Optimal. Leaf size=106 \[ \frac{2}{15} \sqrt{3 x^2+2} (2 x+1)^4+\frac{13}{60} \sqrt{3 x^2+2} (2 x+1)^3-\frac{19}{540} \sqrt{3 x^2+2} (2 x+1)^2-\frac{1}{810} (2073 x+3937) \sqrt{3 x^2+2}+\frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.114004, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1654, 833, 780, 215} \[ \frac{2}{15} \sqrt{3 x^2+2} (2 x+1)^4+\frac{13}{60} \sqrt{3 x^2+2} (2 x+1)^3-\frac{19}{540} \sqrt{3 x^2+2} (2 x+1)^2-\frac{1}{810} (2073 x+3937) \sqrt{3 x^2+2}+\frac{5 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1654
Rule 833
Rule 780
Rule 215
Rubi steps
\begin{align*} \int \frac{(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt{2+3 x^2}} \, dx &=\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*}
Mathematica [A] time = 0.0677779, size = 54, normalized size = 0.51 \[ \frac{1}{405} \left (\sqrt{3 x^2+2} \left (864 x^4+2430 x^3+2292 x^2-135 x-1841\right )+225 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 79, normalized size = 0.8 \begin{align*}{\frac{32\,{x}^{4}}{15}\sqrt{3\,{x}^{2}+2}}+{\frac{764\,{x}^{2}}{135}\sqrt{3\,{x}^{2}+2}}-{\frac{1841}{405}\sqrt{3\,{x}^{2}+2}}+6\,{x}^{3}\sqrt{3\,{x}^{2}+2}-{\frac{x}{3}\sqrt{3\,{x}^{2}+2}}+{\frac{5\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44823, size = 105, normalized size = 0.99 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5656, size = 174, normalized size = 1.64 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.18135, size = 94, normalized size = 0.89 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21552, size = 73, normalized size = 0.69 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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