Optimal. Leaf size=219 \[ -\log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x^2}-2 x\right )}{\sqrt [3]{2}}+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2} x+\sqrt [3]{2} \left (x^3-x^2\right )^{2/3}\right )}{2 \sqrt [3]{2}}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2}+x}\right )}{\sqrt [3]{2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 295, normalized size of antiderivative = 1.35, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2042, 105, 59, 91} \begin {gather*} -\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3-x^2}}+\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x)}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x+1)}{2 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3-x^2}}+\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{x^3-x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 59
Rule 91
Rule 105
Rule 2042
Rubi steps
\begin {align*} \int \frac {x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x} (1+x)} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{\sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} (1+x)} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=-\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{-x^2+x^3}}+\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{-x^2+x^3}}+\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (x)}{2 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (1+x)}{2 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 61, normalized size = 0.28 \begin {gather*} \frac {3 \left ((x-1) x^2\right )^{2/3} \left (2 x^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};1-x\right )-\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x-1}{2 x}\right )\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.46, size = 219, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{\sqrt [3]{2}}-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{\sqrt [3]{2}}+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.42, size = 241, normalized size = 1.10 \begin {gather*} \frac {1}{4} \, {\left (2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}}} - 2^{\frac {2}{3}}\right )} \log \left (\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} x \sqrt {-2^{\frac {1}{3}}} + 2^{\frac {1}{3}} x + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{2 \, x}\right ) - \frac {1}{4} \, {\left (2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}}} + 2^{\frac {2}{3}}\right )} \log \left (-\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} x \sqrt {-2^{\frac {1}{3}}} - 2^{\frac {1}{3}} x - 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{2 \, x}\right ) + \frac {1}{2} \cdot 2^{\frac {2}{3}} \log \left (-\frac {3 \, {\left (2^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{x}\right ) - \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \log \left (-\frac {3 \, {\left (x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{x}\right ) + \frac {1}{2} \, \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.32, size = 149, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{4} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{2} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \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}{2} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 3.60, size = 1000, normalized size = 4.57
method | result | size |
trager | \(-\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+48 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-30 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -16 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-36 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+96 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+18 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -64 x^{2}+16 x}{x}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+24 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -19 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-30 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+48 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+10 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -10 x^{2}+6 x}{x}\right )}{2}+\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \ln \left (\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{2}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x +24 \left (x^{3}-x^{2}\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-48 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +26 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-52 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}-4\right ) x +20 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) x +36 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{x \left (1+x \right )}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{3}-4\right ) \ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{2}-\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -24 \left (x^{3}-x^{2}\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+48 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -44 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}+22 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{3}-4\right ) x -2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) x -60 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{x \left (1+x \right )}\right )}{2}\) | \(1000\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x}{{\left (x^3-x^2\right )}^{1/3}\,\left (x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________