Optimal. Leaf size=60 \[ \frac{70-47 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac{59 x+168}{54 \sqrt{3 x^2+2}}+\frac{16 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]
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Rubi [A] time = 0.0752021, antiderivative size = 60, 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, 12, 215} \[ \frac{70-47 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac{59 x+168}{54 \sqrt{3 x^2+2}}+\frac{16 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 12
Rule 215
Rubi steps
\begin{align*} \int \frac{(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx &=\frac{70-47 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac{1}{6} \int \frac{-\frac{74}{9}-56 x-32 x^2}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{70-47 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac{168+59 x}{54 \sqrt{2+3 x^2}}+\frac{1}{12} \int \frac{64}{3 \sqrt{2+3 x^2}} \, dx\\ &=\frac{70-47 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac{168+59 x}{54 \sqrt{2+3 x^2}}+\frac{16}{9} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{70-47 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac{168+59 x}{54 \sqrt{2+3 x^2}}+\frac{16 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0527414, size = 58, normalized size = 0.97 \[ \frac{-177 x^3-504 x^2+32 \sqrt{3} \left (3 x^2+2\right )^{3/2} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-165 x-266}{54 \left (3 x^2+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 77, normalized size = 1.3 \begin{align*} -{\frac{16\,{x}^{3}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{x}{2}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{16\,\sqrt{3}}{27}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{28\,{x}^{2}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{133}{27} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{37\,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 [B] time = 1.46374, size = 123, normalized size = 2.05 \begin{align*} -\frac{16}{27} \, 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{16}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{37 \, x}{54 \, \sqrt{3 \, x^{2} + 2}} - \frac{28 \, x^{2}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{37 \, x}{18 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{133}{27 \,{\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 = 1.58438, size = 212, normalized size = 3.53 \begin{align*} \frac{16 \, \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 (177 \, x^{3} + 504 \, x^{2} + 165 \, x + 266\right )} \sqrt{3 \, x^{2} + 2}}{54 \,{\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 )^{2} \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 = 1.27473, size = 65, normalized size = 1.08 \begin{align*} -\frac{16}{27} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3 \,{\left ({\left (59 \, x + 168\right )} x + 55\right )} x + 266}{54 \,{\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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