((-6*RootOf(9*_Z^2+3*_Z+1)*x^6-x^6+18*RootO
f(9*_Z^2+3*_Z+1)^2*x^3+9*RootOf(9*_Z^2+3*_Z+1)*(x^6-1)^(2/3)*x-9*RootOf(9*_Z^2+3*_Z+1)*(x^6-1)^(1/3)*x^2+9*Roo
tOf(9*_Z^2+3*_Z+1)*x^3+x^3+6*RootOf(9*_Z^2+3*_Z+1)+1)/(x^6+x^3-1))*RootOf(9*_Z^2+3*_Z+1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7} + x}{{\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^7+x}{{\left (x^6-1\right )}^{2/3}\,\left (x^6+x^3-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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