Optimal. Leaf size=26 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {x^6-x}}\right )}{\sqrt {a}} \]
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Rubi [F] time = 1.19, 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 {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^5}\right ) \int \frac {1+4 x^5}{\sqrt {x} \sqrt {-1+x^5} \left (-1-a x+x^5\right )} \, dx}{\sqrt {-x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {1+4 x^{10}}{\sqrt {-1+x^{10}} \left (-1-a x^2+x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {4}{\sqrt {-1+x^{10}}}+\frac {5+4 a x^2}{\sqrt {-1+x^{10}} \left (-1-a x^2+x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {5+4 a x^2}{\sqrt {-1+x^{10}} \left (-1-a x^2+x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}\\ &=\frac {\left (8 \sqrt {x} \sqrt {1-x^5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {5}{\left (1+a x^2-x^{10}\right ) \sqrt {-1+x^{10}}}-\frac {4 a x^2}{\left (1+a x^2-x^{10}\right ) \sqrt {-1+x^{10}}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}\\ &=\frac {8 x \sqrt {1-x^5} \, _2F_1\left (\frac {1}{10},\frac {1}{2};\frac {11}{10};x^5\right )}{\sqrt {-x+x^6}}-\frac {\left (10 \sqrt {x} \sqrt {-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+a x^2-x^{10}\right ) \sqrt {-1+x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}-\frac {\left (8 a \sqrt {x} \sqrt {-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+a x^2-x^{10}\right ) \sqrt {-1+x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}\\ \end {align*}
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Mathematica [F] time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.66, size = 26, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {-x+x^6}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 123, normalized size = 4.73 \begin {gather*} \left [\frac {\log \left (-\frac {x^{10} + 6 \, a x^{6} - 2 \, x^{5} + a^{2} x^{2} - 4 \, \sqrt {x^{6} - x} {\left (x^{5} + a x - 1\right )} \sqrt {a} - 6 \, a x + 1}{x^{10} - 2 \, a x^{6} - 2 \, x^{5} + a^{2} x^{2} + 2 \, a x + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} - x} \sqrt {-a}}{x^{5} + a x - 1}\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 {4 \, x^{5} + 1}{\sqrt {x^{6} - x} {\left (x^{5} - a x - 1\right )}}\,{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 \[\int \frac {4 x^{5}+1}{\left (x^{5}-a x -1\right ) \sqrt {x^{6}-x}}\, dx\]
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 {4 \, x^{5} + 1}{\sqrt {x^{6} - x} {\left (x^{5} - a x - 1\right )}}\,{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 = -0.04 \begin {gather*} \int -\frac {4\,x^5+1}{\sqrt {x^6-x}\,\left (-x^5+a\,x+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 {4 x^{5} + 1}{\sqrt {x \left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (- a x + x^{5} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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