Optimal. Leaf size=37 \[ \frac {5}{6} \log \left (3 x^2-4 x+11\right )-\frac {103 \tan ^{-1}\left (\frac {2-3 x}{\sqrt {29}}\right )}{3 \sqrt {29}} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {634, 618, 204, 628} \[ \frac {5}{6} \log \left (3 x^2-4 x+11\right )-\frac {103 \tan ^{-1}\left (\frac {2-3 x}{\sqrt {29}}\right )}{3 \sqrt {29}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {31+5 x}{11-4 x+3 x^2} \, dx &=\frac {5}{6} \int \frac {-4+6 x}{11-4 x+3 x^2} \, dx+\frac {103}{3} \int \frac {1}{11-4 x+3 x^2} \, dx\\ &=\frac {5}{6} \log \left (11-4 x+3 x^2\right )-\frac {206}{3} \operatorname {Subst}\left (\int \frac {1}{-116-x^2} \, dx,x,-4+6 x\right )\\ &=-\frac {103 \tan ^{-1}\left (\frac {2-3 x}{\sqrt {29}}\right )}{3 \sqrt {29}}+\frac {5}{6} \log \left (11-4 x+3 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ \frac {5}{6} \log \left (3 x^2-4 x+11\right )+\frac {103 \tan ^{-1}\left (\frac {3 x-2}{\sqrt {29}}\right )}{3 \sqrt {29}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 30, normalized size = 0.81 \[ \frac {103}{87} \, \sqrt {29} \arctan \left (\frac {1}{29} \, \sqrt {29} {\left (3 \, x - 2\right )}\right ) + \frac {5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 30, normalized size = 0.81 \[ \frac {103}{87} \, \sqrt {29} \arctan \left (\frac {1}{29} \, \sqrt {29} {\left (3 \, x - 2\right )}\right ) + \frac {5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.84 \[ \frac {103 \sqrt {29}\, \arctan \left (\frac {\left (6 x -4\right ) \sqrt {29}}{58}\right )}{87}+\frac {5 \ln \left (3 x^{2}-4 x +11\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 30, normalized size = 0.81 \[ \frac {103}{87} \, \sqrt {29} \arctan \left (\frac {1}{29} \, \sqrt {29} {\left (3 \, x - 2\right )}\right ) + \frac {5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 30, normalized size = 0.81 \[ \frac {5\,\ln \left (x^2-\frac {4\,x}{3}+\frac {11}{3}\right )}{6}+\frac {103\,\sqrt {29}\,\mathrm {atan}\left (\frac {3\,\sqrt {29}\,x}{29}-\frac {2\,\sqrt {29}}{29}\right )}{87} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 44, normalized size = 1.19 \[ \frac {5 \log {\left (x^{2} - \frac {4 x}{3} + \frac {11}{3} \right )}}{6} + \frac {103 \sqrt {29} \operatorname {atan}{\left (\frac {3 \sqrt {29} x}{29} - \frac {2 \sqrt {29}}{29} \right )}}{87} \]
Verification of antiderivative is not currently implemented for this CAS.
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