Optimal. Leaf size=102 \[ \frac {1}{27} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{9 \sqrt {3}}-\frac {1}{54} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )+\frac {1}{18} \sqrt [3]{x^3+1} \left (3 x^5+x^2\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {279, 321, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {1}{27} \log \left (1-\frac {x}{\sqrt [3]{x^3+1}}\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{6} \sqrt [3]{x^3+1} x^5+\frac {1}{18} \sqrt [3]{x^3+1} x^2-\frac {1}{54} \log \left (\frac {x}{\sqrt [3]{x^3+1}}+\frac {x^2}{\left (x^3+1\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 279
Rule 292
Rule 321
Rule 331
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int x^4 \sqrt [3]{1+x^3} \, dx &=\frac {1}{6} x^5 \sqrt [3]{1+x^3}+\frac {1}{6} \int \frac {x^4}{\left (1+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{18} x^2 \sqrt [3]{1+x^3}+\frac {1}{6} x^5 \sqrt [3]{1+x^3}-\frac {1}{9} \int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{18} x^2 \sqrt [3]{1+x^3}+\frac {1}{6} x^5 \sqrt [3]{1+x^3}-\frac {1}{9} \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{18} x^2 \sqrt [3]{1+x^3}+\frac {1}{6} x^5 \sqrt [3]{1+x^3}-\frac {1}{27} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{27} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{18} x^2 \sqrt [3]{1+x^3}+\frac {1}{6} x^5 \sqrt [3]{1+x^3}+\frac {1}{27} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{54} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{18} x^2 \sqrt [3]{1+x^3}+\frac {1}{6} x^5 \sqrt [3]{1+x^3}+\frac {1}{27} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{54} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{18} x^2 \sqrt [3]{1+x^3}+\frac {1}{6} x^5 \sqrt [3]{1+x^3}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{27} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{54} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+x^3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.33 \begin {gather*} \frac {1}{6} x^2 \left (\left (x^3+1\right )^{4/3}-\, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-x^3\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 102, normalized size = 1.00 \begin {gather*} \frac {1}{18} \sqrt [3]{1+x^3} \left (x^2+3 x^5\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{9 \sqrt {3}}+\frac {1}{27} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{54} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 94, normalized size = 0.92 \begin {gather*} -\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{18} \, {\left (3 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{27} \, \log \left (-\frac {x - {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{54} \, \log \left (\frac {x^{2} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.62, size = 17, normalized size = 0.17
method | result | size |
meijerg | \(\frac {x^{5} \hypergeom \left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], -x^{3}\right )}{5}\) | \(17\) |
risch | \(\frac {x^{2} \left (3 x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{18}-\frac {x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{18}\) | \(37\) |
trager | \(\frac {x^{2} \left (3 x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{18}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{27}-\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+4 x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{27}-\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+4 x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2\right )}{27}\) | \(313\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 121, normalized size = 1.19 \begin {gather*} -\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}}}{x^{4}}}{18 \, {\left (\frac {2 \, {\left (x^{3} + 1\right )}}{x^{3}} - \frac {{\left (x^{3} + 1\right )}^{2}}{x^{6}} - 1\right )}} - \frac {1}{54} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {1}{27} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,{\left (x^3+1\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.97, size = 31, normalized size = 0.30 \begin {gather*} \frac {x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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