Optimal. Leaf size=56 \[ \frac{1}{18} (14-3 x) \left (3 x^2+2\right )^{3/2}+\frac{23}{3} x \sqrt{3 x^2+2}+\frac{46 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0141275, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {780, 195, 215} \[ \frac{1}{18} (14-3 x) \left (3 x^2+2\right )^{3/2}+\frac{23}{3} x \sqrt{3 x^2+2}+\frac{46 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 780
Rule 195
Rule 215
Rubi steps
\begin{align*} \int (5-x) (3+2 x) \sqrt{2+3 x^2} \, dx &=\frac{1}{18} (14-3 x) \left (2+3 x^2\right )^{3/2}+\frac{46}{3} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{23}{3} x \sqrt{2+3 x^2}+\frac{1}{18} (14-3 x) \left (2+3 x^2\right )^{3/2}+\frac{46}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{23}{3} x \sqrt{2+3 x^2}+\frac{1}{18} (14-3 x) \left (2+3 x^2\right )^{3/2}+\frac{46 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0398892, size = 50, normalized size = 0.89 \[ \frac{1}{18} \left (92 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (9 x^3-42 x^2-132 x-28\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 49, normalized size = 0.9 \begin{align*} -{\frac{x}{6} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{23\,x}{3}\sqrt{3\,{x}^{2}+2}}+{\frac{46\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{7}{9} \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 = 1.46978, size = 65, normalized size = 1.16 \begin{align*} -\frac{1}{6} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{7}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{23}{3} \, \sqrt{3 \, x^{2} + 2} x + \frac{46}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69668, size = 151, normalized size = 2.7 \begin{align*} -\frac{1}{18} \,{\left (9 \, x^{3} - 42 \, x^{2} - 132 \, x - 28\right )} \sqrt{3 \, x^{2} + 2} + \frac{23}{9} \, \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 = 9.36279, size = 94, normalized size = 1.68 \begin{align*} - \frac{3 x^{5}}{2 \sqrt{3 x^{2} + 2}} - \frac{3 x^{3}}{2 \sqrt{3 x^{2} + 2}} + \frac{15 x \sqrt{3 x^{2} + 2}}{2} - \frac{x}{3 \sqrt{3 x^{2} + 2}} + \frac{7 \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{9} + \frac{46 \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 = 1.20328, size = 65, normalized size = 1.16 \begin{align*} -\frac{1}{18} \,{\left (3 \,{\left ({\left (3 \, x - 14\right )} x - 44\right )} x - 28\right )} \sqrt{3 \, x^{2} + 2} - \frac{46}{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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