Optimal. Leaf size=26 \[ \frac {4 \sqrt [4]{x^5+1} \left (x^5+5 x^4+1\right )}{5 x^5} \]
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Rubi [A] time = 0.10, antiderivative size = 44, normalized size of antiderivative = 1.69, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1835, 1585, 12, 261} \begin {gather*} \frac {4 \sqrt [4]{x^5+1}}{x}+\frac {4 \sqrt [4]{x^5+1}}{5 x^5}+\frac {4}{5} \sqrt [4]{x^5+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 261
Rule 1585
Rule 1835
Rubi steps
\begin {align*} \int \frac {\left (-4+x^5\right ) \left (1+x^4+x^5\right )}{x^6 \left (1+x^5\right )^{3/4}} \, dx &=\frac {4 \sqrt [4]{1+x^5}}{5 x^5}-\frac {1}{10} \int \frac {40 x^3-10 x^8-10 x^9}{x^5 \left (1+x^5\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{1+x^5}}{5 x^5}-\frac {1}{10} \int \frac {40-10 x^5-10 x^6}{x^2 \left (1+x^5\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{1+x^5}}{5 x^5}+\frac {4 \sqrt [4]{1+x^5}}{x}+\frac {1}{20} \int \frac {20 x^4}{\left (1+x^5\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{1+x^5}}{5 x^5}+\frac {4 \sqrt [4]{1+x^5}}{x}+\int \frac {x^4}{\left (1+x^5\right )^{3/4}} \, dx\\ &=\frac {4}{5} \sqrt [4]{1+x^5}+\frac {4 \sqrt [4]{1+x^5}}{5 x^5}+\frac {4 \sqrt [4]{1+x^5}}{x}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 107, normalized size = 4.12 \begin {gather*} -\frac {16}{5} \sqrt [4]{x^5+1} \, _2F_1\left (\frac {1}{4},2;\frac {5}{4};x^5+1\right )+\frac {4 \, _2F_1\left (-\frac {1}{5},\frac {3}{4};\frac {4}{5};-x^5\right )}{x}+\frac {1}{4} x^4 \, _2F_1\left (\frac {3}{4},\frac {4}{5};\frac {9}{5};-x^5\right )+\frac {4}{5} \sqrt [4]{x^5+1}+\frac {6}{5} \left (\tan ^{-1}\left (\sqrt [4]{x^5+1}\right )+\tanh ^{-1}\left (\sqrt [4]{x^5+1}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.88, size = 26, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{1+x^5} \left (1+5 x^4+x^5\right )}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 22, normalized size = 0.85 \begin {gather*} \frac {4 \, {\left (x^{5} + 5 \, x^{4} + 1\right )} {\left (x^{5} + 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \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^{5} + x^{4} + 1\right )} {\left (x^{5} - 4\right )}}{{\left (x^{5} + 1\right )}^{\frac {3}{4}} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 23, normalized size = 0.88
method | result | size |
trager | \(\frac {4 \left (x^{5}+1\right )^{\frac {1}{4}} \left (x^{5}+5 x^{4}+1\right )}{5 x^{5}}\) | \(23\) |
risch | \(\frac {\frac {8}{5} x^{5}+\frac {4}{5}+4 x^{9}+4 x^{4}+\frac {4}{5} x^{10}}{\left (x^{5}+1\right )^{\frac {3}{4}} x^{5}}\) | \(33\) |
gosper | \(\frac {4 \left (x^{5}+5 x^{4}+1\right ) \left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}{5 \left (x^{5}+1\right )^{\frac {3}{4}} x^{5}}\) | \(42\) |
meijerg | \(\frac {\hypergeom \left (\left [\frac {3}{4}, 1\right ], \relax [2], -x^{5}\right ) x^{5}}{5}+\frac {\hypergeom \left (\left [\frac {3}{4}, \frac {4}{5}\right ], \left [\frac {9}{5}\right ], -x^{5}\right ) x^{4}}{4}-\frac {3 \left (\left (-3 \ln \relax (2)+\frac {\pi }{2}+5 \ln \relax (x )\right ) \Gamma \left (\frac {3}{4}\right )-\frac {3 \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], -x^{5}\right ) \Gamma \left (\frac {3}{4}\right ) x^{5}}{4}\right )}{5 \Gamma \left (\frac {3}{4}\right )}-\frac {4 \left (-\frac {\Gamma \left (\frac {3}{4}\right )}{x^{5}}-\frac {3 \left (\frac {1}{3}-3 \ln \relax (2)+\frac {\pi }{2}+5 \ln \relax (x )\right ) \Gamma \left (\frac {3}{4}\right )}{4}+\frac {21 \hypergeom \left (\left [1, 1, \frac {11}{4}\right ], \left [2, 3\right ], -x^{5}\right ) \Gamma \left (\frac {3}{4}\right ) x^{5}}{32}\right )}{5 \Gamma \left (\frac {3}{4}\right )}+\frac {4 \hypergeom \left (\left [-\frac {1}{5}, \frac {3}{4}\right ], \left [\frac {4}{5}\right ], -x^{5}\right )}{x}\) | \(143\) |
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*} \frac {4 \, {\left (x^{5} + 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} - \frac {6}{5} \, \arctan \left ({\left (x^{5} + 1\right )}^{\frac {1}{4}}\right ) + \int \frac {{\left (x^{6} + x^{5} - 3 \, x - 4\right )} {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}}}{x^{7} + x^{2}}\,{d x} - \frac {3}{5} \, \log \left ({\left (x^{5} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {3}{5} \, \log \left ({\left (x^{5} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 27, normalized size = 1.04 \begin {gather*} \frac {4\,{\left (x^5+1\right )}^{5/4}+20\,x^4\,{\left (x^5+1\right )}^{1/4}}{5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.41, size = 143, normalized size = 5.50 \begin {gather*} \frac {x^{4} \Gamma \left (\frac {4}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {4}{5} \\ \frac {9}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 \Gamma \left (\frac {9}{5}\right )} + \frac {4 \sqrt [4]{x^{5} + 1}}{5} - \frac {4 \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{5}, \frac {3}{4} \\ \frac {4}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} + \frac {3 \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{5}}} \right )}}{5 x^{\frac {15}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {4 \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{5}}} \right )}}{5 x^{\frac {35}{4}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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