Integrand size = 22, antiderivative size = 271 \[ \int \frac {x^7}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}+\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {1}{4} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {\log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}+\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \]
-1/4*x^2*(-x^3+1)^(2/3)-1/4*x^2*hypergeom([1/3, 2/3],[5/3],x^3)+1/24*ln((1 -x)*(1+x)^2)*2^(2/3)+1/12*ln(1+2^(2/3)*(1-x)^2/(-x^3+1)^(2/3)-2^(1/3)*(1-x )/(-x^3+1)^(1/3))*2^(2/3)-1/6*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(2/3)-1 /8*ln(-1+x+2^(2/3)*(-x^3+1)^(1/3))*2^(2/3)+1/6*arctan(1/3*(1-2*2^(1/3)*(1- x)/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)+1/12*arctan(1/3*(1+2^(1/3)*(1- x)/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 10.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.15 \[ \int \frac {x^7}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {1}{4} x^2 \left (-\left (1-x^3\right )^{2/3}+\operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^3,-x^3\right )\right ) \]
Time = 0.50 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {979, 27, 984, 888, 991, 750, 16, 27, 1142, 25, 27, 1082, 217, 1103, 2574}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^7}{\sqrt [3]{1-x^3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 979 |
| \(\displaystyle \frac {1}{4} \int \frac {2 x \left (1-x^3\right )^{2/3}}{x^3+1}dx-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {x \left (1-x^3\right )^{2/3}}{x^3+1}dx-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 984 |
| \(\displaystyle \frac {1}{2} \left (2 \int \frac {x}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx-\int \frac {x}{\sqrt [3]{1-x^3}}dx\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {1}{2} \left (2 \int \frac {x}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 991 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx-\int \frac {1}{\frac {2 (1-x)^3}{1-x^3}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx-\frac {1}{3} \int \frac {\sqrt [3]{2} \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {1}{3} \int \frac {1}{\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx-\frac {1}{3} \int \frac {\sqrt [3]{2} \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx-\frac {1}{3} \sqrt [3]{2} \int \frac {2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx-\frac {1}{3} \sqrt [3]{2} \left (\frac {3 \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{2} \left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2\ 2^{2/3}}\right )-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx-\frac {1}{3} \sqrt [3]{2} \left (\frac {3 \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{2} \left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2\ 2^{2/3}}\right )-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx-\frac {1}{3} \sqrt [3]{2} \left (\frac {3 \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}+\frac {\int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}\right )-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx-\frac {1}{3} \sqrt [3]{2} \left (\frac {3 \int \frac {1}{-\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}-3}d\left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{2^{2/3}}+\frac {\int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}\right )-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \sqrt [3]{2} \left (\frac {\int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}}\right )-\frac {1}{3} \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx-\frac {1}{3} \sqrt [3]{2} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{2\ 2^{2/3}}\right )-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 2574 |
| \(\displaystyle \frac {1}{2} \left (2 \left (-\frac {1}{3} \sqrt [3]{2} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{2\ 2^{2/3}}\right )+\frac {1}{3} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}-\frac {3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac {\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}}\right )-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}\right )-\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )\right )-\frac {1}{4} x^2 \left (1-x^3\right )^{2/3}\) |
-1/4*(x^2*(1 - x^3)^(2/3)) + (-1/2*(x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x ^3]) + 2*(-1/3*(2^(1/3)*(-((Sqrt[3]*ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x ^3)^(1/3))/Sqrt[3]])/2^(2/3)) - Log[1 + (2^(2/3)*(1 - x)^2)/(1 - x^3)^(2/3 ) - (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(2*2^(2/3)))) - Log[1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(3*2^(1/3)) + ((Sqrt[3]*ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/(2*2^(1/3)) + Log[(1 - x)*(1 + x)^2]/(4*2^( 1/3)) - (3*Log[-1 + x + 2^(2/3)*(1 - x^3)^(1/3)])/(4*2^(1/3)))/3))/2
3.7.18.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Simp[e^(2*n)/(b*d *(m + n*(p + q) + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Sim p[a*c*(m - 2*n + 1) + (a*d*(m + n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x ^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && I GtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x ]
Int[((x_)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol ] :> Simp[b/d Int[x*(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d In t[x*((a + b*x^n)^(p - 1)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, 1, 1, n, p, -1, x]
Int[(x_)/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[b/a, 3]}, Simp[-q^2/(3*d) Int[1/((1 - q*x)*(a + b*x^3)^(1/3) ), x], x] + Simp[q/d Subst[Int[1/(1 + 2*a*x^3), x], x, (1 + q*x)/(a + b*x ^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + a*d, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[ Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b, 3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sq rt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2^(7/3 )*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^ (1/3)])/(2^(7/3)*Rt[b, 3]*c), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]
\[\int \frac {x^{7}}{\left (-x^{3}+1\right )^{\frac {1}{3}} \left (x^{3}+1\right )}d x\]
\[ \int \frac {x^7}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int { \frac {x^{7}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {x^7}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {x^{7}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {x^7}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int { \frac {x^{7}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {x^7}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int { \frac {x^{7}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {x^7}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {x^7}{{\left (1-x^3\right )}^{1/3}\,\left (x^3+1\right )} \,d x \]