Optimal. Leaf size=24 \[ -\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x^6+x}}\right )}{\sqrt {a}} \]
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Rubi [F] time = 0.96, 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 (a-x+a 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 (a-x+a 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 (a-x+a 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 (a-x^2+a 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}{a \sqrt {1+x^{10}}}-\frac {5 a-4 x^2}{a \sqrt {1+x^{10}} \left (a-x^2+a 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 a-4 x^2}{\sqrt {1+x^{10}} \left (a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \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 )}{a \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 )}{a \sqrt {x+x^6}}-\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {5 a}{\sqrt {1+x^{10}} \left (a-x^2+a x^{10}\right )}-\frac {4 x^2}{\sqrt {1+x^{10}} \left (a-x^2+a x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{a \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 )}{a \sqrt {x+x^6}}-\frac {\left (10 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^{10}} \left (a-x^2+a 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 {x^2}{\sqrt {1+x^{10}} \left (a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}\\ \end {align*}
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Mathematica [F] time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+4 x^5}{\left (a-x+a 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 = 24, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x+x^6}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 129, normalized size = 5.38 \begin {gather*} \left [\frac {\log \left (-\frac {a^{2} x^{10} + 2 \, a^{2} x^{5} + 6 \, a x^{6} - 4 \, {\left (a x^{5} + a + x\right )} \sqrt {x^{6} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{10} + 2 \, a^{2} x^{5} - 2 \, a x^{6} + a^{2} - 2 \, a x + x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a}}{a x^{5} + a + x}\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}{{\left (a x^{5} + a - x\right )} \sqrt {x^{6} + x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {4 x^{5}-1}{\left (a \,x^{5}+a -x \right ) \sqrt {x^{6}+x}}\, dx\]
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}{{\left (a x^{5} + a - x\right )} \sqrt {x^{6} + x}}\,{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 (a\,x^5-x+a\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^{5} + a - x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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