Integrand size = 15, antiderivative size = 102 \[ \int (1-x)^{2/3} \sqrt [3]{x} \, dx=\frac {1}{6} (1-x)^{2/3} \sqrt [3]{x}-\frac {1}{2} (1-x)^{5/3} \sqrt [3]{x}+\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x}}\right )}{3 \sqrt {3}}+\frac {1}{6} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{x}}\right )+\frac {\log (x)}{18} \] Output:
1/6*(1-x)^(2/3)*x^(1/3)-1/2*(1-x)^(5/3)*x^(1/3)-1/9*arctan(-1/3*3^(1/2)+2/ 3*(1-x)^(1/3)*3^(1/2)/x^(1/3))*3^(1/2)+1/6*ln(1+(1-x)^(1/3)/x^(1/3))+1/18* ln(x)
Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.12 \[ \int (1-x)^{2/3} \sqrt [3]{x} \, dx=\frac {1}{18} \left (3 (1-x)^{2/3} \sqrt [3]{x} (-2+3 x)+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{1-x}-\sqrt [3]{x}}\right )+2 \log \left (\sqrt [3]{1-x}+\sqrt [3]{x}\right )-\log \left ((1-x)^{2/3}+x^{2/3}-\sqrt [3]{-((-1+x) x)}\right )\right ) \] Input:
Integrate[(1 - x)^(2/3)*x^(1/3),x]
Output:
(3*(1 - x)^(2/3)*x^(1/3)*(-2 + 3*x) + 2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/( 2*(1 - x)^(1/3) - x^(1/3))] + 2*Log[(1 - x)^(1/3) + x^(1/3)] - Log[(1 - x) ^(2/3) + x^(2/3) - (-((-1 + x)*x))^(1/3)])/18
Time = 0.19 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {60, 60, 72}
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.
\(\displaystyle \int (1-x)^{2/3} \sqrt [3]{x} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{6} \int \frac {(1-x)^{2/3}}{x^{2/3}}dx-\frac {1}{2} (1-x)^{5/3} \sqrt [3]{x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{6} \left (\frac {2}{3} \int \frac {1}{\sqrt [3]{1-x} x^{2/3}}dx+(1-x)^{2/3} \sqrt [3]{x}\right )-\frac {1}{2} (1-x)^{5/3} \sqrt [3]{x}\) |
\(\Big \downarrow \) 72 |
\(\displaystyle \frac {1}{6} \left (\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x}}\right )+\frac {3}{2} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x}}+1\right )+\frac {\log (x)}{2}\right )+(1-x)^{2/3} \sqrt [3]{x}\right )-\frac {1}{2} (1-x)^{5/3} \sqrt [3]{x}\) |
Input:
Int[(1 - x)^(2/3)*x^(1/3),x]
Output:
-1/2*((1 - x)^(5/3)*x^(1/3)) + ((1 - x)^(2/3)*x^(1/3) + (2*(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)/6
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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[1/Sqrt[3] - 2*q*((a + b* x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))], 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])] /; F reeQ[{a, b, c, d}, x] && NegQ[d/b]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.13
method | result | size |
meijerg | \(\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], x\right )}{4}\) | \(13\) |
risch | \(-\frac {\left (-2+3 x \right ) x^{\frac {1}{3}} \left (-1+x \right ) \left (x^{2} \left (1-x \right )\right )^{\frac {1}{3}}}{6 \left (-x^{2} \left (-1+x \right )\right )^{\frac {1}{3}} \left (1-x \right )^{\frac {1}{3}}}+\frac {\operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right ) \left (x^{2} \left (1-x \right )\right )^{\frac {1}{3}}}{3 x^{\frac {1}{3}} \left (1-x \right )^{\frac {1}{3}}}\) | \(73\) |
Input:
int((1-x)^(2/3)*x^(1/3),x,method=_RETURNVERBOSE)
Output:
3/4*x^(4/3)*hypergeom([-2/3,4/3],[7/3],x)
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.12 \[ \int (1-x)^{2/3} \sqrt [3]{x} \, dx=\frac {1}{6} \, {\left (3 \, x - 2\right )} x^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x - 1\right )} + 2 \, \sqrt {3} x^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}}}{3 \, {\left (x - 1\right )}}\right ) - \frac {1}{18} \, \log \left (\frac {x - x^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + x^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - 1}{x - 1}\right ) + \frac {1}{9} \, \log \left (-\frac {x - x^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} - 1}{x - 1}\right ) \] Input:
integrate((1-x)^(2/3)*x^(1/3),x, algorithm="fricas")
Output:
1/6*(3*x - 2)*x^(1/3)*(-x + 1)^(2/3) - 1/9*sqrt(3)*arctan(1/3*(sqrt(3)*(x - 1) + 2*sqrt(3)*x^(1/3)*(-x + 1)^(2/3))/(x - 1)) - 1/18*log((x - x^(2/3)* (-x + 1)^(1/3) + x^(1/3)*(-x + 1)^(2/3) - 1)/(x - 1)) + 1/9*log(-(x - x^(1 /3)*(-x + 1)^(2/3) - 1)/(x - 1))
Result contains complex when optimal does not.
Time = 1.95 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.30 \[ \int (1-x)^{2/3} \sqrt [3]{x} \, dx=\frac {x^{\frac {4}{3}} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {x e^{2 i \pi }} \right )}}{\Gamma \left (\frac {7}{3}\right )} \] Input:
integrate((1-x)**(2/3)*x**(1/3),x)
Output:
x**(4/3)*gamma(4/3)*hyper((-2/3, 4/3), (7/3,), x*exp_polar(2*I*pi))/gamma( 7/3)
Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.15 \[ \int (1-x)^{2/3} \sqrt [3]{x} \, dx=-\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (-x + 1\right )}^{\frac {1}{3}}}{x^{\frac {1}{3}}} - 1\right )}\right ) + \frac {\frac {{\left (-x + 1\right )}^{\frac {2}{3}}}{x^{\frac {2}{3}}} - \frac {2 \, {\left (-x + 1\right )}^{\frac {5}{3}}}{x^{\frac {5}{3}}}}{6 \, {\left (\frac {{\left (x - 1\right )}^{2}}{x^{2}} - \frac {2 \, {\left (x - 1\right )}}{x} + 1\right )}} + \frac {1}{9} \, \log \left (\frac {{\left (-x + 1\right )}^{\frac {1}{3}}}{x^{\frac {1}{3}}} + 1\right ) - \frac {1}{18} \, \log \left (-\frac {{\left (-x + 1\right )}^{\frac {1}{3}}}{x^{\frac {1}{3}}} + \frac {{\left (-x + 1\right )}^{\frac {2}{3}}}{x^{\frac {2}{3}}} + 1\right ) \] Input:
integrate((1-x)^(2/3)*x^(1/3),x, algorithm="maxima")
Output:
-1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x + 1)^(1/3)/x^(1/3) - 1)) + 1/6*((-x + 1)^(2/3)/x^(2/3) - 2*(-x + 1)^(5/3)/x^(5/3))/((x - 1)^2/x^2 - 2*(x - 1) /x + 1) + 1/9*log((-x + 1)^(1/3)/x^(1/3) + 1) - 1/18*log(-(-x + 1)^(1/3)/x ^(1/3) + (-x + 1)^(2/3)/x^(2/3) + 1)
\[ \int (1-x)^{2/3} \sqrt [3]{x} \, dx=\int { x^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}} \,d x } \] Input:
integrate((1-x)^(2/3)*x^(1/3),x, algorithm="giac")
Output:
integrate(x^(1/3)*(-x + 1)^(2/3), x)
Timed out. \[ \int (1-x)^{2/3} \sqrt [3]{x} \, dx=\int x^{1/3}\,{\left (1-x\right )}^{2/3} \,d x \] Input:
int(x^(1/3)*(1 - x)^(2/3),x)
Output:
int(x^(1/3)*(1 - x)^(2/3), x)
\[ \int (1-x)^{2/3} \sqrt [3]{x} \, dx=\frac {x^{\frac {4}{3}} \left (1-x \right )^{\frac {2}{3}}}{2}-\frac {x^{\frac {1}{3}} \left (1-x \right )^{\frac {2}{3}}}{3}-\frac {\left (\int \frac {\left (1-x \right )^{\frac {2}{3}}}{x^{\frac {5}{3}}-x^{\frac {2}{3}}}d x \right )}{9} \] Input:
int((1-x)^(2/3)*x^(1/3),x)
Output:
(9*x**(1/3)*( - x + 1)**(2/3)*x - 6*x**(1/3)*( - x + 1)**(2/3) - 2*int(( - x + 1)**(2/3)/(x**(2/3)*x - x**(2/3)),x))/18