Optimal. Leaf size=82 \[ \frac{1}{6} \sqrt{3 x^2+2} (2 x+1)^3+\frac{5}{18} \sqrt{3 x^2+2} (2 x+1)^2-\frac{1}{27} (3 x+61) \sqrt{3 x^2+2}-\sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
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Rubi [A] time = 0.0884072, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1654, 833, 780, 215} \[ \frac{1}{6} \sqrt{3 x^2+2} (2 x+1)^3+\frac{5}{18} \sqrt{3 x^2+2} (2 x+1)^2-\frac{1}{27} (3 x+61) \sqrt{3 x^2+2}-\sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
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)^2 \left (1+3 x+4 x^2\right )}{\sqrt{2+3 x^2}} \, dx &=\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*}
Mathematica [A] time = 0.0426069, size = 48, normalized size = 0.59 \[ \frac{1}{27} \sqrt{3 x^2+2} \left (36 x^3+84 x^2+54 x-49\right )-\sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 65, normalized size = 0.8 \begin{align*}{\frac{4\,{x}^{3}}{3}\sqrt{3\,{x}^{2}+2}}+2\,x\sqrt{3\,{x}^{2}+2}-{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) \sqrt{3}+{\frac{28\,{x}^{2}}{9}\sqrt{3\,{x}^{2}+2}}-{\frac{49}{27}\sqrt{3\,{x}^{2}+2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49621, size = 86, normalized size = 1.05 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52905, size = 147, normalized size = 1.79 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.15238, size = 75, normalized size = 0.91 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1942, size = 65, normalized size = 0.79 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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