Integrand size = 20, antiderivative size = 54 \[ \int (-2+3 x) \sqrt {8+12 x+9 x^2} \, dx=-\frac {2}{3} (2+3 x) \sqrt {8+12 x+9 x^2}+\frac {1}{9} \left (8+12 x+9 x^2\right )^{3/2}-\frac {8}{3} \text {arcsinh}\left (1+\frac {3 x}{2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 626, 633, 221} \[ \int (-2+3 x) \sqrt {8+12 x+9 x^2} \, dx=-\frac {8}{3} \text {arcsinh}\left (\frac {3 x}{2}+1\right )+\frac {1}{9} \left (9 x^2+12 x+8\right )^{3/2}-\frac {2}{3} (3 x+2) \sqrt {9 x^2+12 x+8} \]
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Rule 221
Rule 626
Rule 633
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \left (8+12 x+9 x^2\right )^{3/2}-4 \int \sqrt {8+12 x+9 x^2} \, dx \\ & = -\frac {2}{3} (2+3 x) \sqrt {8+12 x+9 x^2}+\frac {1}{9} \left (8+12 x+9 x^2\right )^{3/2}-8 \int \frac {1}{\sqrt {8+12 x+9 x^2}} \, dx \\ & = -\frac {2}{3} (2+3 x) \sqrt {8+12 x+9 x^2}+\frac {1}{9} \left (8+12 x+9 x^2\right )^{3/2}-\frac {2}{9} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{144}}} \, dx,x,12+18 x\right ) \\ & = -\frac {2}{3} (2+3 x) \sqrt {8+12 x+9 x^2}+\frac {1}{9} \left (8+12 x+9 x^2\right )^{3/2}-\frac {8}{3} \sinh ^{-1}\left (1+\frac {3 x}{2}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int (-2+3 x) \sqrt {8+12 x+9 x^2} \, dx=\frac {1}{9} \left (-4-6 x+9 x^2\right ) \sqrt {8+12 x+9 x^2}+\frac {8}{3} \log \left (-2-3 x+\sqrt {8+12 x+9 x^2}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {\left (9 x^{2}-6 x -4\right ) \sqrt {9 x^{2}+12 x +8}}{9}-\frac {8 \,\operatorname {arcsinh}\left (1+\frac {3 x}{2}\right )}{3}\) | \(34\) |
default | \(-\frac {\left (18 x +12\right ) \sqrt {9 x^{2}+12 x +8}}{9}-\frac {8 \,\operatorname {arcsinh}\left (1+\frac {3 x}{2}\right )}{3}+\frac {\left (9 x^{2}+12 x +8\right )^{\frac {3}{2}}}{9}\) | \(43\) |
trager | \(\left (x^{2}-\frac {2}{3} x -\frac {4}{9}\right ) \sqrt {9 x^{2}+12 x +8}+\frac {8 \ln \left (\sqrt {9 x^{2}+12 x +8}-2-3 x \right )}{3}\) | \(43\) |
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Time = 0.31 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int (-2+3 x) \sqrt {8+12 x+9 x^2} \, dx=\frac {1}{9} \, \sqrt {9 \, x^{2} + 12 \, x + 8} {\left (9 \, x^{2} - 6 \, x - 4\right )} + \frac {8}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} + 12 \, x + 8} - 2\right ) \]
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Time = 0.45 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.67 \[ \int (-2+3 x) \sqrt {8+12 x+9 x^2} \, dx=\left (x^{2} - \frac {2 x}{3} - \frac {4}{9}\right ) \sqrt {9 x^{2} + 12 x + 8} - \frac {8 \operatorname {asinh}{\left (\frac {3 x}{2} + 1 \right )}}{3} \]
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Time = 0.44 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96 \[ \int (-2+3 x) \sqrt {8+12 x+9 x^2} \, dx=\frac {1}{9} \, {\left (9 \, x^{2} + 12 \, x + 8\right )}^{\frac {3}{2}} - 2 \, \sqrt {9 \, x^{2} + 12 \, x + 8} x - \frac {4}{3} \, \sqrt {9 \, x^{2} + 12 \, x + 8} - \frac {8}{3} \, \operatorname {arsinh}\left (\frac {3}{2} \, x + 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int (-2+3 x) \sqrt {8+12 x+9 x^2} \, dx=\frac {1}{9} \, {\left (3 \, {\left (3 \, x - 2\right )} x - 4\right )} \sqrt {9 \, x^{2} + 12 \, x + 8} + \frac {8}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} + 12 \, x + 8} - 2\right ) \]
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Time = 11.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.56 \[ \int (-2+3 x) \sqrt {8+12 x+9 x^2} \, dx=\frac {\sqrt {9\,x^2+12\,x+8}\,\left (648\,x^2+216\,x+144\right )}{648}-\frac {4\,\ln \left (x+\frac {\sqrt {9\,x^2+12\,x+8}}{3}+\frac {2}{3}\right )}{3}-2\,\left (\frac {x}{2}+\frac {1}{3}\right )\,\sqrt {9\,x^2+12\,x+8}-\frac {4\,\ln \left (3\,x+\sqrt {9\,x^2+12\,x+8}+2\right )}{3} \]
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