Optimal. Leaf size=250 \[ \frac {2 \sqrt {-1+x^5}}{5 x^5}+\frac {\left (\sqrt {2} \sqrt {a}+\sqrt {2} \sqrt {1-4 a} \sqrt {a}-4 \sqrt {2} a^{3/2}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1-\sqrt {1-4 a}+2 a}}\right )}{5 \sqrt {1-4 a} \sqrt {-1-\sqrt {1-4 a}+2 a}}+\frac {\left (-\sqrt {2} \sqrt {a}+\sqrt {2} \sqrt {1-4 a} \sqrt {a}+4 \sqrt {2} a^{3/2}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1+\sqrt {1-4 a}+2 a}}\right )}{5 \sqrt {1-4 a} \sqrt {-1+\sqrt {1-4 a}+2 a}} \]
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Rubi [A]
time = 0.51, antiderivative size = 97, normalized size of antiderivative = 0.39, number of steps
used = 16, number of rules used = 11, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {6860, 272,
43, 65, 209, 52, 6847, 838, 840, 1178, 642} \begin {gather*} \frac {1}{5} \sqrt {a} \log \left (-\sqrt {a} \left (1-x^5\right )+\sqrt {a}-\sqrt {x^5-1}\right )-\frac {1}{5} \sqrt {a} \log \left (-\sqrt {a} \left (1-x^5\right )+\sqrt {a}+\sqrt {x^5-1}\right )+\frac {2 \sqrt {x^5-1}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 209
Rule 272
Rule 642
Rule 838
Rule 840
Rule 1178
Rule 6847
Rule 6860
Rubi steps
\begin {align*} \int \frac {\left (-2+x^5\right ) \sqrt {-1+x^5}}{x^6 \left (1-x^5+a x^{10}\right )} \, dx &=\int \left (-\frac {2 \sqrt {-1+x^5}}{x^6}-\frac {\sqrt {-1+x^5}}{x}+\frac {x^4 \sqrt {-1+x^5} \left (-1+2 a+a x^5\right )}{1-x^5+a x^{10}}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {-1+x^5}}{x^6} \, dx\right )-\int \frac {\sqrt {-1+x^5}}{x} \, dx+\int \frac {x^4 \sqrt {-1+x^5} \left (-1+2 a+a x^5\right )}{1-x^5+a x^{10}} \, dx\\ &=-\left (\frac {1}{5} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^5\right )\right )+\frac {1}{5} \text {Subst}\left (\int \frac {\sqrt {-1+x} (-1+2 a+a x)}{1-x+a x^2} \, dx,x,x^5\right )-\frac {2}{5} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^5\right )\\ &=\frac {2 \sqrt {-1+x^5}}{5 x^5}+\frac {\text {Subst}\left (\int \frac {-2 a^2+a^2 x}{\sqrt {-1+x} \left (1-x+a x^2\right )} \, dx,x,x^5\right )}{5 a}\\ &=\frac {2 \sqrt {-1+x^5}}{5 x^5}+\frac {2 \text {Subst}\left (\int \frac {-a^2+a^2 x^2}{a+(-1+2 a) x^2+a x^4} \, dx,x,\sqrt {-1+x^5}\right )}{5 a}\\ &=\frac {2 \sqrt {-1+x^5}}{5 x^5}+\frac {1}{5} \sqrt {a} \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 )+\frac {1}{5} \sqrt {a} \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 )\\ &=\frac {2 \sqrt {-1+x^5}}{5 x^5}+\frac {1}{5} \sqrt {a} \log \left (\sqrt {a}-\sqrt {a} \left (1-x^5\right )-\sqrt {-1+x^5}\right )-\frac {1}{5} \sqrt {a} \log \left (\sqrt {a}-\sqrt {a} \left (1-x^5\right )+\sqrt {-1+x^5}\right )\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 199, normalized size = 0.80 \begin {gather*} \frac {1}{5} \left (\frac {2 \sqrt {-1+x^5}}{x^5}+\frac {\left (1+\sqrt {1-4 a}-4 a\right ) \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1-\sqrt {1-4 a}+2 a}}\right )}{\sqrt {\frac {1}{2}-2 a} \sqrt {-1-\sqrt {1-4 a}+2 a}}+\frac {\sqrt {a} \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 {\frac {1}{2}-2 a} \sqrt {-1+\sqrt {1-4 a}+2 a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{5}-2\right ) \sqrt {x^{5}-1}}{x^{6} \left (a \,x^{10}-x^{5}+1\right )}\, dx\]
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 = 0.38, size = 103, normalized size = 0.41 \begin {gather*} \left [\frac {\sqrt {a} x^{5} \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 ) + 2 \, \sqrt {x^{5} - 1}}{5 \, x^{5}}, \frac {2 \, {\left (\sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {-a} x^{5}}{\sqrt {x^{5} - 1}}\right ) + \sqrt {x^{5} - 1}\right )}}{5 \, x^{5}}\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 = 0.52, size = 55, normalized size = 0.22 \begin {gather*} \frac {{\left (\pi + 2 \, \arctan \left (\frac {{\left (x^{5} - 1\right )} a + a}{\sqrt {x^{5} - 1} \sqrt {-a}}\right )\right )} a}{5 \, \sqrt {-a}} + \frac {2}{5 \, {\left (\sqrt {x^{5} - 1} + \frac {1}{\sqrt {x^{5} - 1}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.54, size = 60, normalized size = 0.24 \begin {gather*} \frac {2\,\sqrt {x^5-1}}{5\,x^5}+\frac {\sqrt {a}\,\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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