Optimal. Leaf size=24 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {x^5-1}}\right )}{\sqrt {a}} \]
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Rubi [F] time = 0.70, 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 {2+3 x^5}{\sqrt {-1+x^5} \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 {2+3 x^5}{\sqrt {-1+x^5} \left (-1-a x^2+x^5\right )} \, dx &=\int \left (\frac {3}{\sqrt {-1+x^5}}+\frac {5+3 a x^2}{\sqrt {-1+x^5} \left (-1-a x^2+x^5\right )}\right ) \, dx\\ &=3 \int \frac {1}{\sqrt {-1+x^5}} \, dx+\int \frac {5+3 a x^2}{\sqrt {-1+x^5} \left (-1-a x^2+x^5\right )} \, dx\\ &=\frac {\left (3 \sqrt {1-x^5}\right ) \int \frac {1}{\sqrt {1-x^5}} \, dx}{\sqrt {-1+x^5}}+\int \left (-\frac {5}{\left (1+a x^2-x^5\right ) \sqrt {-1+x^5}}-\frac {3 a x^2}{\left (1+a x^2-x^5\right ) \sqrt {-1+x^5}}\right ) \, dx\\ &=\frac {3 x \sqrt {1-x^5} \, _2F_1\left (\frac {1}{5},\frac {1}{2};\frac {6}{5};x^5\right )}{\sqrt {-1+x^5}}-5 \int \frac {1}{\left (1+a x^2-x^5\right ) \sqrt {-1+x^5}} \, dx-(3 a) \int \frac {x^2}{\left (1+a x^2-x^5\right ) \sqrt {-1+x^5}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+3 x^5}{\sqrt {-1+x^5} \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 = 5.03, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {-1+x^5}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 138, normalized size = 5.75 \begin {gather*} \left [\frac {\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 )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (x^{5} + a x^{2} - 1\right )} \sqrt {x^{5} - 1} \sqrt {-a}}{2 \, {\left (a x^{6} - a x\right )}}\right )}{a}\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 {3 \, x^{5} + 2}{{\left (x^{5} - a x^{2} - 1\right )} \sqrt {x^{5} - 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {3 x^{5}+2}{\sqrt {x^{5}-1}\, \left (x^{5}-a \,x^{2}-1\right )}\, dx\]
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 {3 \, x^{5} + 2}{{\left (x^{5} - a x^{2} - 1\right )} \sqrt {x^{5} - 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 44, normalized size = 1.83 \begin {gather*} \frac {\ln \left (\frac {a\,x^2+x^5-2\,\sqrt {a}\,x\,\sqrt {x^5-1}-1}{-4\,x^5+4\,a\,x^2+4}\right )}{\sqrt {a}} \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 {3 x^{5} + 2}{\sqrt {\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \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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