Integrand size = 27, antiderivative size = 111 \[ \int \frac {\left (2+x^3\right ) \left (1+2 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\frac {\left (-4-3 x^3\right ) \left (1+2 x^3\right )^{2/3}}{10 x^5}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+2 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{1+2 x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+2 x^3}+\left (1+2 x^3\right )^{2/3}\right ) \]
1/10*(-3*x^3-4)*(2*x^3+1)^(2/3)/x^5-1/3*arctan(3^(1/2)*x/(x+2*(2*x^3+1)^(1 /3)))*3^(1/2)+1/3*ln(-x+(2*x^3+1)^(1/3))-1/6*ln(x^2+x*(2*x^3+1)^(1/3)+(2*x ^3+1)^(2/3))
Time = 0.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x^3\right ) \left (1+2 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\frac {\left (-4-3 x^3\right ) \left (1+2 x^3\right )^{2/3}}{10 x^5}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+2 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{1+2 x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+2 x^3}+\left (1+2 x^3\right )^{2/3}\right ) \]
((-4 - 3*x^3)*(1 + 2*x^3)^(2/3))/(10*x^5) - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + 2*x^3)^(1/3))]/Sqrt[3] + Log[-x + (1 + 2*x^3)^(1/3)]/3 - Log[x^2 + x*(1 + 2*x^3)^(1/3) + (1 + 2*x^3)^(2/3)]/6
Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1050, 1053, 27, 901}
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.
| \(\displaystyle \int \frac {\left (x^3+2\right ) \left (2 x^3+1\right )^{2/3}}{x^6 \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle \frac {1}{5} \int \frac {3-2 x^3}{x^3 \left (x^3+1\right ) \sqrt [3]{2 x^3+1}}dx-\frac {2 \left (2 x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{5} \left (-\frac {1}{2} \int \frac {10}{\left (x^3+1\right ) \sqrt [3]{2 x^3+1}}dx-\frac {3 \left (2 x^3+1\right )^{2/3}}{2 x^2}\right )-\frac {2 \left (2 x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (-5 \int \frac {1}{\left (x^3+1\right ) \sqrt [3]{2 x^3+1}}dx-\frac {3 \left (2 x^3+1\right )^{2/3}}{2 x^2}\right )-\frac {2 \left (2 x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {1}{5} \left (-5 \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{2 x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (x^3+1\right )-\frac {1}{2} \log \left (x-\sqrt [3]{2 x^3+1}\right )\right )-\frac {3 \left (2 x^3+1\right )^{2/3}}{2 x^2}\right )-\frac {2 \left (2 x^3+1\right )^{2/3}}{5 x^5}\) |
(-2*(1 + 2*x^3)^(2/3))/(5*x^5) + ((-3*(1 + 2*x^3)^(2/3))/(2*x^2) - 5*(ArcT an[(1 + (2*x)/(1 + 2*x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + Log[1 + x^3]/6 - Log[x - (1 + 2*x^3)^(1/3)]/2))/5
3.17.40.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S qrt[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]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^n*(m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1 ))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n, 0] && G tQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Time = 2.48 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07
| method | result | size |
| pseudoelliptic | \(\frac {10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (2 x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-5 \ln \left (\frac {x^{2}+x \left (2 x^{3}+1\right )^{\frac {1}{3}}+\left (2 x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+10 \ln \left (\frac {-x +\left (2 x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}-9 \left (2 x^{3}+1\right )^{\frac {2}{3}} x^{3}-12 \left (2 x^{3}+1\right )^{\frac {2}{3}}}{30 x^{5}}\) | \(119\) |
| risch | \(-\frac {6 x^{6}+11 x^{3}+4}{10 x^{5} \left (2 x^{3}+1\right )^{\frac {1}{3}}}+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x +15 \left (2 x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+4 \left (2 x^{3}+1\right )^{\frac {2}{3}} x +4 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+9 x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )-\frac {\ln \left (-\frac {-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x +15 \left (2 x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+\left (2 x^{3}+1\right )^{\frac {2}{3}} x +\left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}-\ln \left (-\frac {-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x +15 \left (2 x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+\left (2 x^{3}+1\right )^{\frac {2}{3}} x +\left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(488\) |
| trager | \(-\frac {\left (3 x^{3}+4\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}}}{10 x^{5}}+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {33786 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+48096 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x +48096 \left (2 x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+108513 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-17091 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -17091 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-13690 x^{3}-33786 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+37893 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4107}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )-\frac {\ln \left (-\frac {-99 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x +45 \left (2 x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+204 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+177 \left (2 x^{3}+1\right )^{\frac {2}{3}} x +177 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+279 x^{3}+99 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+225 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+124}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}-\ln \left (-\frac {-99 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x +45 \left (2 x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+204 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+177 \left (2 x^{3}+1\right )^{\frac {2}{3}} x +177 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+279 x^{3}+99 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+225 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+124}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(532\) |
1/30*(10*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(2*x^3+1)^(1/3)))*x^5-5*ln((x^2 +x*(2*x^3+1)^(1/3)+(2*x^3+1)^(2/3))/x^2)*x^5+10*ln((-x+(2*x^3+1)^(1/3))/x) *x^5-9*(2*x^3+1)^(2/3)*x^3-12*(2*x^3+1)^(2/3))/x^5
Time = 0.62 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.20 \[ \int \frac {\left (2+x^3\right ) \left (1+2 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {4 \, \sqrt {3} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2 \, x^{3} + 1\right )}}{10 \, x^{3} + 1}\right ) - 5 \, x^{5} \log \left (\frac {x^{3} + 3 \, {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{3} + 1}\right ) + 3 \, {\left (3 \, x^{3} + 4\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \]
-1/30*(10*sqrt(3)*x^5*arctan(-(4*sqrt(3)*(2*x^3 + 1)^(1/3)*x^2 - 2*sqrt(3) *(2*x^3 + 1)^(2/3)*x + sqrt(3)*(2*x^3 + 1))/(10*x^3 + 1)) - 5*x^5*log((x^3 + 3*(2*x^3 + 1)^(1/3)*x^2 - 3*(2*x^3 + 1)^(2/3)*x + 1)/(x^3 + 1)) + 3*(3* x^3 + 4)*(2*x^3 + 1)^(2/3))/x^5
\[ \int \frac {\left (2+x^3\right ) \left (1+2 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int \frac {\left (x^{3} + 2\right ) \left (2 x^{3} + 1\right )^{\frac {2}{3}}}{x^{6} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {\left (2+x^3\right ) \left (1+2 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int { \frac {{\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 2\right )}}{{\left (x^{3} + 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (2+x^3\right ) \left (1+2 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int { \frac {{\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 2\right )}}{{\left (x^{3} + 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (2+x^3\right ) \left (1+2 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int \frac {\left (x^3+2\right )\,{\left (2\,x^3+1\right )}^{2/3}}{x^6\,\left (x^3+1\right )} \,d x \]