Integrand size = 29, antiderivative size = 82 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {5}{18} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {1}{6} (1+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{27} (61+3 x) \sqrt {2+3 x^2}-\sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1668, 847, 794, 221} \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=-\sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\frac {1}{6} \sqrt {3 x^2+2} (2 x+1)^3+\frac {5}{18} \sqrt {3 x^2+2} (2 x+1)^2-\frac {1}{27} (3 x+61) \sqrt {3 x^2+2} \]
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Rule 221
Rule 794
Rule 847
Rule 1668
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {1}{48} \int \frac {(1+2 x)^2 (-48+120 x)}{\sqrt {2+3 x^2}} \, dx \\ & = \frac {5}{18} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {1}{6} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {1}{432} \int \frac {(-1392-144 x) (1+2 x)}{\sqrt {2+3 x^2}} \, dx \\ & = \frac {5}{18} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {1}{6} (1+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{27} (61+3 x) \sqrt {2+3 x^2}-3 \int \frac {1}{\sqrt {2+3 x^2}} \, dx \\ & = \frac {5}{18} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {1}{6} (1+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{27} (61+3 x) \sqrt {2+3 x^2}-\sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.71 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{27} \sqrt {2+3 x^2} \left (-49+54 x+84 x^2+36 x^3\right )+\sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right ) \]
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Time = 0.50 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.49
method | result | size |
risch | \(\frac {\left (36 x^{3}+84 x^{2}+54 x -49\right ) \sqrt {3 x^{2}+2}}{27}-\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}\) | \(40\) |
trager | \(\left (\frac {4}{3} x^{3}+\frac {28}{9} x^{2}+2 x -\frac {49}{27}\right ) \sqrt {3 x^{2}+2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )\) | \(56\) |
default | \(-\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}-\frac {49 \sqrt {3 x^{2}+2}}{27}+\frac {4 x^{3} \sqrt {3 x^{2}+2}}{3}+2 x \sqrt {3 x^{2}+2}+\frac {28 x^{2} \sqrt {3 x^{2}+2}}{9}\) | \(65\) |
meijerg | \(\frac {\sqrt {3}\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{3}+\frac {20 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \sqrt {\frac {3 x^{2}}{2}+1}}{2}-\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )\right )}{9 \sqrt {\pi }}+\frac {7 \sqrt {2}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {\frac {3 x^{2}}{2}+1}\right )}{6 \sqrt {\pi }}+\frac {28 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{6}\right )}{9 \sqrt {\pi }}+\frac {32 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \left (-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{40}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{4}\right )}{27 \sqrt {\pi }}\) | \(177\) |
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Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.66 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{27} \, {\left (36 \, x^{3} + 84 \, x^{2} + 54 \, x - 49\right )} \sqrt {3 \, x^{2} + 2} + \frac {1}{2} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
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Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {4 x^{3} \sqrt {3 x^{2} + 2}}{3} + \frac {28 x^{2} \sqrt {3 x^{2} + 2}}{9} + 2 x \sqrt {3 x^{2} + 2} - \frac {49 \sqrt {3 x^{2} + 2}}{27} - \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )} \]
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Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.78 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {4}{3} \, \sqrt {3 \, x^{2} + 2} x^{3} + \frac {28}{9} \, \sqrt {3 \, x^{2} + 2} x^{2} + 2 \, \sqrt {3 \, x^{2} + 2} x - \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {49}{27} \, \sqrt {3 \, x^{2} + 2} \]
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.59 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{27} \, {\left (6 \, {\left (2 \, {\left (3 \, x + 7\right )} x + 9\right )} x - 49\right )} \sqrt {3 \, x^{2} + 2} + \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
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Time = 12.85 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.49 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (4\,x^3+\frac {28\,x^2}{3}+6\,x-\frac {49}{9}\right )}{3}-\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right ) \]
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