Integrand size = 13, antiderivative size = 113 \[ \int x \sqrt [3]{x^2+x^3} \, dx=\frac {1}{54} \left (-5+3 x+18 x^2\right ) \sqrt [3]{x^2+x^3}-\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )}{27 \sqrt {3}}-\frac {5}{81} \log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {5}{162} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right ) \]
1/54*(18*x^2+3*x-5)*(x^3+x^2)^(1/3)-5/81*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^ 3+x^2)^(1/3)))-5/81*ln(-x+(x^3+x^2)^(1/3))+5/162*ln(x^2+x*(x^3+x^2)^(1/3)+ (x^3+x^2)^(2/3))
Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.35 \[ \int x \sqrt [3]{x^2+x^3} \, dx=\frac {x^{4/3} (1+x)^{2/3} \left (-15 x^{2/3} \sqrt [3]{1+x}+9 x^{5/3} \sqrt [3]{1+x}+54 x^{8/3} \sqrt [3]{1+x}-10 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )-10 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )+5 \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{162 \left (x^2 (1+x)\right )^{2/3}} \]
(x^(4/3)*(1 + x)^(2/3)*(-15*x^(2/3)*(1 + x)^(1/3) + 9*x^(5/3)*(1 + x)^(1/3 ) + 54*x^(8/3)*(1 + x)^(1/3) - 10*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3 ) + 2*(1 + x)^(1/3))] - 10*Log[-x^(1/3) + (1 + x)^(1/3)] + 5*Log[x^(2/3) + x^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)]))/(162*(x^2*(1 + x))^(2/3))
Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.28, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1927, 1930, 1930, 1938, 71}
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.
| \(\displaystyle \int x \sqrt [3]{x^3+x^2} \, dx\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle \frac {1}{9} \int \frac {x^3}{\left (x^3+x^2\right )^{2/3}}dx+\frac {1}{3} \sqrt [3]{x^3+x^2} x^2\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{2} x \sqrt [3]{x^3+x^2}-\frac {5}{6} \int \frac {x^2}{\left (x^3+x^2\right )^{2/3}}dx\right )+\frac {1}{3} \sqrt [3]{x^3+x^2} x^2\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{2} x \sqrt [3]{x^3+x^2}-\frac {5}{6} \left (\sqrt [3]{x^3+x^2}-\frac {2}{3} \int \frac {x}{\left (x^3+x^2\right )^{2/3}}dx\right )\right )+\frac {1}{3} \sqrt [3]{x^3+x^2} x^2\) |
| \(\Big \downarrow \) 1938 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{2} x \sqrt [3]{x^3+x^2}-\frac {5}{6} \left (\sqrt [3]{x^3+x^2}-\frac {2 x^{4/3} (x+1)^{2/3} \int \frac {1}{\sqrt [3]{x} (x+1)^{2/3}}dx}{3 \left (x^3+x^2\right )^{2/3}}\right )\right )+\frac {1}{3} \sqrt [3]{x^3+x^2} x^2\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{2} x \sqrt [3]{x^3+x^2}-\frac {5}{6} \left (\sqrt [3]{x^3+x^2}-\frac {2 x^{4/3} (x+1)^{2/3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )-\frac {1}{2} \log (x+1)-\frac {3}{2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x+1}}-1\right )\right )}{3 \left (x^3+x^2\right )^{2/3}}\right )\right )+\frac {1}{3} \sqrt [3]{x^3+x^2} x^2\) |
(x^2*(x^2 + x^3)^(1/3))/3 + ((x*(x^2 + x^3)^(1/3))/2 - (5*((x^2 + x^3)^(1/ 3) - (2*x^(4/3)*(1 + x)^(2/3)*(-(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(S qrt[3]*(1 + x)^(1/3))]) - Log[1 + x]/2 - (3*Log[-1 + x^(1/3)/(1 + x)^(1/3) ])/2))/(3*(x^2 + x^3)^(2/3))))/6)/9
3.17.81.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a* (n - j)*(p/(c^j*(m + n*p + 1))) Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1) , x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Int egersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))) I nt[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && Gt Q[m + j*p - n + j + 1, 0] && NeQ[m + n*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])) Int[x^(m + j* p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !Inte gerQ[p] && NeQ[n, j] && PosQ[n - j]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.65 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.13
| method | result | size |
| meijerg | \(\frac {3 x^{\frac {8}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], -x \right )}{8}\) | \(15\) |
| pseudoelliptic | \(\frac {x^{6} \left (54 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x^{2}+10 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+9 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +5 \ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-10 \ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}-x}{x}\right )-15 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}\right )}{162 {\left (\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}\right )}^{3} {\left (\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}-x \right )}^{3}}\) | \(163\) |
| risch | \(\frac {\left (18 x^{2}+3 x -5\right ) \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{54}+\frac {\left (-\frac {5 \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+48 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-36 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+96 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}-14 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -64 x^{2}+96 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-112 x -48}{1+x}\right )}{81}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -19 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}-28 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -30 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+48 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -10 x^{2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+48 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}-14 x -4}{1+x}\right )}{162}\right ) \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} \left (\left (1+x \right )^{2} x \right )^{\frac {1}{3}}}{x \left (1+x \right )}\) | \(455\) |
| trager | \(\left (\frac {1}{3} x^{2}+\frac {1}{18} x -\frac {5}{54}\right ) \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-\frac {5 \ln \left (-\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+72 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -30 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-15 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+25 x^{2}+10 x}{x}\right )}{81}+\frac {5 \ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-27 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+48 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +24 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-15 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-4 x^{2}-3 x}{x}\right )}{81}-\frac {5 \ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-27 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+48 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +24 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-15 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-4 x^{2}-3 x}{x}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{27}\) | \(478\) |
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.93 \[ \int x \sqrt [3]{x^2+x^3} \, dx=\frac {5}{81} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{54} \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (18 \, x^{2} + 3 \, x - 5\right )} - \frac {5}{81} \, \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {5}{162} \, \log \left (\frac {x^{2} + {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
5/81*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + x^2)^(1/3))/x) + 1/5 4*(x^3 + x^2)^(1/3)*(18*x^2 + 3*x - 5) - 5/81*log(-(x - (x^3 + x^2)^(1/3)) /x) + 5/162*log((x^2 + (x^3 + x^2)^(1/3)*x + (x^3 + x^2)^(2/3))/x^2)
\[ \int x \sqrt [3]{x^2+x^3} \, dx=\int x \sqrt [3]{x^{2} \left (x + 1\right )}\, dx \]
\[ \int x \sqrt [3]{x^2+x^3} \, dx=\int { {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x \,d x } \]
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78 \[ \int x \sqrt [3]{x^2+x^3} \, dx=-\frac {1}{54} \, {\left (5 \, {\left (\frac {1}{x} + 1\right )}^{\frac {7}{3}} - 13 \, {\left (\frac {1}{x} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )} x^{3} + \frac {5}{81} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {5}{162} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {5}{81} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
-1/54*(5*(1/x + 1)^(7/3) - 13*(1/x + 1)^(4/3) - 10*(1/x + 1)^(1/3))*x^3 + 5/81*sqrt(3)*arctan(1/3*sqrt(3)*(2*(1/x + 1)^(1/3) + 1)) + 5/162*log((1/x + 1)^(2/3) + (1/x + 1)^(1/3) + 1) - 5/81*log(abs((1/x + 1)^(1/3) - 1))
Timed out. \[ \int x \sqrt [3]{x^2+x^3} \, dx=\int x\,{\left (x^3+x^2\right )}^{1/3} \,d x \]