Integrand size = 14, antiderivative size = 49 \[ \int \sqrt {-8+6 x+9 x^2} \, dx=\frac {1}{6} (1+3 x) \sqrt {-8+6 x+9 x^2}-\frac {3}{2} \text {arctanh}\left (\frac {1+3 x}{\sqrt {-8+6 x+9 x^2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {626, 635, 212} \[ \int \sqrt {-8+6 x+9 x^2} \, dx=\frac {1}{6} (3 x+1) \sqrt {9 x^2+6 x-8}-\frac {3}{2} \text {arctanh}\left (\frac {3 x+1}{\sqrt {9 x^2+6 x-8}}\right ) \]
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Rule 212
Rule 626
Rule 635
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} (1+3 x) \sqrt {-8+6 x+9 x^2}-\frac {9}{2} \int \frac {1}{\sqrt {-8+6 x+9 x^2}} \, dx \\ & = \frac {1}{6} (1+3 x) \sqrt {-8+6 x+9 x^2}-9 \text {Subst}\left (\int \frac {1}{36-x^2} \, dx,x,\frac {6+18 x}{\sqrt {-8+6 x+9 x^2}}\right ) \\ & = \frac {1}{6} (1+3 x) \sqrt {-8+6 x+9 x^2}-\frac {3}{2} \tanh ^{-1}\left (\frac {1+3 x}{\sqrt {-8+6 x+9 x^2}}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \sqrt {-8+6 x+9 x^2} \, dx=\frac {1}{6} (1+3 x) \sqrt {-8+6 x+9 x^2}-3 \text {arctanh}\left (\frac {\sqrt {-8+6 x+9 x^2}}{-2+3 x}\right ) \]
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Time = 2.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82
method | result | size |
trager | \(\left (\frac {x}{2}+\frac {1}{6}\right ) \sqrt {9 x^{2}+6 x -8}-\frac {3 \ln \left (\sqrt {9 x^{2}+6 x -8}+1+3 x \right )}{2}\) | \(40\) |
default | \(\frac {\left (18 x +6\right ) \sqrt {9 x^{2}+6 x -8}}{36}-\frac {\ln \left (\frac {\left (9 x +3\right ) \sqrt {9}}{9}+\sqrt {9 x^{2}+6 x -8}\right ) \sqrt {9}}{2}\) | \(50\) |
risch | \(\frac {\left (3 x +1\right ) \sqrt {9 x^{2}+6 x -8}}{6}-\frac {\ln \left (\frac {\left (9 x +3\right ) \sqrt {9}}{9}+\sqrt {9 x^{2}+6 x -8}\right ) \sqrt {9}}{2}\) | \(50\) |
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Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int \sqrt {-8+6 x+9 x^2} \, dx=\frac {1}{6} \, \sqrt {9 \, x^{2} + 6 \, x - 8} {\left (3 \, x + 1\right )} + \frac {3}{2} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} + 6 \, x - 8} - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \sqrt {-8+6 x+9 x^2} \, dx=\left (\frac {x}{2} + \frac {1}{6}\right ) \sqrt {9 x^{2} + 6 x - 8} - \frac {3 \log {\left (18 x + 6 \sqrt {9 x^{2} + 6 x - 8} + 6 \right )}}{2} \]
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Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.06 \[ \int \sqrt {-8+6 x+9 x^2} \, dx=\frac {1}{2} \, \sqrt {9 \, x^{2} + 6 \, x - 8} x + \frac {1}{6} \, \sqrt {9 \, x^{2} + 6 \, x - 8} - \frac {3}{2} \, \log \left (18 \, x + 6 \, \sqrt {9 \, x^{2} + 6 \, x - 8} + 6\right ) \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \sqrt {-8+6 x+9 x^2} \, dx=\frac {1}{6} \, \sqrt {9 \, x^{2} + 6 \, x - 8} {\left (3 \, x + 1\right )} + \frac {3}{2} \, \log \left ({\left | -3 \, x + \sqrt {9 \, x^{2} + 6 \, x - 8} - 1 \right |}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int \sqrt {-8+6 x+9 x^2} \, dx=\left (\frac {x}{2}+\frac {1}{6}\right )\,\sqrt {9\,x^2+6\,x-8}-\frac {3\,\ln \left (3\,x+\sqrt {9\,x^2+6\,x-8}+1\right )}{2} \]
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