Optimal. Leaf size=27 \[ x+\frac {3}{8} \tan ^{-1}\left (\frac {1}{2}+x\right )+\frac {1}{8} \log \left (5+4 x+4 x^2\right ) \]
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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 = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1671, 648, 632,
210, 642} \begin {gather*} \frac {3}{8} \text {ArcTan}\left (x+\frac {1}{2}\right )+\frac {1}{8} \log \left (4 x^2+4 x+5\right )+x \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 {7+5 x+4 x^2}{5+4 x+4 x^2} \, dx &=\int \left (1+\frac {2+x}{5+4 x+4 x^2}\right ) \, dx\\ &=x+\int \frac {2+x}{5+4 x+4 x^2} \, dx\\ &=x+\frac {1}{8} \int \frac {4+8 x}{5+4 x+4 x^2} \, dx+\frac {3}{2} \int \frac {1}{5+4 x+4 x^2} \, dx\\ &=x+\frac {1}{8} \log \left (5+4 x+4 x^2\right )-3 \text {Subst}\left (\int \frac {1}{-64-x^2} \, dx,x,4+8 x\right )\\ &=x+\frac {3}{8} \tan ^{-1}\left (\frac {1}{2}+x\right )+\frac {1}{8} \log \left (5+4 x+4 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 31, normalized size = 1.15 \begin {gather*} x+\frac {3}{8} \tan ^{-1}\left (\frac {1}{2} (1+2 x)\right )+\frac {1}{8} \log \left (5+4 x+4 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 22, normalized size = 0.81
method | result | size |
default | \(x +\frac {3 \arctan \left (x +\frac {1}{2}\right )}{8}+\frac {\ln \left (4 x^{2}+4 x +5\right )}{8}\) | \(22\) |
risch | \(x +\frac {3 \arctan \left (x +\frac {1}{2}\right )}{8}+\frac {\ln \left (4 x^{2}+4 x +5\right )}{8}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.03, size = 21, normalized size = 0.78 \begin {gather*} x + \frac {3}{8} \, \arctan \left (x + \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.17, size = 21, normalized size = 0.78 \begin {gather*} x + \frac {3}{8} \, \arctan \left (x + \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 22, normalized size = 0.81 \begin {gather*} x + \frac {\log {\left (x^{2} + x + \frac {5}{4} \right )}}{8} + \frac {3 \operatorname {atan}{\left (x + \frac {1}{2} \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 21, normalized size = 0.78 \begin {gather*} x + \frac {3}{8} \, \arctan \left (x + \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (4 \, x^{2} + 4 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 17, normalized size = 0.63 \begin {gather*} x+\frac {\ln \left (x^2+x+\frac {5}{4}\right )}{8}+\frac {3\,\mathrm {atan}\left (x+\frac {1}{2}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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