Optimal. Leaf size=98 \[ \frac {4 \sqrt [4]{x^5-1}}{x}-2 \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4-1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^5-1}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)-\log \left (\sqrt [4]{x^5-1}-\text {$\#$1} x\right )+\log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ] \]
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Rubi [F] time = 3.61, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx &=\int \left (\frac {6}{\left (-1+x^5\right )^{3/4}}+\frac {4}{x^2 \left (-1+x^5\right )^{3/4}}+\frac {4 x}{\left (-1+x^5\right )^{3/4}}+\frac {2 x^2}{\left (-1+x^5\right )^{3/4}}+\frac {x^3}{\left (-1+x^5\right )^{3/4}}+\frac {2 \left (-3-2 x-5 x^2-3 x^4+4 x^5+7 x^6+5 x^7+3 x^8+5 x^9\right )}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}\right ) \, dx\\ &=2 \int \frac {x^2}{\left (-1+x^5\right )^{3/4}} \, dx+2 \int \frac {-3-2 x-5 x^2-3 x^4+4 x^5+7 x^6+5 x^7+3 x^8+5 x^9}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+4 \int \frac {1}{x^2 \left (-1+x^5\right )^{3/4}} \, dx+4 \int \frac {x}{\left (-1+x^5\right )^{3/4}} \, dx+6 \int \frac {1}{\left (-1+x^5\right )^{3/4}} \, dx+\int \frac {x^3}{\left (-1+x^5\right )^{3/4}} \, dx\\ &=2 \int \left (-\frac {3}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}-\frac {2 x}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}-\frac {5 x^2}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}-\frac {3 x^4}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {4 x^5}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {7 x^6}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {5 x^7}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {3 x^8}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {5 x^9}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}\right ) \, dx+\frac {\left (1-x^5\right )^{3/4} \int \frac {x^3}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (2 \left (1-x^5\right )^{3/4}\right ) \int \frac {x^2}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (4 \left (1-x^5\right )^{3/4}\right ) \int \frac {1}{x^2 \left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (4 \left (1-x^5\right )^{3/4}\right ) \int \frac {x}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (6 \left (1-x^5\right )^{3/4}\right ) \int \frac {1}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}\\ &=-\frac {4 \left (1-x^5\right )^{3/4} \, _2F_1\left (-\frac {1}{5},\frac {3}{4};\frac {4}{5};x^5\right )}{x \left (-1+x^5\right )^{3/4}}+\frac {6 x \left (1-x^5\right )^{3/4} \, _2F_1\left (\frac {1}{5},\frac {3}{4};\frac {6}{5};x^5\right )}{\left (-1+x^5\right )^{3/4}}+\frac {2 x^2 \left (1-x^5\right )^{3/4} \, _2F_1\left (\frac {2}{5},\frac {3}{4};\frac {7}{5};x^5\right )}{\left (-1+x^5\right )^{3/4}}+\frac {2 x^3 \left (1-x^5\right )^{3/4} \, _2F_1\left (\frac {3}{5},\frac {3}{4};\frac {8}{5};x^5\right )}{3 \left (-1+x^5\right )^{3/4}}+\frac {x^4 \left (1-x^5\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {4}{5};\frac {9}{5};x^5\right )}{4 \left (-1+x^5\right )^{3/4}}-4 \int \frac {x}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx-6 \int \frac {1}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx-6 \int \frac {x^4}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+6 \int \frac {x^8}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+8 \int \frac {x^5}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx-10 \int \frac {x^2}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+10 \int \frac {x^7}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+10 \int \frac {x^9}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+14 \int \frac {x^6}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.76, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.00, size = 98, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{-1+x^5}}{x}-2 \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{-1+x^5}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-1+x^5}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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^{10} + x^{9} + x^{8} - 2 \, x^{5} - x^{4} + 1\right )} {\left (x^{5} + 4\right )}}{{\left (x^{10} - x^{9} - x^{8} - 2 \, x^{5} + x^{4} + 1\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{5}+4\right ) \left (x^{10}+x^{9}+x^{8}-2 x^{5}-x^{4}+1\right )}{x^{2} \left (x^{5}-1\right )^{\frac {3}{4}} \left (x^{10}-x^{9}-x^{8}-2 x^{5}+x^{4}+1\right )}\, dx\]
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*} \int \frac {{\left (x^{10} + x^{9} + x^{8} - 2 \, x^{5} - x^{4} + 1\right )} {\left (x^{5} + 4\right )}}{{\left (x^{10} - x^{9} - x^{8} - 2 \, x^{5} + x^{4} + 1\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \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 \frac {\left (x^5+4\right )\,\left (x^{10}+x^9+x^8-2\,x^5-x^4+1\right )}{x^2\,{\left (x^5-1\right )}^{3/4}\,\left (x^{10}-x^9-x^8-2\,x^5+x^4+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{5} + 4\right ) \left (x^{10} + x^{9} + x^{8} - 2 x^{5} - x^{4} + 1\right )}{x^{2} \left (\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )\right )^{\frac {3}{4}} \left (x^{10} - x^{9} - x^{8} - 2 x^{5} + x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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