Integrand size = 15, antiderivative size = 118 \[ \int x^2 \sqrt [3]{x^2+x^3} \, dx=\frac {1}{324} \sqrt [3]{x^2+x^3} \left (20-12 x+9 x^2+81 x^3\right )+\frac {10 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )}{81 \sqrt {3}}+\frac {10}{243} \log \left (-x+\sqrt [3]{x^2+x^3}\right )-\frac {5}{243} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right ) \] Output:
1/324*(x^3+x^2)^(1/3)*(81*x^3+9*x^2-12*x+20)+10/243*3^(1/2)*arctan(3^(1/2) *x/(x+2*(x^3+x^2)^(1/3)))+10/243*ln(-x+(x^3+x^2)^(1/3))-5/243*ln(x^2+x*(x^ 3+x^2)^(1/3)+(x^3+x^2)^(2/3))
Time = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.36 \[ \int x^2 \sqrt [3]{x^2+x^3} \, dx=\frac {60 x^2+24 x^3-9 x^4+270 x^5+243 x^6+40 \sqrt {3} x^{4/3} (1+x)^{2/3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )+40 x^{4/3} (1+x)^{2/3} \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )-20 x^{4/3} (1+x)^{2/3} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )}{972 \left (x^2 (1+x)\right )^{2/3}} \] Input:
Integrate[x^2*(x^2 + x^3)^(1/3),x]
Output:
(60*x^2 + 24*x^3 - 9*x^4 + 270*x^5 + 243*x^6 + 40*Sqrt[3]*x^(4/3)*(1 + x)^ (2/3)*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(1 + x)^(1/3))] + 40*x^(4/3)*( 1 + x)^(2/3)*Log[-x^(1/3) + (1 + x)^(1/3)] - 20*x^(4/3)*(1 + x)^(2/3)*Log[ x^(2/3) + x^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)])/(972*(x^2*(1 + x))^(2/3) )
Time = 0.31 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.42, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1927, 1930, 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.
\(\displaystyle \int x^2 \sqrt [3]{x^3+x^2} \, dx\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle \frac {1}{12} \int \frac {x^4}{\left (x^3+x^2\right )^{2/3}}dx+\frac {1}{4} \sqrt [3]{x^3+x^2} x^3\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{3} x^2 \sqrt [3]{x^3+x^2}-\frac {8}{9} \int \frac {x^3}{\left (x^3+x^2\right )^{2/3}}dx\right )+\frac {1}{4} \sqrt [3]{x^3+x^2} x^3\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{3} x^2 \sqrt [3]{x^3+x^2}-\frac {8}{9} \left (\frac {1}{2} x \sqrt [3]{x^3+x^2}-\frac {5}{6} \int \frac {x^2}{\left (x^3+x^2\right )^{2/3}}dx\right )\right )+\frac {1}{4} \sqrt [3]{x^3+x^2} x^3\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{3} x^2 \sqrt [3]{x^3+x^2}-\frac {8}{9} \left (\frac {1}{2} x \sqrt [3]{x^3+x^2}-\frac {5}{6} \left (\sqrt [3]{x^3+x^2}-\frac {2}{3} \int \frac {x}{\left (x^3+x^2\right )^{2/3}}dx\right )\right )\right )+\frac {1}{4} \sqrt [3]{x^3+x^2} x^3\) |
\(\Big \downarrow \) 1938 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{3} x^2 \sqrt [3]{x^3+x^2}-\frac {8}{9} \left (\frac {1}{2} x \sqrt [3]{x^3+x^2}-\frac {5}{6} \left (\sqrt [3]{x^3+x^2}-\frac {2 x^{4/3} (x+1)^{2/3} \int \frac {1}{\sqrt [3]{x} (x+1)^{2/3}}dx}{3 \left (x^3+x^2\right )^{2/3}}\right )\right )\right )+\frac {1}{4} \sqrt [3]{x^3+x^2} x^3\) |
\(\Big \downarrow \) 71 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{3} x^2 \sqrt [3]{x^3+x^2}-\frac {8}{9} \left (\frac {1}{2} x \sqrt [3]{x^3+x^2}-\frac {5}{6} \left (\sqrt [3]{x^3+x^2}-\frac {2 x^{4/3} (x+1)^{2/3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )-\frac {1}{2} \log (x+1)-\frac {3}{2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x+1}}-1\right )\right )}{3 \left (x^3+x^2\right )^{2/3}}\right )\right )\right )+\frac {1}{4} \sqrt [3]{x^3+x^2} x^3\) |
Input:
Int[x^2*(x^2 + x^3)^(1/3),x]
Output:
(x^3*(x^2 + x^3)^(1/3))/4 + ((x^2*(x^2 + x^3)^(1/3))/3 - (8*((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]*A rcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[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)/12
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 = 0.73 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.13
method | result | size |
meijerg | \(\frac {3 x^{\frac {11}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], -x \right )}{11}\) | \(15\) |
pseudoelliptic | \(-\frac {x^{8} \left (-243 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x^{3}-27 x^{2} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+40 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+36 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x -40 \ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}-x}{x}\right )+20 \ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-60 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}\right )}{972 {\left (\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}\right )}^{4} {\left (\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}-x \right )}^{4}}\) | \(177\) |
trager | \(\left (\frac {1}{4} x^{3}+\frac {1}{36} x^{2}-\frac {1}{27} x +\frac {5}{81}\right ) \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+\frac {10 \ln \left (\frac {-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {1}{3}} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +24 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+24 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -16 x^{2}-12 x}{x}\right )}{243}+\frac {10 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {1}{3}} x +45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+24 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -20 x^{2}-8 x}{x}\right )}{81}\) | \(335\) |
risch | \(\frac {\left (81 x^{3}+9 x^{2}-12 x +20\right ) \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{324}+\frac {\left (\frac {10 \ln \left (-\frac {x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{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 )}{243}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \ln \left (\frac {-2 x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{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}+30 \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}}-48 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}-28 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x +10 x^{2}-48 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )+14 x +4}{1+x}\right )}{243}\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 )}\) | \(461\) |
Input:
int(x^2*(x^3+x^2)^(1/3),x,method=_RETURNVERBOSE)
Output:
3/11*x^(11/3)*hypergeom([-1/3,11/3],[14/3],-x)
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.93 \[ \int x^2 \sqrt [3]{x^2+x^3} \, dx=-\frac {10}{243} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{324} \, {\left (81 \, x^{3} + 9 \, x^{2} - 12 \, x + 20\right )} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} + \frac {10}{243} \, \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {5}{243} \, \log \left (\frac {x^{2} + {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \] Input:
integrate(x^2*(x^3+x^2)^(1/3),x, algorithm="fricas")
Output:
-10/243*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + x^2)^(1/3))/x) + 1/324*(81*x^3 + 9*x^2 - 12*x + 20)*(x^3 + x^2)^(1/3) + 10/243*log(-(x - (x ^3 + x^2)^(1/3))/x) - 5/243*log((x^2 + (x^3 + x^2)^(1/3)*x + (x^3 + x^2)^( 2/3))/x^2)
\[ \int x^2 \sqrt [3]{x^2+x^3} \, dx=\int x^{2} \sqrt [3]{x^{2} \left (x + 1\right )}\, dx \] Input:
integrate(x**2*(x**3+x**2)**(1/3),x)
Output:
Integral(x**2*(x**2*(x + 1))**(1/3), x)
\[ \int x^2 \sqrt [3]{x^2+x^3} \, dx=\int { {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x^{2} \,d x } \] Input:
integrate(x^2*(x^3+x^2)^(1/3),x, algorithm="maxima")
Output:
integrate((x^3 + x^2)^(1/3)*x^2, x)
Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82 \[ \int x^2 \sqrt [3]{x^2+x^3} \, dx=\frac {1}{324} \, {\left (20 \, {\left (\frac {1}{x} + 1\right )}^{\frac {10}{3}} - 72 \, {\left (\frac {1}{x} + 1\right )}^{\frac {7}{3}} + 93 \, {\left (\frac {1}{x} + 1\right )}^{\frac {4}{3}} + 40 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )} x^{4} - \frac {10}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5}{243} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {10}{243} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \] Input:
integrate(x^2*(x^3+x^2)^(1/3),x, algorithm="giac")
Output:
1/324*(20*(1/x + 1)^(10/3) - 72*(1/x + 1)^(7/3) + 93*(1/x + 1)^(4/3) + 40* (1/x + 1)^(1/3))*x^4 - 10/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(1/x + 1)^(1/3 ) + 1)) - 5/243*log((1/x + 1)^(2/3) + (1/x + 1)^(1/3) + 1) + 10/243*log(ab s((1/x + 1)^(1/3) - 1))
Timed out. \[ \int x^2 \sqrt [3]{x^2+x^3} \, dx=\int x^2\,{\left (x^3+x^2\right )}^{1/3} \,d x \] Input:
int(x^2*(x^2 + x^3)^(1/3),x)
Output:
int(x^2*(x^2 + x^3)^(1/3), x)
\[ \int x^2 \sqrt [3]{x^2+x^3} \, dx=\frac {x^{\frac {11}{3}} \left (x +1\right )^{\frac {1}{3}}}{4}+\frac {x^{\frac {8}{3}} \left (x +1\right )^{\frac {1}{3}}}{36}-\frac {x^{\frac {5}{3}} \left (x +1\right )^{\frac {1}{3}}}{27}+\frac {5 x^{\frac {2}{3}} \left (x +1\right )^{\frac {1}{3}}}{81}-\frac {10 \left (\int \frac {\left (x +1\right )^{\frac {1}{3}}}{x^{\frac {4}{3}}+x^{\frac {1}{3}}}d x \right )}{243} \] Input:
int(x^2*(x^3+x^2)^(1/3),x)
Output:
(243*x**(2/3)*(x + 1)**(1/3)*x**3 + 27*x**(2/3)*(x + 1)**(1/3)*x**2 - 36*x **(2/3)*(x + 1)**(1/3)*x + 60*x**(2/3)*(x + 1)**(1/3) - 40*int((x + 1)**(1 /3)/(x**(1/3)*x + x**(1/3)),x))/972