Optimal. Leaf size=58 \[ -\frac {1}{40} \left (1+2 x^5\right ) \sqrt {1+x^5+x^{10}}+\frac {1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac {3}{80} \sinh ^{-1}\left (\frac {1+2 x^5}{\sqrt {3}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1371, 654, 626,
633, 221} \begin {gather*} -\frac {3}{80} \sinh ^{-1}\left (\frac {2 x^5+1}{\sqrt {3}}\right )+\frac {1}{15} \left (x^{10}+x^5+1\right )^{3/2}-\frac {1}{40} \left (2 x^5+1\right ) \sqrt {x^{10}+x^5+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 626
Rule 633
Rule 654
Rule 1371
Rubi steps
\begin {align*} \int x^9 \sqrt {1+x^5+x^{10}} \, dx &=\frac {1}{5} \text {Subst}\left (\int x \sqrt {1+x+x^2} \, dx,x,x^5\right )\\ &=\frac {1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac {1}{10} \text {Subst}\left (\int \sqrt {1+x+x^2} \, dx,x,x^5\right )\\ &=-\frac {1}{40} \left (1+2 x^5\right ) \sqrt {1+x^5+x^{10}}+\frac {1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac {3}{80} \text {Subst}\left (\int \frac {1}{\sqrt {1+x+x^2}} \, dx,x,x^5\right )\\ &=-\frac {1}{40} \left (1+2 x^5\right ) \sqrt {1+x^5+x^{10}}+\frac {1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac {1}{80} \sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x^5\right )\\ &=-\frac {1}{40} \left (1+2 x^5\right ) \sqrt {1+x^5+x^{10}}+\frac {1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac {3}{80} \sinh ^{-1}\left (\frac {1+2 x^5}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 55, normalized size = 0.95 \begin {gather*} \frac {1}{120} \sqrt {1+x^5+x^{10}} \left (5+2 x^5+8 x^{10}\right )+\frac {3}{80} \log \left (-1-2 x^5+2 \sqrt {1+x^5+x^{10}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 47, normalized size = 0.81
method | result | size |
trager | \(\left (\frac {1}{15} x^{10}+\frac {1}{60} x^{5}+\frac {1}{24}\right ) \sqrt {x^{10}+x^{5}+1}-\frac {3 \ln \left (2 x^{5}+2 \sqrt {x^{10}+x^{5}+1}+1\right )}{80}\) | \(47\) |
risch | \(\frac {\left (8 x^{10}+2 x^{5}+5\right ) \sqrt {x^{10}+x^{5}+1}}{120}+\frac {3 \ln \left (-2 x^{5}+2 \sqrt {x^{10}+x^{5}+1}-1\right )}{80}\) | \(48\) |
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 = 0.78, size = 47, normalized size = 0.81 \begin {gather*} \frac {1}{120} \, {\left (8 \, x^{10} + 2 \, x^{5} + 5\right )} \sqrt {x^{10} + x^{5} + 1} + \frac {3}{80} \, \log \left (-2 \, x^{5} + 2 \, \sqrt {x^{10} + x^{5} + 1} - 1\right ) \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 x^{9} \sqrt {\left (x^{2} + x + 1\right ) \left (x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.50, size = 49, normalized size = 0.84 \begin {gather*} \frac {1}{120} \, \sqrt {x^{10} + x^{5} + 1} {\left (2 \, {\left (4 \, x^{5} + 1\right )} x^{5} + 5\right )} + \frac {3}{80} \, \log \left (-2 \, x^{5} + 2 \, \sqrt {x^{10} + x^{5} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 43, normalized size = 0.74 \begin {gather*} \frac {\sqrt {x^{10}+x^5+1}\,\left (8\,x^{10}+2\,x^5+5\right )}{120}-\frac {3\,\ln \left (\sqrt {x^{10}+x^5+1}+x^5+\frac {1}{2}\right )}{80} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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