Optimal. Leaf size=55 \[ \frac{2-51 x}{18 \sqrt{3 x^2+2}}+\frac{8}{9} \sqrt{3 x^2+2}+\frac{10 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0444929, antiderivative size = 55, 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 = {1814, 641, 215} \[ \frac{2-51 x}{18 \sqrt{3 x^2+2}}+\frac{8}{9} \sqrt{3 x^2+2}+\frac{10 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 641
Rule 215
Rubi steps
\begin{align*} \int \frac{(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx &=\frac{2-51 x}{18 \sqrt{2+3 x^2}}-\frac{1}{2} \int \frac{-\frac{20}{3}-\frac{16 x}{3}}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{2-51 x}{18 \sqrt{2+3 x^2}}+\frac{8}{9} \sqrt{2+3 x^2}+\frac{10}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{2-51 x}{18 \sqrt{2+3 x^2}}+\frac{8}{9} \sqrt{2+3 x^2}+\frac{10 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0304869, size = 48, normalized size = 0.87 \[ \frac{48 x^2+20 \sqrt{9 x^2+6} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-51 x+34}{18 \sqrt{3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 51, normalized size = 0.9 \begin{align*}{\frac{8\,{x}^{2}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{17}{9}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{17\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{10\,\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 = 1.47274, size = 68, normalized size = 1.24 \begin{align*} \frac{8 \, x^{2}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{10}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{17 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} + \frac{17}{9 \, \sqrt{3 \, x^{2} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85049, size = 170, normalized size = 3.09 \begin{align*} \frac{10 \, \sqrt{3}{\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) +{\left (48 \, x^{2} - 51 \, x + 34\right )} \sqrt{3 \, x^{2} + 2}}{18 \,{\left (3 \, x^{2} + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 12.906, size = 114, normalized size = 2.07 \begin{align*} \frac{30 \sqrt{3} x^{2} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27 x^{2} + 18} + \frac{8 x^{2}}{3 \sqrt{3 x^{2} + 2}} - \frac{30 x \sqrt{3 x^{2} + 2}}{27 x^{2} + 18} + \frac{x}{2 \sqrt{3 x^{2} + 2}} + \frac{20 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27 x^{2} + 18} + \frac{17}{9 \sqrt{3 x^{2} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20877, size = 59, normalized size = 1.07 \begin{align*} -\frac{10}{9} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{3 \,{\left (16 \, x - 17\right )} x + 34}{18 \, \sqrt{3 \, x^{2} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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