Integrand size = 13, antiderivative size = 109 \[ \int x^7 \sqrt [3]{1+x^3} \, dx=\frac {1}{162} \sqrt [3]{1+x^3} \left (-5 x^2+3 x^5+18 x^8\right )-\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{81 \sqrt {3}}-\frac {5}{243} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {5}{486} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
1/162*(x^3+1)^(1/3)*(18*x^8+3*x^5-5*x^2)-5/243*arctan(3^(1/2)*x/(x+2*(x^3+ 1)^(1/3)))*3^(1/2)-5/243*ln(-x+(x^3+1)^(1/3))+5/486*ln(x^2+x*(x^3+1)^(1/3) +(x^3+1)^(2/3))
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int x^7 \sqrt [3]{1+x^3} \, dx=\frac {1}{486} \left (3 x^2 \sqrt [3]{1+x^3} \left (-5+3 x^3+18 x^6\right )-10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )-10 \log \left (-x+\sqrt [3]{1+x^3}\right )+5 \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right ) \]
(3*x^2*(1 + x^3)^(1/3)*(-5 + 3*x^3 + 18*x^6) - 10*Sqrt[3]*ArcTan[(Sqrt[3]* x)/(x + 2*(1 + x^3)^(1/3))] - 10*Log[-x + (1 + x^3)^(1/3)] + 5*Log[x^2 + x *(1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/486
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {811, 843, 843, 853}
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.
\(\displaystyle \int x^7 \sqrt [3]{x^3+1} \, dx\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {1}{9} \int \frac {x^7}{\left (x^3+1\right )^{2/3}}dx+\frac {1}{9} \sqrt [3]{x^3+1} x^8\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{6} x^5 \sqrt [3]{x^3+1}-\frac {5}{6} \int \frac {x^4}{\left (x^3+1\right )^{2/3}}dx\right )+\frac {1}{9} \sqrt [3]{x^3+1} x^8\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{6} x^5 \sqrt [3]{x^3+1}-\frac {5}{6} \left (\frac {1}{3} x^2 \sqrt [3]{x^3+1}-\frac {2}{3} \int \frac {x}{\left (x^3+1\right )^{2/3}}dx\right )\right )+\frac {1}{9} \sqrt [3]{x^3+1} x^8\) |
\(\Big \downarrow \) 853 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{6} x^5 \sqrt [3]{x^3+1}-\frac {5}{6} \left (\frac {1}{3} x^2 \sqrt [3]{x^3+1}-\frac {2}{3} \left (-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3+1}\right )\right )\right )\right )+\frac {1}{9} \sqrt [3]{x^3+1} x^8\) |
(x^8*(1 + x^3)^(1/3))/9 + ((x^5*(1 + x^3)^(1/3))/6 - (5*((x^2*(1 + x^3)^(1 /3))/3 - (2*(-(ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3]) - Log[ x - (1 + x^3)^(1/3)]/2))/3))/6)/9
3.16.86.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp [Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.97 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.16
method | result | size |
meijerg | \(\frac {x^{8} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], -x^{3}\right )}{8}\) | \(17\) |
risch | \(\frac {x^{2} \left (18 x^{6}+3 x^{3}-5\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{162}+\frac {5 x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{162}\) | \(42\) |
pseudoelliptic | \(\frac {-5 \ln \left (\frac {x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )+10 \ln \left (\frac {-x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )+\left (-54 x^{8}-9 x^{5}+15 x^{2}\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{486 \left (x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right )^{3} {\left (x -\left (x^{3}+1\right )^{\frac {1}{3}}\right )}^{3}}\) | \(133\) |
trager | \(\frac {x^{2} \left (18 x^{6}+3 x^{3}-5\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{162}-\frac {5 \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{243}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}-2 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{243}\) | \(215\) |
Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int x^7 \sqrt [3]{1+x^3} \, dx=\frac {5}{243} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{162} \, {\left (18 \, x^{8} + 3 \, x^{5} - 5 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \frac {5}{243} \, \log \left (-\frac {x - {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {5}{486} \, \log \left (\frac {x^{2} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
5/243*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + 1)^(1/3))/x) + 1/16 2*(18*x^8 + 3*x^5 - 5*x^2)*(x^3 + 1)^(1/3) - 5/243*log(-(x - (x^3 + 1)^(1/ 3))/x) + 5/486*log((x^2 + (x^3 + 1)^(1/3)*x + (x^3 + 1)^(2/3))/x^2)
Result contains complex when optimal does not.
Time = 3.49 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.28 \[ \int x^7 \sqrt [3]{1+x^3} \, dx=\frac {x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \]
Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.33 \[ \int x^7 \sqrt [3]{1+x^3} \, dx=\frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {\frac {10 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {13 \, {\left (x^{3} + 1\right )}^{\frac {4}{3}}}{x^{4}} - \frac {5 \, {\left (x^{3} + 1\right )}^{\frac {7}{3}}}{x^{7}}}{162 \, {\left (\frac {3 \, {\left (x^{3} + 1\right )}}{x^{3}} - \frac {3 \, {\left (x^{3} + 1\right )}^{2}}{x^{6}} + \frac {{\left (x^{3} + 1\right )}^{3}}{x^{9}} - 1\right )}} + \frac {5}{486} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {5}{243} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
5/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3)/x + 1)) + 1/162*(10*(x ^3 + 1)^(1/3)/x + 13*(x^3 + 1)^(4/3)/x^4 - 5*(x^3 + 1)^(7/3)/x^7)/(3*(x^3 + 1)/x^3 - 3*(x^3 + 1)^2/x^6 + (x^3 + 1)^3/x^9 - 1) + 5/486*log((x^3 + 1)^ (1/3)/x + (x^3 + 1)^(2/3)/x^2 + 1) - 5/243*log((x^3 + 1)^(1/3)/x - 1)
\[ \int x^7 \sqrt [3]{1+x^3} \, dx=\int { {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{7} \,d x } \]
Timed out. \[ \int x^7 \sqrt [3]{1+x^3} \, dx=\int x^7\,{\left (x^3+1\right )}^{1/3} \,d x \]