Integrand size = 18, antiderivative size = 37 \[ \int \frac {31+5 x}{11-4 x+3 x^2} \, dx=-\frac {103 \arctan \left (\frac {2-3 x}{\sqrt {29}}\right )}{3 \sqrt {29}}+\frac {5}{6} \log \left (11-4 x+3 x^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {648, 632, 210, 642} \[ \int \frac {31+5 x}{11-4 x+3 x^2} \, dx=\frac {5}{6} \log \left (3 x^2-4 x+11\right )-\frac {103 \arctan \left (\frac {2-3 x}{\sqrt {29}}\right )}{3 \sqrt {29}} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \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} \text {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*}
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {31+5 x}{11-4 x+3 x^2} \, dx=\frac {103 \arctan \left (\frac {-2+3 x}{\sqrt {29}}\right )}{3 \sqrt {29}}+\frac {5}{6} \log \left (11-4 x+3 x^2\right ) \]
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Time = 1.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {5 \ln \left (3 x^{2}-4 x +11\right )}{6}+\frac {103 \sqrt {29}\, \arctan \left (\frac {\left (6 x -4\right ) \sqrt {29}}{58}\right )}{87}\) | \(31\) |
risch | \(\frac {5 \ln \left (9 x^{2}-12 x +33\right )}{6}+\frac {103 \sqrt {29}\, \arctan \left (\frac {\left (3 x -2\right ) \sqrt {29}}{29}\right )}{87}\) | \(31\) |
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Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81 \[ \int \frac {31+5 x}{11-4 x+3 x^2} \, dx=\frac {103}{87} \, \sqrt {29} \arctan \left (\frac {1}{29} \, \sqrt {29} {\left (3 \, x - 2\right )}\right ) + \frac {5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \]
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Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.19 \[ \int \frac {31+5 x}{11-4 x+3 x^2} \, dx=\frac {5 \log {\left (x^{2} - \frac {4 x}{3} + \frac {11}{3} \right )}}{6} + \frac {103 \sqrt {29} \operatorname {atan}{\left (\frac {3 \sqrt {29} x}{29} - \frac {2 \sqrt {29}}{29} \right )}}{87} \]
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Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81 \[ \int \frac {31+5 x}{11-4 x+3 x^2} \, dx=\frac {103}{87} \, \sqrt {29} \arctan \left (\frac {1}{29} \, \sqrt {29} {\left (3 \, x - 2\right )}\right ) + \frac {5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \]
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Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81 \[ \int \frac {31+5 x}{11-4 x+3 x^2} \, dx=\frac {103}{87} \, \sqrt {29} \arctan \left (\frac {1}{29} \, \sqrt {29} {\left (3 \, x - 2\right )}\right ) + \frac {5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \]
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Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81 \[ \int \frac {31+5 x}{11-4 x+3 x^2} \, dx=\frac {5\,\ln \left (x^2-\frac {4\,x}{3}+\frac {11}{3}\right )}{6}+\frac {103\,\sqrt {29}\,\mathrm {atan}\left (\frac {3\,\sqrt {29}\,x}{29}-\frac {2\,\sqrt {29}}{29}\right )}{87} \]
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