Optimal. Leaf size=23 \[ \frac {2 \tan ^{-1}\left (\frac {1+2 x^5}{\sqrt {7}}\right )}{5 \sqrt {7}} \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1366, 632, 210}
\begin {gather*} \frac {2 \text {ArcTan}\left (\frac {2 x^5+1}{\sqrt {7}}\right )}{5 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 1366
Rubi steps
\begin {align*} \int \frac {x^4}{2+x^5+x^{10}} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{2+x+x^2} \, dx,x,x^5\right )\\ &=-\left (\frac {2}{5} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,1+2 x^5\right )\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {1+2 x^5}{\sqrt {7}}\right )}{5 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {1+2 x^5}{\sqrt {7}}\right )}{5 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 19, normalized size = 0.83
method | result | size |
default | \(\frac {2 \arctan \left (\frac {\left (2 x^{5}+1\right ) \sqrt {7}}{7}\right ) \sqrt {7}}{35}\) | \(19\) |
risch | \(\frac {2 \arctan \left (\frac {\left (2 x^{5}+1\right ) \sqrt {7}}{7}\right ) \sqrt {7}}{35}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 18, normalized size = 0.78 \begin {gather*} \frac {2}{35} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x^{5} + 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 18, normalized size = 0.78 \begin {gather*} \frac {2}{35} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x^{5} + 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 27, normalized size = 1.17 \begin {gather*} \frac {2 \sqrt {7} \operatorname {atan}{\left (\frac {2 \sqrt {7} x^{5}}{7} + \frac {\sqrt {7}}{7} \right )}}{35} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.21, size = 18, normalized size = 0.78 \begin {gather*} \frac {2}{35} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x^{5} + 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.33, size = 20, normalized size = 0.87 \begin {gather*} \frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {2\,\sqrt {7}\,x^5}{7}+\frac {\sqrt {7}}{7}\right )}{35} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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