Optimal. Leaf size=112 \[ \frac {1}{54} \log \left (\sqrt [3]{x^6+1}-x^2\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6+1}+x^2}\right )}{18 \sqrt {3}}+\frac {1}{36} \left (x^6+1\right )^{2/3} \left (3 x^8+2 x^2\right )-\frac {1}{108} \log \left (\left (x^6+1\right )^{2/3}+x^4+\sqrt [3]{x^6+1} x^2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {275, 279, 321, 239} \begin {gather*} \frac {1}{12} \left (x^6+1\right )^{2/3} x^8+\frac {1}{18} \left (x^6+1\right )^{2/3} x^2+\frac {1}{36} \log \left (x^2-\sqrt [3]{x^6+1}\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6+1}}+1}{\sqrt {3}}\right )}{18 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 275
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^7 \left (1+x^6\right )^{2/3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^3 \left (1+x^3\right )^{2/3} \, dx,x,x^2\right )\\ &=\frac {1}{12} x^8 \left (1+x^6\right )^{2/3}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{18} x^2 \left (1+x^6\right )^{2/3}+\frac {1}{12} x^8 \left (1+x^6\right )^{2/3}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{18} x^2 \left (1+x^6\right )^{2/3}+\frac {1}{12} x^8 \left (1+x^6\right )^{2/3}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{1+x^6}}}{\sqrt {3}}\right )}{18 \sqrt {3}}+\frac {1}{36} \log \left (x^2-\sqrt [3]{1+x^6}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.30 \begin {gather*} \frac {1}{12} x^2 \left (\left (x^6+1\right )^{5/3}-\, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};-x^6\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.01, size = 112, normalized size = 1.00 \begin {gather*} \frac {1}{54} \log \left (\sqrt [3]{x^6+1}-x^2\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6+1}+x^2}\right )}{18 \sqrt {3}}+\frac {1}{36} \left (x^6+1\right )^{2/3} \left (3 x^8+2 x^2\right )-\frac {1}{108} \log \left (\left (x^6+1\right )^{2/3}+x^4+\sqrt [3]{x^6+1} x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 102, normalized size = 0.91 \begin {gather*} \frac {1}{36} \, {\left (3 \, x^{8} + 2 \, x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) + \frac {1}{54} \, \log \left (-\frac {x^{2} - {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{108} \, \log \left (\frac {x^{4} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}}\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^{6} + 1\right )}^{\frac {2}{3}} x^{7}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 37, normalized size = 0.33 \begin {gather*} \frac {x^{2} \left (3 x^{6}+2\right ) \left (x^{6}+1\right )^{\frac {2}{3}}}{36}-\frac {x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{6}\right )}{18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 121, normalized size = 1.08 \begin {gather*} \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {\frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}} + \frac {2 \, {\left (x^{6} + 1\right )}^{\frac {5}{3}}}{x^{10}}}{36 \, {\left (\frac {2 \, {\left (x^{6} + 1\right )}}{x^{6}} - \frac {{\left (x^{6} + 1\right )}^{2}}{x^{12}} - 1\right )}} - \frac {1}{108} \, \log \left (\frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) + \frac {1}{54} \, \log \left (\frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} - 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^6+1\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.17, size = 31, normalized size = 0.28 \begin {gather*} \frac {x^{8} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {7}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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