Integrand size = 18, antiderivative size = 74 \[ \int \frac {x}{\sqrt {-1+x^3} \left (8+x^3\right )} \, dx=\frac {1}{18} \arctan \left (\frac {(1-x)^2}{3 \sqrt {-1+x^3}}\right )+\frac {1}{18} \arctan \left (\frac {1}{3} \sqrt {-1+x^3}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} (1-x)}{\sqrt {-1+x^3}}\right )}{6 \sqrt {3}} \]
1/18*arctan(1/3*(1-x)^2/(x^3-1)^(1/2))+1/18*arctan(1/3*(x^3-1)^(1/2))-1/18 *arctanh((1-x)*3^(1/2)/(x^3-1)^(1/2))*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.65 \[ \int \frac {x}{\sqrt {-1+x^3} \left (8+x^3\right )} \, dx=\frac {x^2 \sqrt {1-x^3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},x^3,-\frac {x^3}{8}\right )}{16 \sqrt {-1+x^3}} \]
Time = 0.43 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {988, 25, 946, 73, 216, 2563, 216, 2570, 220}
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.
| \(\displaystyle \int \frac {x}{\sqrt {x^3-1} \left (x^3+8\right )} \, dx\) |
| \(\Big \downarrow \) 988 |
| \(\displaystyle -\frac {1}{12} \int \frac {1-x}{(x+2) \sqrt {x^3-1}}dx-\frac {1}{12} \int -\frac {-x^2+2 x+2}{\left (x^2-2 x+4\right ) \sqrt {x^3-1}}dx-\frac {1}{4} \int -\frac {x^2}{\sqrt {x^3-1} \left (x^3+8\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{12} \int \frac {1-x}{(x+2) \sqrt {x^3-1}}dx+\frac {1}{12} \int \frac {-x^2+2 x+2}{\left (x^2-2 x+4\right ) \sqrt {x^3-1}}dx+\frac {1}{4} \int \frac {x^2}{\sqrt {x^3-1} \left (x^3+8\right )}dx\) |
| \(\Big \downarrow \) 946 |
| \(\displaystyle -\frac {1}{12} \int \frac {1-x}{(x+2) \sqrt {x^3-1}}dx+\frac {1}{12} \int \frac {1}{\sqrt {x^3-1} \left (x^3+8\right )}dx^3+\frac {1}{12} \int \frac {-x^2+2 x+2}{\left (x^2-2 x+4\right ) \sqrt {x^3-1}}dx\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {1}{12} \int \frac {1-x}{(x+2) \sqrt {x^3-1}}dx+\frac {1}{6} \int \frac {1}{x^6+9}d\sqrt {x^3-1}+\frac {1}{12} \int \frac {-x^2+2 x+2}{\left (x^2-2 x+4\right ) \sqrt {x^3-1}}dx\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {1}{12} \int \frac {1-x}{(x+2) \sqrt {x^3-1}}dx+\frac {1}{12} \int \frac {-x^2+2 x+2}{\left (x^2-2 x+4\right ) \sqrt {x^3-1}}dx+\frac {1}{18} \arctan \left (\frac {\sqrt {x^3-1}}{3}\right )\) |
| \(\Big \downarrow \) 2563 |
| \(\displaystyle \frac {1}{6} \int \frac {1}{\frac {(1-x)^4}{x^3-1}+9}d\frac {(1-x)^2}{\sqrt {x^3-1}}+\frac {1}{12} \int \frac {-x^2+2 x+2}{\left (x^2-2 x+4\right ) \sqrt {x^3-1}}dx+\frac {1}{18} \arctan \left (\frac {\sqrt {x^3-1}}{3}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{12} \int \frac {-x^2+2 x+2}{\left (x^2-2 x+4\right ) \sqrt {x^3-1}}dx+\frac {1}{18} \arctan \left (\frac {(1-x)^2}{3 \sqrt {x^3-1}}\right )+\frac {1}{18} \arctan \left (\frac {\sqrt {x^3-1}}{3}\right )\) |
| \(\Big \downarrow \) 2570 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{\frac {6 (1-x)^2}{x^3-1}-2}d\frac {1-x}{\sqrt {x^3-1}}+\frac {1}{18} \arctan \left (\frac {(1-x)^2}{3 \sqrt {x^3-1}}\right )+\frac {1}{18} \arctan \left (\frac {\sqrt {x^3-1}}{3}\right )\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {1}{18} \arctan \left (\frac {(1-x)^2}{3 \sqrt {x^3-1}}\right )+\frac {1}{18} \arctan \left (\frac {\sqrt {x^3-1}}{3}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} (1-x)}{\sqrt {x^3-1}}\right )}{6 \sqrt {3}}\) |
ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])]/18 + ArcTan[Sqrt[-1 + x^3]/3]/18 - Ar cTanh[(Sqrt[3]*(1 - x))/Sqrt[-1 + x^3]]/(6*Sqrt[3])
3.1.75.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[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]
Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> Wi th[{q = Rt[d/c, 3]}, Simp[d*(q/(4*b)) Int[x^2/((8*c - d*x^3)*Sqrt[c + d*x ^3]), x], x] + (-Simp[q^2/(12*b) Int[(1 + q*x)/((2 - q*x)*Sqrt[c + d*x^3] ), x], x] + Simp[1/(12*b*c) Int[(2*c*q^2 - 2*d*x - d*q*x^2)/((4 + 2*q*x + q^2*x^2)*Sqrt[c + d*x^3]), x], x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[8*b*c + a*d, 0]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[-2*(e/d) Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & & EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)* Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Simp[-2*g*h Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /; Fre eQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8 *a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 3.02 (sec) , antiderivative size = 421, normalized size of antiderivative = 5.69
| method | result | size |
| default | \(-\frac {\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \sqrt {x^{3}-1}}+\frac {i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1+i \sqrt {3}\right ) \sqrt {3}}{6}+\frac {i \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \sqrt {x^{3}-1}}-\frac {i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1-i \sqrt {3}\right ) \sqrt {3}}{6}-\frac {2 i \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \sqrt {x^{3}-1}}\) | \(421\) |
| trager | \(324 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {-629856 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{5} x^{2}+2519424 x \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{5}+972 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{3} x^{2}-11664 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{3} x +324 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-1}-3888 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{3}+12 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right ) x -\sqrt {x^{3}-1}+6 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )}{\left (162 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{2} x -1\right ) \left (2+x \right )}\right )-\operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {-629856 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{5} x^{2}+2519424 x \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{5}+972 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{3} x^{2}-11664 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{3} x +324 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-1}-3888 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{3}+12 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right ) x -\sqrt {x^{3}-1}+6 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )}{\left (162 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{2} x -1\right ) \left (2+x \right )}\right )-\operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {104976 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{4} x^{2}-419904 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{4} x -1296 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{2} x +108 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )-648 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{2}-x^{2}-2}{\left (324 \operatorname {RootOf}\left (104976 \textit {\_Z}^{4}-324 \textit {\_Z}^{2}+1\right )^{2} x -x +2\right ) \left (2+x \right )}\right )\) | \(558\) |
| elliptic | \(\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \sqrt {3}\, \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1+i \sqrt {3}\right ) \sqrt {3}}{6}+\frac {i \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{6 \sqrt {x^{3}-1}}+\frac {\sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1+i \sqrt {3}\right ) \sqrt {3}}{6}+\frac {i \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{6 \sqrt {x^{3}-1}}+\frac {\sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1-i \sqrt {3}\right ) \sqrt {3}}{6}-\frac {2 i \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}+\frac {\sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{6 \sqrt {x^{3}-1}}+\frac {i \sqrt {-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{18 \sqrt {x^{3}-1}}\) | \(889\) |
-1/9*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2* I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^ (1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2) ,1/6*I*3^(1/2)+1/2,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+1/9*I* (1/2-1/2*I*3^(1/2))*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/ 2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2) )/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*3^(1/2)*EllipticPi(((-1+x)/(-3/ 2-1/2*I*3^(1/2)))^(1/2),1/6*I*(1+I*3^(1/2))*3^(1/2)+1/3*I*3^(1/2),((3/2+1/ 2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))-1/9*I*(1/2+1/2*I*3^(1/2))*(-3/2-1 /2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/( 3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2 )/(x^3-1)^(1/2)*3^(1/2)*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),1/6 *I*(1-I*3^(1/2))*3^(1/2)-2/3*I*3^(1/2),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^( 1/2)))^(1/2))
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 723, normalized size of antiderivative = 9.77 \[ \int \frac {x}{\sqrt {-1+x^3} \left (8+x^3\right )} \, dx =\text {Too large to display} \]
-1/216*sqrt(2)*sqrt(I*sqrt(3) + 1)*log((2*x^6 + 6*x^5 - 150*x^4 + 176*x^3 + 12*x^2 - 3*sqrt(x^3 - 1)*(sqrt(3)*sqrt(2)*(7*I*x^3 - 12*I*x^2 - 12*I*x + 8*I) + sqrt(2)*(2*x^4 - 5*x^3 - 24*x^2 + 28*x + 8))*sqrt(I*sqrt(3) + 1) - 18*sqrt(3)*(-I*x^5 + I*x^4 + 6*I*x^3 - 2*I*x^2 - 4*I*x) + 168*x - 160)/(x ^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)) + 1/216*sqrt(2)*sqrt(I *sqrt(3) + 1)*log((2*x^6 + 6*x^5 - 150*x^4 + 176*x^3 + 12*x^2 - 3*sqrt(x^3 - 1)*(sqrt(3)*sqrt(2)*(-7*I*x^3 + 12*I*x^2 + 12*I*x - 8*I) - sqrt(2)*(2*x ^4 - 5*x^3 - 24*x^2 + 28*x + 8))*sqrt(I*sqrt(3) + 1) - 18*sqrt(3)*(-I*x^5 + I*x^4 + 6*I*x^3 - 2*I*x^2 - 4*I*x) + 168*x - 160)/(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)) + 1/216*sqrt(2)*sqrt(-I*sqrt(3) + 1)*log(( 2*x^6 + 6*x^5 - 150*x^4 + 176*x^3 + 12*x^2 - 3*sqrt(x^3 - 1)*(sqrt(3)*sqrt (2)*(7*I*x^3 - 12*I*x^2 - 12*I*x + 8*I) - sqrt(2)*(2*x^4 - 5*x^3 - 24*x^2 + 28*x + 8))*sqrt(-I*sqrt(3) + 1) - 18*sqrt(3)*(I*x^5 - I*x^4 - 6*I*x^3 + 2*I*x^2 + 4*I*x) + 168*x - 160)/(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)) - 1/216*sqrt(2)*sqrt(-I*sqrt(3) + 1)*log((2*x^6 + 6*x^5 - 150* x^4 + 176*x^3 + 12*x^2 - 3*sqrt(x^3 - 1)*(sqrt(3)*sqrt(2)*(-7*I*x^3 + 12*I *x^2 + 12*I*x - 8*I) + sqrt(2)*(2*x^4 - 5*x^3 - 24*x^2 + 28*x + 8))*sqrt(- I*sqrt(3) + 1) - 18*sqrt(3)*(I*x^5 - I*x^4 - 6*I*x^3 + 2*I*x^2 + 4*I*x) + 168*x - 160)/(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)) + 1/54* arctan(1/6*(x^3 - 12*x^2 - 6*x - 10)*sqrt(x^3 - 1)/(x^4 - x^3 - x + 1))
\[ \int \frac {x}{\sqrt {-1+x^3} \left (8+x^3\right )} \, dx=\int \frac {x}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 2\right ) \left (x^{2} - 2 x + 4\right )}\, dx \]
\[ \int \frac {x}{\sqrt {-1+x^3} \left (8+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} + 8\right )} \sqrt {x^{3} - 1}} \,d x } \]
\[ \int \frac {x}{\sqrt {-1+x^3} \left (8+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} + 8\right )} \sqrt {x^{3} - 1}} \,d x } \]
Time = 0.21 (sec) , antiderivative size = 533, normalized size of antiderivative = 7.20 \[ \int \frac {x}{\sqrt {-1+x^3} \left (8+x^3\right )} \, dx =\text {Too large to display} \]
(((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2 ))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi((3^(1/2)*1i)/6 + 1/2, asin((-( x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1 i)/2 - 3/2)))/(9*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^( 1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2)) - (3^(1/2)*((3 ^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^( 1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/( (3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(-(3^(1/2)*((3^(1/2)*1i)/2 + 3/2)*1 i)/3, asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/ 2)/((3^(1/2)*1i)/2 - 3/2))*2i)/(9*(3^(1/2)*1i - 1)*(((3^(1/2)*1i)/2 - 1/2) *((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2)) - (3^(1/2)*((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/ 2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/ 2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi( (3^(1/2)*((3^(1/2)*1i)/2 + 3/2)*1i)/3, asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/ 2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))*2i)/(9*(3^(1/2 )*1i + 1)*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i )/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2))
\[ \int \frac {x}{\sqrt {-1+x^3} \left (8+x^3\right )} \, dx=\int \frac {\sqrt {x^{3}-1}\, x}{x^{6}+7 x^{3}-8}d x \]