Optimal. Leaf size=38 \[ \frac {3 \sqrt [3]{x^4+1} \left (4 x^8+7 x^7+8 x^4+7 x^3+4\right )}{28 x^7} \]
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Rubi [A] time = 0.08, antiderivative size = 33, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1833, 1584, 446, 74, 1478, 449} \begin {gather*} \frac {3 \left (x^4+1\right )^{4/3}}{4 x^4}+\frac {3 \left (x^4+1\right )^{7/3}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 446
Rule 449
Rule 1478
Rule 1584
Rule 1833
Rubi steps
\begin {align*} \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4} \left (1+x^3+x^4\right )}{x^8} \, dx &=\int \left (\frac {\sqrt [3]{1+x^4} \left (-3 x^2+x^6\right )}{x^7}+\frac {\sqrt [3]{1+x^4} \left (-3-2 x^4+x^8\right )}{x^8}\right ) \, dx\\ &=\int \frac {\sqrt [3]{1+x^4} \left (-3 x^2+x^6\right )}{x^7} \, dx+\int \frac {\sqrt [3]{1+x^4} \left (-3-2 x^4+x^8\right )}{x^8} \, dx\\ &=\int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^5} \, dx+\int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{4/3}}{x^8} \, dx\\ &=\frac {3 \left (1+x^4\right )^{7/3}}{7 x^7}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {(-3+x) \sqrt [3]{1+x}}{x^2} \, dx,x,x^4\right )\\ &=\frac {3 \left (1+x^4\right )^{4/3}}{4 x^4}+\frac {3 \left (1+x^4\right )^{7/3}}{7 x^7}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 169, normalized size = 4.45 \begin {gather*} \frac {1}{16} \left (-9 \left (x^4+1\right )^{4/3} \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};x^4+1\right )+16 x \, _2F_1\left (-\frac {1}{3},\frac {1}{4};\frac {5}{4};-x^4\right )+12 \sqrt [3]{x^4+1}+4 \log \left (1-\sqrt [3]{x^4+1}\right )-2 \log \left (\left (x^4+1\right )^{2/3}+\sqrt [3]{x^4+1}+1\right )-4 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x^4+1}+1}{\sqrt {3}}\right )\right )+\frac {3 \, _2F_1\left (-\frac {7}{4},-\frac {1}{3};-\frac {3}{4};-x^4\right )}{7 x^7}+\frac {2 \, _2F_1\left (-\frac {3}{4},-\frac {1}{3};\frac {1}{4};-x^4\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 28, normalized size = 0.74 \begin {gather*} \frac {3 \left (1+x^4\right )^{4/3} \left (4+7 x^3+4 x^4\right )}{28 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 34, normalized size = 0.89 \begin {gather*} \frac {3 \, {\left (4 \, x^{8} + 7 \, x^{7} + 8 \, x^{4} + 7 \, x^{3} + 4\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \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 \frac {{\left (x^{4} + x^{3} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}} {\left (x^{4} - 3\right )}}{x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 25, normalized size = 0.66
method | result | size |
gosper | \(\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}} \left (4 x^{4}+7 x^{3}+4\right )}{28 x^{7}}\) | \(25\) |
trager | \(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}} \left (4 x^{8}+7 x^{7}+8 x^{4}+7 x^{3}+4\right )}{28 x^{7}}\) | \(35\) |
risch | \(\frac {\frac {9}{7} x^{8}+\frac {9}{7} x^{4}+\frac {3}{7}+\frac {3}{2} x^{7}+\frac {3}{4} x^{3}+\frac {3}{7} x^{12}+\frac {3}{4} x^{11}}{x^{7} \left (x^{4}+1\right )^{\frac {2}{3}}}\) | \(45\) |
meijerg | \(\frac {2 \hypergeom \left (\left [-\frac {3}{4}, -\frac {1}{3}\right ], \left [\frac {1}{4}\right ], -x^{4}\right )}{3 x^{3}}+\frac {\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{4}}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {\hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{4}\right ) \Gamma \left (\frac {2}{3}\right ) x^{4}}{3}}{4 \Gamma \left (\frac {2}{3}\right )}+\frac {3 \hypergeom \left (\left [-\frac {7}{4}, -\frac {1}{3}\right ], \left [-\frac {3}{4}\right ], -x^{4}\right )}{7 x^{7}}+\hypergeom \left (\left [-\frac {1}{3}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -x^{4}\right ) x -\frac {-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )-\hypergeom \left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], -x^{4}\right ) \Gamma \left (\frac {2}{3}\right ) x^{4}}{12 \Gamma \left (\frac {2}{3}\right )}\) | \(148\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 34, normalized size = 0.89 \begin {gather*} \frac {3 \, {\left (4 \, x^{8} + 7 \, x^{7} + 8 \, x^{4} + 7 \, x^{3} + 4\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 50, normalized size = 1.32 \begin {gather*} \left (\frac {3\,x}{7}+\frac {3}{4}\right )\,{\left (x^4+1\right )}^{1/3}+\frac {6\,{\left (x^4+1\right )}^{1/3}}{7\,x^3}+\frac {3\,{\left (x^4+1\right )}^{1/3}}{4\,x^4}+\frac {3\,{\left (x^4+1\right )}^{1/3}}{7\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.81, size = 178, normalized size = 4.68 \begin {gather*} - \frac {x^{\frac {4}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 \Gamma \left (\frac {2}{3}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{3} \\ \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{2 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {3 \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{3} \\ - \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {3 \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 x^{\frac {8}{3}} \Gamma \left (\frac {5}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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