Optimal. Leaf size=73 \[ \frac{279 x+398}{162 \left (3 x^2+2\right )^{3/2}}+\frac{32}{27} \sqrt{3 x^2+2}-\frac{465 x+152}{54 \sqrt{3 x^2+2}}+\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0849701, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {1814, 641, 215} \[ \frac{279 x+398}{162 \left (3 x^2+2\right )^{3/2}}+\frac{32}{27} \sqrt{3 x^2+2}-\frac{465 x+152}{54 \sqrt{3 x^2+2}}+\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 641
Rule 215
Rubi steps
\begin{align*} \int \frac{(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx &=\frac{398+279 x}{162 \left (2+3 x^2\right )^{3/2}}-\frac{1}{6} \int \frac{\frac{22}{3}-\frac{280 x}{3}-144 x^2-64 x^3}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{398+279 x}{162 \left (2+3 x^2\right )^{3/2}}-\frac{152+465 x}{54 \sqrt{2+3 x^2}}+\frac{1}{12} \int \frac{96+\frac{128 x}{3}}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{398+279 x}{162 \left (2+3 x^2\right )^{3/2}}-\frac{152+465 x}{54 \sqrt{2+3 x^2}}+\frac{32}{27} \sqrt{2+3 x^2}+8 \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{398+279 x}{162 \left (2+3 x^2\right )^{3/2}}-\frac{152+465 x}{54 \sqrt{2+3 x^2}}+\frac{32}{27} \sqrt{2+3 x^2}+\frac{8 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0738641, size = 63, normalized size = 0.86 \[ \frac{1728 x^4-4185 x^3+936 x^2+432 \sqrt{3} \left (3 x^2+2\right )^{3/2} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-2511 x+254}{162 \left (3 x^2+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 91, normalized size = 1.3 \begin{align*}{\frac{32\,{x}^{4}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{52\,{x}^{2}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{127}{81} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-8\,{\frac{{x}^{3}}{ \left ( 3\,{x}^{2}+2 \right ) ^{3/2}}}-{\frac{107\,x}{18}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{8\,\sqrt{3}}{3}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{65\,x}{18} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50444, size = 142, normalized size = 1.95 \begin{align*} \frac{32 \, x^{4}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{8}{3} \, x{\left (\frac{9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{4}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}\right )} + \frac{8}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{11 \, x}{18 \, \sqrt{3 \, x^{2} + 2}} + \frac{52 \, x^{2}}{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{65 \, x}{18 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{127}{81 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54365, size = 232, normalized size = 3.18 \begin{align*} \frac{216 \, \sqrt{3}{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) +{\left (1728 \, x^{4} - 4185 \, x^{3} + 936 \, x^{2} - 2511 \, x + 254\right )} \sqrt{3 \, x^{2} + 2}}{162 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (2 x + 1\right )^{3} \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} + 2\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21445, size = 72, normalized size = 0.99 \begin{align*} -\frac{8}{3} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{9 \,{\left ({\left (3 \,{\left (64 \, x - 155\right )} x + 104\right )} x - 279\right )} x + 254}{162 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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