Optimal. Leaf size=49 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {x^5-1}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {x^5-1}}\right )}{\sqrt [4]{a}} \]
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Rubi [F] time = 0.54, 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^5} \left (2+3 x^5\right )}{1-a x^4-2 x^5+x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^5} \left (2+3 x^5\right )}{1-a x^4-2 x^5+x^{10}} \, dx &=\int \left (-\frac {2 \sqrt {-1+x^5}}{-1+a x^4+2 x^5-x^{10}}+\frac {3 x^5 \sqrt {-1+x^5}}{1-a x^4-2 x^5+x^{10}}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {-1+x^5}}{-1+a x^4+2 x^5-x^{10}} \, dx\right )+3 \int \frac {x^5 \sqrt {-1+x^5}}{1-a x^4-2 x^5+x^{10}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-1+x^5} \left (2+3 x^5\right )}{1-a x^4-2 x^5+x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.59, size = 49, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {-1+x^5}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {-1+x^5}}\right )}{\sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 195, normalized size = 3.98 \begin {gather*} -\frac {\arctan \left (\frac {a^{\frac {1}{4}} x}{\sqrt {x^{5} - 1}}\right )}{a^{\frac {1}{4}}} - \frac {\log \left (\frac {x^{10} + a x^{4} - 2 \, x^{5} + 2 \, \sqrt {x^{5} - 1} {\left (a^{\frac {3}{4}} x^{3} + \frac {a x^{6} - a x}{a^{\frac {3}{4}}}\right )} + \frac {2 \, {\left (a x^{7} - a x^{2}\right )}}{\sqrt {a}} + 1}{x^{10} - a x^{4} - 2 \, x^{5} + 1}\right )}{4 \, a^{\frac {1}{4}}} + \frac {\log \left (\frac {x^{10} + a x^{4} - 2 \, x^{5} - 2 \, \sqrt {x^{5} - 1} {\left (a^{\frac {3}{4}} x^{3} + \frac {a x^{6} - a x}{a^{\frac {3}{4}}}\right )} + \frac {2 \, {\left (a x^{7} - a x^{2}\right )}}{\sqrt {a}} + 1}{x^{10} - a x^{4} - 2 \, x^{5} + 1}\right )}{4 \, a^{\frac {1}{4}}} \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} - 1}}{x^{10} - a x^{4} - 2 \, x^{5} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{5}-1}\, \left (3 x^{5}+2\right )}{x^{10}-a \,x^{4}-2 x^{5}+1}\, dx\]
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 (3 \, x^{5} + 2\right )} \sqrt {x^{5} - 1}}{x^{10} - a x^{4} - 2 \, x^{5} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.02, size = 98, normalized size = 2.00 \begin {gather*} \frac {\ln \left (\frac {x^5+\sqrt {a}\,x^2-2\,a^{1/4}\,x\,\sqrt {x^5-1}-1}{\sqrt {a}\,x^2-x^5+1}\right )}{2\,a^{1/4}}+\frac {\ln \left (\frac {x^5-\sqrt {a}\,x^2-1+a^{1/4}\,x\,\sqrt {x^5-1}\,2{}\mathrm {i}}{x^5+\sqrt {a}\,x^2-1}\right )\,1{}\mathrm {i}}{2\,a^{1/4}} \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 {\sqrt {\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (3 x^{5} + 2\right )}{- a x^{4} + x^{10} - 2 x^{5} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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