Optimal. Leaf size=62 \[ \frac{2}{9} \sqrt{3 x^2+2} (2 x+1)^2+\frac{7}{27} (3 x+1) \sqrt{3 x^2+2}-\frac{7 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0516871, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1654, 780, 215} \[ \frac{2}{9} \sqrt{3 x^2+2} (2 x+1)^2+\frac{7}{27} (3 x+1) \sqrt{3 x^2+2}-\frac{7 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1654
Rule 780
Rule 215
Rubi steps
\begin{align*} \int \frac{(1+2 x) \left (1+3 x+4 x^2\right )}{\sqrt{2+3 x^2}} \, dx &=\frac{2}{9} (1+2 x)^2 \sqrt{2+3 x^2}+\frac{1}{36} \int \frac{(1+2 x) (-28+84 x)}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{2}{9} (1+2 x)^2 \sqrt{2+3 x^2}+\frac{7}{27} (1+3 x) \sqrt{2+3 x^2}-\frac{7}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{2}{9} (1+2 x)^2 \sqrt{2+3 x^2}+\frac{7}{27} (1+3 x) \sqrt{2+3 x^2}-\frac{7 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0283797, size = 44, normalized size = 0.71 \[ \frac{1}{27} \left (\sqrt{3 x^2+2} \left (24 x^2+45 x+13\right )-21 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 51, normalized size = 0.8 \begin{align*}{\frac{8\,{x}^{2}}{9}\sqrt{3\,{x}^{2}+2}}+{\frac{13}{27}\sqrt{3\,{x}^{2}+2}}+{\frac{5\,x}{3}\sqrt{3\,{x}^{2}+2}}-{\frac{7\,\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 = 1.46932, size = 68, normalized size = 1.1 \begin{align*} \frac{8}{9} \, \sqrt{3 \, x^{2} + 2} x^{2} + \frac{5}{3} \, \sqrt{3 \, x^{2} + 2} x - \frac{7}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{13}{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 = 1.5361, size = 136, normalized size = 2.19 \begin{align*} \frac{1}{27} \,{\left (24 \, x^{2} + 45 \, x + 13\right )} \sqrt{3 \, x^{2} + 2} + \frac{7}{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 = 0.569814, size = 63, normalized size = 1.02 \begin{align*} \frac{8 x^{2} \sqrt{3 x^{2} + 2}}{9} + \frac{5 x \sqrt{3 x^{2} + 2}}{3} + \frac{13 \sqrt{3 x^{2} + 2}}{27} - \frac{7 \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 = 1.21317, size = 59, normalized size = 0.95 \begin{align*} \frac{1}{27} \,{\left (3 \,{\left (8 \, x + 15\right )} x + 13\right )} \sqrt{3 \, x^{2} + 2} + \frac{7}{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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