Integrand size = 29, antiderivative size = 106 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=-\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}-\frac {1}{810} (3937+2073 x) \sqrt {2+3 x^2}+\frac {5 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
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Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1668, 847, 794, 221} \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {5 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}+\frac {2}{15} \sqrt {3 x^2+2} (2 x+1)^4+\frac {13}{60} \sqrt {3 x^2+2} (2 x+1)^3-\frac {19}{540} \sqrt {3 x^2+2} (2 x+1)^2-\frac {1}{810} (2073 x+3937) \sqrt {3 x^2+2} \]
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Rule 221
Rule 794
Rule 847
Rule 1668
Rubi steps \begin{align*} \text {integral}& = \frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}+\frac {1}{60} \int \frac {(1+2 x)^3 (-68+156 x)}{\sqrt {2+3 x^2}} \, dx \\ & = \frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}+\frac {1}{720} \int \frac {(-2688-228 x) (1+2 x)^2}{\sqrt {2+3 x^2}} \, dx \\ & = -\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}+\frac {\int \frac {(-22368-49752 x) (1+2 x)}{\sqrt {2+3 x^2}} \, dx}{6480} \\ & = -\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}-\frac {1}{810} (3937+2073 x) \sqrt {2+3 x^2}+\frac {5}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx \\ & = -\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}-\frac {1}{810} (3937+2073 x) \sqrt {2+3 x^2}+\frac {5 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{405} \sqrt {2+3 x^2} \left (-1841-135 x+2292 x^2+2430 x^3+864 x^4\right )-\frac {5 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{3 \sqrt {3}} \]
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Time = 0.54 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.42
method | result | size |
risch | \(\frac {\left (864 x^{4}+2430 x^{3}+2292 x^{2}-135 x -1841\right ) \sqrt {3 x^{2}+2}}{405}+\frac {5 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{9}\) | \(45\) |
trager | \(\left (\frac {32}{15} x^{4}+6 x^{3}+\frac {764}{135} x^{2}-\frac {1}{3} x -\frac {1841}{405}\right ) \sqrt {3 x^{2}+2}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{9}\) | \(61\) |
default | \(\frac {5 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{9}-\frac {1841 \sqrt {3 x^{2}+2}}{405}+\frac {32 x^{4} \sqrt {3 x^{2}+2}}{15}+\frac {764 x^{2} \sqrt {3 x^{2}+2}}{135}+6 x^{3} \sqrt {3 x^{2}+2}-\frac {x \sqrt {3 x^{2}+2}}{3}\) | \(79\) |
meijerg | \(\frac {\sqrt {3}\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{3}+\frac {34 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \sqrt {\frac {3 x^{2}}{2}+1}}{2}-\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )\right )}{9 \sqrt {\pi }}+\frac {3 \sqrt {2}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {\frac {3 x^{2}}{2}+1}\right )}{2 \sqrt {\pi }}+\frac {68 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{6}\right )}{9 \sqrt {\pi }}+\frac {16 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \left (-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{40}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{4}\right )}{3 \sqrt {\pi }}+\frac {64 \sqrt {2}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {27}{2} x^{4}-12 x^{2}+16\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}\right )}{27 \sqrt {\pi }}\) | \(217\) |
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.57 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{405} \, {\left (864 \, x^{4} + 2430 \, x^{3} + 2292 \, x^{2} - 135 \, x - 1841\right )} \sqrt {3 \, x^{2} + 2} + \frac {5}{18} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
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Time = 0.34 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.89 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {32 x^{4} \sqrt {3 x^{2} + 2}}{15} + 6 x^{3} \sqrt {3 x^{2} + 2} + \frac {764 x^{2} \sqrt {3 x^{2} + 2}}{135} - \frac {x \sqrt {3 x^{2} + 2}}{3} - \frac {1841 \sqrt {3 x^{2} + 2}}{405} + \frac {5 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{9} \]
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Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {32}{15} \, \sqrt {3 \, x^{2} + 2} x^{4} + 6 \, \sqrt {3 \, x^{2} + 2} x^{3} + \frac {764}{135} \, \sqrt {3 \, x^{2} + 2} x^{2} - \frac {1}{3} \, \sqrt {3 \, x^{2} + 2} x + \frac {5}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {1841}{405} \, \sqrt {3 \, x^{2} + 2} \]
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Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.51 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{405} \, {\left (3 \, {\left (2 \, {\left (9 \, {\left (16 \, x + 45\right )} x + 382\right )} x - 45\right )} x - 1841\right )} \sqrt {3 \, x^{2} + 2} - \frac {5}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.42 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {5\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {32\,x^4}{5}+18\,x^3+\frac {764\,x^2}{45}-x-\frac {1841}{135}\right )}{3} \]
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