Optimal. Leaf size=27 \[ \frac {x^3}{3}-3 \tan ^{-1}(3+x)+\frac {1}{2} \log \left (10+6 x+x^2\right ) \]
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Rubi [A]
time = 0.02, 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 = {1671, 648, 632,
210, 642} \begin {gather*} -3 \text {ArcTan}(x+3)+\frac {x^3}{3}+\frac {1}{2} \log \left (x^2+6 x+10\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1671
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 \text {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 \begin {gather*} \frac {x^3}{3}-3 \tan ^{-1}(3+x)+\frac {1}{2} \log \left (10+6 x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 24, normalized size = 0.89
method | result | size |
default | \(\frac {x^{3}}{3}-3 \arctan \left (3+x \right )+\frac {\ln \left (x^{2}+6 x +10\right )}{2}\) | \(24\) |
risch | \(\frac {x^{3}}{3}-3 \arctan \left (3+x \right )+\frac {\ln \left (x^{2}+6 x +10\right )}{2}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, x^{3} - 3 \, \arctan \left (x + 3\right ) + \frac {1}{2} \, \log \left (x^{2} + 6 \, x + 10\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, x^{3} - 3 \, \arctan \left (x + 3\right ) + \frac {1}{2} \, \log \left (x^{2} + 6 \, x + 10\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 22, normalized size = 0.81 \begin {gather*} \frac {x^{3}}{3} + \frac {\log {\left (x^{2} + 6 x + 10 \right )}}{2} - 3 \operatorname {atan}{\left (x + 3 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.15, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, x^{3} - 3 \, \arctan \left (x + 3\right ) + \frac {1}{2} \, \log \left (x^{2} + 6 \, x + 10\right ) \end {gather*}
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 \begin {gather*} \frac {\ln \left (x^2+6\,x+10\right )}{2}-3\,\mathrm {atan}\left (x+3\right )+\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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