Optimal. Leaf size=250 \[ \frac {2 \sqrt {1+x^5}}{5 x^5}+\frac {\left (\sqrt {2} \sqrt {a}+4 \sqrt {2} a^{3/2}+\sqrt {2} \sqrt {a} \sqrt {1+4 a}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {1+x^5}}{\sqrt {-1-2 a-\sqrt {1+4 a}}}\right )}{5 \sqrt {1+4 a} \sqrt {-1-2 a-\sqrt {1+4 a}}}+\frac {\left (-\sqrt {2} \sqrt {a}-4 \sqrt {2} a^{3/2}+\sqrt {2} \sqrt {a} \sqrt {1+4 a}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {1+x^5}}{\sqrt {-1-2 a+\sqrt {1+4 a}}}\right )}{5 \sqrt {1+4 a} \sqrt {-1-2 a+\sqrt {1+4 a}}} \]
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Rubi [A]
time = 0.55, antiderivative size = 89, normalized size of antiderivative = 0.36, number of steps
used = 18, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6860, 272,
43, 65, 213, 52, 6847, 838, 840, 1175, 632, 212} \begin {gather*} \frac {2}{5} \sqrt {a} \tanh ^{-1}\left (\sqrt {4 a+1}-2 \sqrt {a} \sqrt {x^5+1}\right )-\frac {2}{5} \sqrt {a} \tanh ^{-1}\left (2 \sqrt {a} \sqrt {x^5+1}+\sqrt {4 a+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 212
Rule 213
Rule 272
Rule 632
Rule 838
Rule 840
Rule 1175
Rule 6847
Rule 6860
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^5} \left (2+x^5\right )}{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\\ &=\frac {1}{5} \text {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^5\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} \text {Subst}\left (\int \frac {1}{1-\frac {\sqrt {1+4 a} x}{\sqrt {a}}+x^2} \, dx,x,\sqrt {1+x^5}\right )+\frac {1}{5} \text {Subst}\left (\int \frac {1}{1+\frac {\sqrt {1+4 a} x}{\sqrt {a}}+x^2} \, dx,x,\sqrt {1+x^5}\right )\\ &=\frac {2 \sqrt {1+x^5}}{5 x^5}-\frac {2}{5} \text {Subst}\left (\int \frac {1}{\frac {1}{a}-x^2} \, dx,x,-\frac {\sqrt {1+4 a}}{\sqrt {a}}+2 \sqrt {1+x^5}\right )-\frac {2}{5} \text {Subst}\left (\int \frac {1}{\frac {1}{a}-x^2} \, dx,x,\frac {\sqrt {1+4 a}}{\sqrt {a}}+2 \sqrt {1+x^5}\right )\\ &=\frac {2 \sqrt {1+x^5}}{5 x^5}+\frac {2}{5} \sqrt {a} \tanh ^{-1}\left (\sqrt {a} \left (\frac {\sqrt {1+4 a}}{\sqrt {a}}-2 \sqrt {1+x^5}\right )\right )-\frac {2}{5} \sqrt {a} \tanh ^{-1}\left (\sqrt {a} \left (\frac {\sqrt {1+4 a}}{\sqrt {a}}+2 \sqrt {1+x^5}\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 205, normalized size = 0.82 \begin {gather*} \frac {1}{5} \left (\frac {2 \sqrt {1+x^5}}{x^5}+\frac {\sqrt {2} \sqrt {a} \left (1+4 a+\sqrt {1+4 a}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {1+x^5}}{\sqrt {-1-2 a-\sqrt {1+4 a}}}\right )}{\sqrt {1+4 a} \sqrt {-1-2 a-\sqrt {1+4 a}}}+\frac {\sqrt {2} \sqrt {a} \left (-1-4 a+\sqrt {1+4 a}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {1+x^5}}{\sqrt {-1-2 a+\sqrt {1+4 a}}}\right )}{\sqrt {1+4 a} \sqrt {-1-2 a+\sqrt {1+4 a}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{5}+1}\, \left (x^{5}+2\right )}{x^{6} \left (a \,x^{10}-x^{5}-1\right )}\, dx\]
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 = 0.43, 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 = 0.57, size = 59, normalized size = 0.24 \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 = 2.56, 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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