Optimal. Leaf size=45 \[ 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^5+x^2+1}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^5+x^2+1}}\right ) \]
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Rubi [F] time = 1.64, 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 {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx &=\int \left (\frac {\sqrt {1+x^2+x^5}}{1+x}+\frac {\left (-1+2 x-3 x^2-x^3\right ) \sqrt {1+x^2+x^5}}{1-x+x^2-x^3+x^4}+\frac {\left (-2+5 x^3\right ) \sqrt {1+x^2+x^5}}{1-x^2+x^5}\right ) \, dx\\ &=\int \frac {\sqrt {1+x^2+x^5}}{1+x} \, dx+\int \frac {\left (-1+2 x-3 x^2-x^3\right ) \sqrt {1+x^2+x^5}}{1-x+x^2-x^3+x^4} \, dx+\int \frac {\left (-2+5 x^3\right ) \sqrt {1+x^2+x^5}}{1-x^2+x^5} \, dx\\ &=\int \frac {\sqrt {1+x^2+x^5}}{1+x} \, dx+\int \left (\frac {\sqrt {1+x^2+x^5}}{-1+x-x^2+x^3-x^4}+\frac {2 x \sqrt {1+x^2+x^5}}{1-x+x^2-x^3+x^4}-\frac {3 x^2 \sqrt {1+x^2+x^5}}{1-x+x^2-x^3+x^4}-\frac {x^3 \sqrt {1+x^2+x^5}}{1-x+x^2-x^3+x^4}\right ) \, dx+\int \left (-\frac {2 \sqrt {1+x^2+x^5}}{1-x^2+x^5}+\frac {5 x^3 \sqrt {1+x^2+x^5}}{1-x^2+x^5}\right ) \, dx\\ &=2 \int \frac {x \sqrt {1+x^2+x^5}}{1-x+x^2-x^3+x^4} \, dx-2 \int \frac {\sqrt {1+x^2+x^5}}{1-x^2+x^5} \, dx-3 \int \frac {x^2 \sqrt {1+x^2+x^5}}{1-x+x^2-x^3+x^4} \, dx+5 \int \frac {x^3 \sqrt {1+x^2+x^5}}{1-x^2+x^5} \, dx+\int \frac {\sqrt {1+x^2+x^5}}{1+x} \, dx+\int \frac {\sqrt {1+x^2+x^5}}{-1+x-x^2+x^3-x^4} \, dx-\int \frac {x^3 \sqrt {1+x^2+x^5}}{1-x+x^2-x^3+x^4} \, dx\\ \end {align*}
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Mathematica [F] time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.66, size = 45, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^5+x^2+1}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^5+x^2+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 115, normalized size = 2.56 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{10} + 14 \, x^{7} + 2 \, x^{5} + 17 \, x^{4} - 4 \, \sqrt {2} {\left (x^{6} + 3 \, x^{3} + x\right )} \sqrt {x^{5} + x^{2} + 1} + 14 \, x^{2} + 1}{x^{10} - 2 \, x^{7} + 2 \, x^{5} + x^{4} - 2 \, x^{2} + 1}\right ) + \log \left (\frac {x^{5} + 2 \, x^{2} + 2 \, \sqrt {x^{5} + x^{2} + 1} x + 1}{x^{5} + 1}\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 \frac {{\left (3 \, x^{5} - 2\right )} \sqrt {x^{5} + x^{2} + 1}}{{\left (x^{5} - x^{2} + 1\right )} {\left (x^{5} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.84, size = 121, normalized size = 2.69 \begin {gather*} -\ln \left (-\frac {-x^{5}+2 \sqrt {x^{5}+x^{2}+1}\, x -2 x^{2}-1}{\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{5}-3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{5}+x^{2}+1}\, x -\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{5}-x^{2}+1}\right ) \end {gather*}
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 (3 \, x^{5} - 2\right )} \sqrt {x^{5} + x^{2} + 1}}{{\left (x^{5} - x^{2} + 1\right )} {\left (x^{5} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 77, normalized size = 1.71 \begin {gather*} \ln \left (\frac {2\,x\,\sqrt {x^5+x^2+1}+2\,x^2+x^5+1}{x^5+1}\right )+\sqrt {2}\,\ln \left (\frac {3\,x^2+x^5-2\,\sqrt {2}\,x\,\sqrt {x^5+x^2+1}+1}{x^5-x^2+1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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