Optimal. Leaf size=100 \[ -\frac {3 \sqrt [3]{x^2+x^3}}{x}-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 147, normalized size of antiderivative = 1.47, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2045, 2057, 61}
\begin {gather*} -\frac {\sqrt {3} (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 )}{\left (x^3+x^2\right )^{2/3}}-\frac {3 \sqrt [3]{x^3+x^2}}{x}-\frac {(x+1)^{2/3} x^{4/3} \log (x+1)}{2 \left (x^3+x^2\right )^{2/3}}-\frac {3 (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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 61
Rule 2045
Rule 2057
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx &=-\frac {3 \sqrt [3]{x^2+x^3}}{x}+\int \frac {x}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=-\frac {3 \sqrt [3]{x^2+x^3}}{x}+\frac {\left (x^{4/3} (1+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (1+x)^{2/3}} \, dx}{\left (x^2+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x^2+x^3}}{x}-\frac {\sqrt {3} x^{4/3} (1+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{\left (x^2+x^3\right )^{2/3}}-\frac {x^{4/3} (1+x)^{2/3} \log (1+x)}{2 \left (x^2+x^3\right )^{2/3}}-\frac {3 x^{4/3} (1+x)^{2/3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{2 \left (x^2+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 131, normalized size = 1.31 \begin {gather*} -\frac {x (1+x)^{2/3} \left (6 \sqrt [3]{1+x}+2 \sqrt {3} \sqrt [3]{x} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )+2 \sqrt [3]{x} \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )-\sqrt [3]{x} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{2 \left (x^2 (1+x)\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 0.79, size = 15, normalized size = 0.15
method | result | size |
meijerg | \(-\frac {3 \hypergeom \left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x \right )}{x^{\frac {1}{3}}}\) | \(15\) |
risch | \(-\frac {3 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}+\frac {\left (-\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}+2 x^{2}+x \right )^{\frac {2}{3}}+30 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +16 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}+30 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+14 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +36 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-96 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +64 x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-96 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+112 x +48}{1+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}+2 x^{2}+x \right )^{\frac {2}{3}}-9 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -19 \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 \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 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}-28 \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 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-14 x -4}{1+x}\right )}{2}\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 )}\) | \(450\) |
trager | \(-\frac {3 \left (x^{3}+x^{2}\right )^{\frac {1}{3}}}{x}-3 \ln \left (\frac {-36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+45 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +57 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+27 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +5 x^{2}+2 x}{x}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-\frac {36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+45 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +33 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-24 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-24 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+51 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -20 x^{2}-15 x}{x}\right )+\ln \left (\frac {-36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+45 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +57 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+27 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +5 x^{2}+2 x}{x}\right )\) | \(481\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.33, size = 102, normalized size = 1.02 \begin {gather*} \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 {1}{3}}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x^{2} \left (x + 1\right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 63, normalized size = 0.63 \begin {gather*} \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - 3 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + \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.
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Mupad [B]
time = 1.03, size = 27, normalized size = 0.27 \begin {gather*} -\frac {3\,{\left (x^2\,\left (x+1\right )\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},-\frac {1}{3};\ \frac {2}{3};\ -x\right )}{x\,{\left (x+1\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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