Integrand size = 15, antiderivative size = 110 \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=\frac {\left (-x^2+x^3\right )^{2/3}}{x}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \]
(x^3-x^2)^(2/3)/x+1/3*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^3-x^2)^(1/3)))-1/3* ln(-x+(x^3-x^2)^(1/3))+1/6*ln(x^2+x*(x^3-x^2)^(1/3)+(x^3-x^2)^(2/3))
Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.25 \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=\frac {x^{2/3} \left (-6 \sqrt [3]{x}+6 x^{4/3}+2 \sqrt {3} \sqrt [3]{-1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2 \sqrt [3]{-1+x} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )+\sqrt [3]{-1+x} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )\right )}{6 \sqrt [3]{(-1+x) x^2}} \]
(x^(2/3)*(-6*x^(1/3) + 6*x^(4/3) + 2*Sqrt[3]*(-1 + x)^(1/3)*ArcTan[(Sqrt[3 ]*x^(1/3))/(2*(-1 + x)^(1/3) + x^(1/3))] - 2*(-1 + x)^(1/3)*Log[(-1 + x)^( 1/3) - x^(1/3)] + (-1 + x)^(1/3)*Log[(-1 + x)^(2/3) + (-1 + x)^(1/3)*x^(1/ 3) + x^(2/3)]))/(6*((-1 + x)*x^2)^(1/3))
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1930, 1917, 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.
| \(\displaystyle \int \frac {x}{\sqrt [3]{x^3-x^2}} \, dx\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{\sqrt [3]{x^3-x^2}}dx+\frac {\left (x^3-x^2\right )^{2/3}}{x}\) |
| \(\Big \downarrow \) 1917 |
| \(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \int \frac {1}{\sqrt [3]{x-1} x^{2/3}}dx}{3 \sqrt [3]{x^3-x^2}}+\frac {\left (x^3-x^2\right )^{2/3}}{x}\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )-\frac {\log (x)}{2}\right )}{3 \sqrt [3]{x^3-x^2}}+\frac {\left (x^3-x^2\right )^{2/3}}{x}\) |
(-x^2 + x^3)^(2/3)/x + ((-1 + x)^(1/3)*x^(2/3)*(-(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))]) - (3*Log[-1 + (-1 + x)^(1/3)/x^( 1/3)])/2 - Log[x]/2))/(3*(-x^2 + x^3)^(1/3))
3.17.8.3.1 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[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]) Int[ x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && !Integ erQ[p] && NeQ[n, j] && PosQ[n - j]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.25
| method | result | size |
| meijerg | \(\frac {3 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], x\right )}{4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(27\) |
| risch | \(\frac {x \left (-1+x \right )}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}+\frac {\left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\operatorname {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(41\) |
| pseudoelliptic | \(-\frac {x \left (x \sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+x \ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x}{x}\right )-\frac {x \ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}-3 \left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}\right )}{3 \left (-\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \left (\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+x \left (x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}\right )\right )}\) | \(135\) |
| trager | \(\frac {\left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{x}+\frac {\ln \left (-\frac {-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{2}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +174 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{2}-60 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-60 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-138 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -80 x^{2}+48 x}{x}\right )}{3}-\frac {\ln \left (-\frac {-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{2}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +174 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{2}-60 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-60 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-138 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -80 x^{2}+48 x}{x}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )}{2}+\frac {\operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{2}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +114 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{2}-36 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-36 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -16 x^{2}+4 x}{x}\right )}{2}\) | \(509\) |
Time = 0.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.03 \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=-\frac {2 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 2 \, x \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - x \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 6 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{6 \, x} \]
-1/6*(2*sqrt(3)*x*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - x^2)^(1/3))/x) + 2*x*log(-(x - (x^3 - x^2)^(1/3))/x) - x*log((x^2 + (x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2) - 6*(x^3 - x^2)^(2/3))/x
\[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=\int \frac {x}{\sqrt [3]{x^{2} \left (x - 1\right )}}\, dx \]
\[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {x}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.67 \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=x {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{6} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
x*(-1/x + 1)^(2/3) - 1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-1/x + 1)^(1/3) + 1)) + 1/6*log((-1/x + 1)^(2/3) + (-1/x + 1)^(1/3) + 1) - 1/3*log(abs((-1/x + 1)^(1/3) - 1))
Timed out. \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=\int \frac {x}{{\left (x^3-x^2\right )}^{1/3}} \,d x \]