Optimal. Leaf size=39 \[ \frac {2 \sqrt {x^5-1}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {x^5-1}}\right ) \]
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Rubi [F] time = 0.79, 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 )}{x^2 \left (-1-a 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^5} \left (2+3 x^5\right )}{x^2 \left (-1-a x^2+x^5\right )} \, dx &=\int \left (-\frac {2 \sqrt {-1+x^5}}{x^2}+\frac {\left (2 a-5 x^3\right ) \sqrt {-1+x^5}}{1+a x^2-x^5}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {-1+x^5}}{x^2} \, dx\right )+\int \frac {\left (2 a-5 x^3\right ) \sqrt {-1+x^5}}{1+a x^2-x^5} \, dx\\ &=\frac {2 \sqrt {-1+x^5}}{x}-5 \int \frac {x^3}{\sqrt {-1+x^5}} \, dx+\int \left (\frac {2 a \sqrt {-1+x^5}}{1+a x^2-x^5}+\frac {5 x^3 \sqrt {-1+x^5}}{-1-a x^2+x^5}\right ) \, dx\\ &=\frac {2 \sqrt {-1+x^5}}{x}+5 \int \frac {x^3 \sqrt {-1+x^5}}{-1-a x^2+x^5} \, dx+(2 a) \int \frac {\sqrt {-1+x^5}}{1+a x^2-x^5} \, dx-\frac {\left (5 \sqrt {1-x^5}\right ) \int \frac {x^3}{\sqrt {1-x^5}} \, dx}{\sqrt {-1+x^5}}\\ &=\frac {2 \sqrt {-1+x^5}}{x}-\frac {5 x^4 \sqrt {1-x^5} \, _2F_1\left (\frac {1}{2},\frac {4}{5};\frac {9}{5};x^5\right )}{4 \sqrt {-1+x^5}}+5 \int \frac {x^3 \sqrt {-1+x^5}}{-1-a x^2+x^5} \, dx+(2 a) \int \frac {\sqrt {-1+x^5}}{1+a x^2-x^5} \, dx\\ \end {align*}
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Mathematica [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-1+x^5} \left (2+3 x^5\right )}{x^2 \left (-1-a x^2+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.02, size = 39, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-1+x^5}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {-1+x^5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.96, size = 157, normalized size = 4.03 \begin {gather*} \left [\frac {\sqrt {a} x \log \left (-\frac {x^{10} + 6 \, a x^{7} + a^{2} x^{4} - 2 \, x^{5} - 6 \, a x^{2} - 4 \, {\left (x^{6} + a x^{3} - x\right )} \sqrt {x^{5} - 1} \sqrt {a} + 1}{x^{10} - 2 \, a x^{7} + a^{2} x^{4} - 2 \, x^{5} + 2 \, a x^{2} + 1}\right ) + 4 \, \sqrt {x^{5} - 1}}{2 \, x}, \frac {\sqrt {-a} x \arctan \left (\frac {2 \, \sqrt {x^{5} - 1} \sqrt {-a} x}{x^{5} + a x^{2} - 1}\right ) + 2 \, \sqrt {x^{5} - 1}}{x}\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} - 1}}{{\left (x^{5} - a x^{2} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{5}-1}\, \left (3 x^{5}+2\right )}{x^{2} \left (x^{5}-a \,x^{2}-1\right )}\, dx\]
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 (3 \, x^{5} + 2\right )} \sqrt {x^{5} - 1}}{{\left (x^{5} - a x^{2} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 56, normalized size = 1.44 \begin {gather*} \sqrt {a}\,\ln \left (\frac {a\,x^2+x^5-2\,\sqrt {a}\,x\,\sqrt {x^5-1}-1}{-x^5+a\,x^2+1}\right )+\frac {2\,\sqrt {x^5-1}}{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 {\sqrt {\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (3 x^{5} + 2\right )}{x^{2} \left (- a x^{2} + x^{5} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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