Optimal. Leaf size=193 \[ \frac {\sqrt {2} \left (1+\sqrt {1-4 a}-4 a\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1-\sqrt {1-4 a}+2 a}}\right )}{5 \sqrt {1-4 a} \sqrt {a} \sqrt {-1-\sqrt {1-4 a}+2 a}}+\frac {\sqrt {2} \left (-1+\sqrt {1-4 a}+4 a\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1+\sqrt {1-4 a}+2 a}}\right )}{5 \sqrt {1-4 a} \sqrt {a} \sqrt {-1+\sqrt {1-4 a}+2 a}} \]
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Rubi [A]
time = 0.25, antiderivative size = 81, normalized size of antiderivative = 0.42, number of steps
used = 5, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6847, 840,
1178, 642} \begin {gather*} \frac {\log \left (-\sqrt {a} \left (1-x^5\right )+\sqrt {a}-\sqrt {x^5-1}\right )}{5 \sqrt {a}}-\frac {\log \left (-\sqrt {a} \left (1-x^5\right )+\sqrt {a}+\sqrt {x^5-1}\right )}{5 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 642
Rule 840
Rule 1178
Rule 6847
Rubi steps
\begin {align*} \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {-2+x}{\sqrt {-1+x} \left (1-x+a x^2\right )} \, dx,x,x^5\right )\\ &=\frac {2}{5} \text {Subst}\left (\int \frac {-1+x^2}{a+(-1+2 a) x^2+a x^4} \, dx,x,\sqrt {-1+x^5}\right )\\ &=\frac {\text {Subst}\left (\int \frac {\frac {1}{\sqrt {a}}+2 x}{-1-\frac {x}{\sqrt {a}}-x^2} \, dx,x,\sqrt {-1+x^5}\right )}{5 \sqrt {a}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{\sqrt {a}}-2 x}{-1+\frac {x}{\sqrt {a}}-x^2} \, dx,x,\sqrt {-1+x^5}\right )}{5 \sqrt {a}}\\ &=\frac {\log \left (\sqrt {a}-\sqrt {a} \left (1-x^5\right )-\sqrt {-1+x^5}\right )}{5 \sqrt {a}}-\frac {\log \left (\sqrt {a}-\sqrt {a} \left (1-x^5\right )+\sqrt {-1+x^5}\right )}{5 \sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 169, normalized size = 0.88 \begin {gather*} \frac {\frac {\left (1+\sqrt {1-4 a}-4 a\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1-\sqrt {1-4 a}+2 a}}\right )}{\sqrt {-1-\sqrt {1-4 a}+2 a}}+\frac {\left (-1+\sqrt {1-4 a}+4 a\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1+\sqrt {1-4 a}+2 a}}\right )}{\sqrt {-1+\sqrt {1-4 a}+2 a}}}{5 \sqrt {\frac {1}{2}-2 a} \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (x^{5}-2\right )}{\sqrt {x^{5}-1}\, \left (a \,x^{10}-x^{5}+1\right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 74, normalized size = 0.38 \begin {gather*} \left [\frac {\log \left (\frac {a x^{10} - 2 \, \sqrt {x^{5} - 1} \sqrt {a} x^{5} + x^{5} - 1}{a x^{10} - x^{5} + 1}\right )}{5 \, \sqrt {a}}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{5}}{\sqrt {x^{5} - 1}}\right )}{5 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 34, normalized size = 0.18 \begin {gather*} \frac {\pi + 2 \, \arctan \left (\frac {{\left (x^{5} - 1\right )} a + a}{\sqrt {x^{5} - 1} \sqrt {-a}}\right )}{5 \, \sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.24, size = 47, normalized size = 0.24 \begin {gather*} \frac {\ln \left (\frac {a\,x^{10}+x^5-2\,\sqrt {a}\,x^5\,\sqrt {x^5-1}-1}{4\,a\,x^{10}-4\,x^5+4}\right )}{5\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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