Optimal. Leaf size=27 \[ \frac {x^3}{3}+\frac {1}{2} \log \left (x^2+6 x+10\right )-3 \tan ^{-1}(x+3) \]
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Rubi [A] time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1657, 634, 618, 204, 628} \[ \frac {x^3}{3}+\frac {1}{2} \log \left (x^2+6 x+10\right )-3 \tan ^{-1}(x+3) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1657
Rubi steps
\begin {align*} \int \frac {x+10 x^2+6 x^3+x^4}{10+6 x+x^2} \, dx &=\int \left (x^2+\frac {x}{10+6 x+x^2}\right ) \, dx\\ &=\frac {x^3}{3}+\int \frac {x}{10+6 x+x^2} \, dx\\ &=\frac {x^3}{3}+\frac {1}{2} \int \frac {6+2 x}{10+6 x+x^2} \, dx-3 \int \frac {1}{10+6 x+x^2} \, dx\\ &=\frac {x^3}{3}+\frac {1}{2} \log \left (10+6 x+x^2\right )+6 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,6+2 x\right )\\ &=\frac {x^3}{3}-3 \tan ^{-1}(3+x)+\frac {1}{2} \log \left (10+6 x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ \frac {x^3}{3}+\frac {1}{2} \log \left (x^2+6 x+10\right )-3 \tan ^{-1}(x+3) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 23, normalized size = 0.85 \[ \frac {1}{3} \, x^{3} - 3 \, \arctan \left (x + 3\right ) + \frac {1}{2} \, \log \left (x^{2} + 6 \, x + 10\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 23, normalized size = 0.85 \[ \frac {1}{3} \, x^{3} - 3 \, \arctan \left (x + 3\right ) + \frac {1}{2} \, \log \left (x^{2} + 6 \, x + 10\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.89 \[ \frac {x^{3}}{3}-3 \arctan \left (x +3\right )+\frac {\ln \left (x^{2}+6 x +10\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.08, size = 23, normalized size = 0.85 \[ \frac {1}{3} \, x^{3} - 3 \, \arctan \left (x + 3\right ) + \frac {1}{2} \, \log \left (x^{2} + 6 \, x + 10\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.14, size = 23, normalized size = 0.85 \[ \frac {\ln \left (x^2+6\,x+10\right )}{2}-3\,\mathrm {atan}\left (x+3\right )+\frac {x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 22, normalized size = 0.81 \[ \frac {x^{3}}{3} + \frac {\log {\left (x^{2} + 6 x + 10 \right )}}{2} - 3 \operatorname {atan}{\left (x + 3 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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