Optimal. Leaf size=106 \[ \sqrt [3]{-x^2+x^3}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-x^2+x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 149, normalized size of antiderivative = 1.41, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2046, 2057, 61}
\begin {gather*} \frac {(x-1)^{2/3} x^{4/3} \text {ArcTan}\left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \left (x^3-x^2\right )^{2/3}}+\sqrt [3]{x^3-x^2}+\frac {(x-1)^{2/3} x^{4/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{2 \left (x^3-x^2\right )^{2/3}}+\frac {(x-1)^{2/3} x^{4/3} \log (x-1)}{6 \left (x^3-x^2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 61
Rule 2046
Rule 2057
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-x^2+x^3}}{x} \, dx &=\sqrt [3]{-x^2+x^3}-\frac {1}{3} \int \frac {x}{\left (-x^2+x^3\right )^{2/3}} \, dx\\ &=\sqrt [3]{-x^2+x^3}-\frac {\left ((-1+x)^{2/3} x^{4/3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, dx}{3 \left (-x^2+x^3\right )^{2/3}}\\ &=\sqrt [3]{-x^2+x^3}+\frac {(-1+x)^{2/3} x^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3} \left (-x^2+x^3\right )^{2/3}}+\frac {(-1+x)^{2/3} x^{4/3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{2 \left (-x^2+x^3\right )^{2/3}}+\frac {(-1+x)^{2/3} x^{4/3} \log (-1+x)}{6 \left (-x^2+x^3\right )^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 125, normalized size = 1.18 \begin {gather*} \frac {(-1+x)^{2/3} x^{4/3} \left (6 \sqrt [3]{-1+x} x^{2/3}+2 \sqrt {3} \text {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}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )\right )}{6 \left ((-1+x) x^2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.83, size = 27, normalized size = 0.25
method | result | size |
meijerg | \(\frac {3 \mathrm {signum}\left (-1+x \right )^{\frac {1}{3}} x^{\frac {2}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x\right )}{2 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}}}\) | \(27\) |
trager | \(\left (x^{3}-x^{2}\right )^{\frac {1}{3}}-\frac {\ln \left (\frac {180 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{2}-360 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x -144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-144 \left (x^{3}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x -114 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+18 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x -9 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-4 x^{2}+x}{x}\right )}{3}-2 \ln \left (\frac {180 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{2}-360 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x -144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-144 \left (x^{3}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x -114 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+18 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x -9 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-4 x^{2}+x}{x}\right ) \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )+2 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \ln \left (\frac {180 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{2}-360 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x +144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \left (x^{3}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x +174 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-138 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x +15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+20 x^{2}-12 x}{x}\right )\) | \(499\) |
risch | \(\text {Expression too large to display}\) | \(638\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.43, size = 103, normalized size = 0.97 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} + \frac {1}{3} \, \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{6} \, \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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x^{2} \left (x - 1\right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 74, normalized size = 0.70 \begin {gather*} -\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 ) + x {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-x^2\right )}^{1/3}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________