Optimal. Leaf size=109 \[ -\frac {5}{243} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{81 \sqrt {3}}+\frac {5}{486} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )+\frac {1}{162} \sqrt [3]{x^3+1} \left (18 x^8+3 x^5-5 x^2\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 129, normalized size of antiderivative = 1.18, number of steps used = 10, 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 {5}{243} \log \left (1-\frac {x}{\sqrt [3]{x^3+1}}\right )-\frac {5 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {1}{9} \sqrt [3]{x^3+1} x^8+\frac {1}{54} \sqrt [3]{x^3+1} x^5-\frac {5}{162} \sqrt [3]{x^3+1} x^2+\frac {5}{486} \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^7 \sqrt [3]{1+x^3} \, dx &=\frac {1}{9} x^8 \sqrt [3]{1+x^3}+\frac {1}{9} \int \frac {x^7}{\left (1+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{54} x^5 \sqrt [3]{1+x^3}+\frac {1}{9} x^8 \sqrt [3]{1+x^3}-\frac {5}{54} \int \frac {x^4}{\left (1+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{162} x^2 \sqrt [3]{1+x^3}+\frac {1}{54} x^5 \sqrt [3]{1+x^3}+\frac {1}{9} x^8 \sqrt [3]{1+x^3}+\frac {5}{81} \int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{162} x^2 \sqrt [3]{1+x^3}+\frac {1}{54} x^5 \sqrt [3]{1+x^3}+\frac {1}{9} x^8 \sqrt [3]{1+x^3}+\frac {5}{81} \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {5}{162} x^2 \sqrt [3]{1+x^3}+\frac {1}{54} x^5 \sqrt [3]{1+x^3}+\frac {1}{9} x^8 \sqrt [3]{1+x^3}+\frac {5}{243} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {5}{243} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {5}{162} x^2 \sqrt [3]{1+x^3}+\frac {1}{54} x^5 \sqrt [3]{1+x^3}+\frac {1}{9} x^8 \sqrt [3]{1+x^3}-\frac {5}{243} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {5}{486} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {5}{162} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {5}{162} x^2 \sqrt [3]{1+x^3}+\frac {1}{54} x^5 \sqrt [3]{1+x^3}+\frac {1}{9} x^8 \sqrt [3]{1+x^3}-\frac {5}{243} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {5}{486} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {5}{81} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {5}{162} x^2 \sqrt [3]{1+x^3}+\frac {1}{54} x^5 \sqrt [3]{1+x^3}+\frac {1}{9} x^8 \sqrt [3]{1+x^3}-\frac {5 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {5}{243} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {5}{486} \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 = 0.02, size = 45, normalized size = 0.41 \begin {gather*} \frac {1}{54} x^2 \left (5 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-x^3\right )+\sqrt [3]{x^3+1} \left (6 x^6+x^3-5\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 109, normalized size = 1.00 \begin {gather*} \frac {1}{162} \sqrt [3]{1+x^3} \left (-5 x^2+3 x^5+18 x^8\right )-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{81 \sqrt {3}}-\frac {5}{243} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {5}{486} \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 = 0.46, size = 101, normalized size = 0.93 \begin {gather*} \frac {5}{243} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{162} \, {\left (18 \, x^{8} + 3 \, x^{5} - 5 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \frac {5}{243} \, \log \left (-\frac {x - {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {5}{486} \, \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^{7}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.82, size = 17, normalized size = 0.16
method | result | size |
meijerg | \(\frac {x^{8} \hypergeom \left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], -x^{3}\right )}{8}\) | \(17\) |
risch | \(\frac {x^{2} \left (18 x^{6}+3 x^{3}-5\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{162}+\frac {5 x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{162}\) | \(42\) |
trager | \(\frac {x^{2} \left (18 x^{6}+3 x^{3}-5\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{162}-\frac {5 \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+x^{3}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{243}+\frac {5 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-5 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+2 x^{3}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{243}\) | \(209\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 145, normalized size = 1.33 \begin {gather*} \frac {5}{243} \, \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 {10 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {13 \, {\left (x^{3} + 1\right )}^{\frac {4}{3}}}{x^{4}} - \frac {5 \, {\left (x^{3} + 1\right )}^{\frac {7}{3}}}{x^{7}}}{162 \, {\left (\frac {3 \, {\left (x^{3} + 1\right )}}{x^{3}} - \frac {3 \, {\left (x^{3} + 1\right )}^{2}}{x^{6}} + \frac {{\left (x^{3} + 1\right )}^{3}}{x^{9}} - 1\right )}} + \frac {5}{486} \, \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 {5}{243} \, \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^7\,{\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 = 1.20, size = 31, normalized size = 0.28 \begin {gather*} \frac {x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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