Optimal. Leaf size=84 \[ -\frac{1}{12} \sqrt{3 x^2+2} (2 x+3)^3+\frac{31}{36} \sqrt{3 x^2+2} (2 x+3)^2+\frac{5}{54} (171 x+809) \sqrt{3 x^2+2}+\frac{275 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0428908, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {833, 780, 215} \[ -\frac{1}{12} \sqrt{3 x^2+2} (2 x+3)^3+\frac{31}{36} \sqrt{3 x^2+2} (2 x+3)^2+\frac{5}{54} (171 x+809) \sqrt{3 x^2+2}+\frac{275 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 833
Rule 780
Rule 215
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)^3}{\sqrt{2+3 x^2}} \, dx &=-\frac{1}{12} (3+2 x)^3 \sqrt{2+3 x^2}+\frac{1}{12} \int \frac{(3+2 x)^2 (192+93 x)}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{31}{36} (3+2 x)^2 \sqrt{2+3 x^2}-\frac{1}{12} (3+2 x)^3 \sqrt{2+3 x^2}+\frac{1}{108} \int \frac{(3+2 x) (4440+5130 x)}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{31}{36} (3+2 x)^2 \sqrt{2+3 x^2}-\frac{1}{12} (3+2 x)^3 \sqrt{2+3 x^2}+\frac{5}{54} (809+171 x) \sqrt{2+3 x^2}+\frac{275}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{31}{36} (3+2 x)^2 \sqrt{2+3 x^2}-\frac{1}{12} (3+2 x)^3 \sqrt{2+3 x^2}+\frac{5}{54} (809+171 x) \sqrt{2+3 x^2}+\frac{275 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0463444, size = 50, normalized size = 0.6 \[ \frac{1}{27} \left (825 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (18 x^3-12 x^2-585 x-2171\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 65, normalized size = 0.8 \begin{align*} -{\frac{2\,{x}^{3}}{3}\sqrt{3\,{x}^{2}+2}}+{\frac{65\,x}{3}\sqrt{3\,{x}^{2}+2}}+{\frac{275\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{4\,{x}^{2}}{9}\sqrt{3\,{x}^{2}+2}}+{\frac{2171}{27}\sqrt{3\,{x}^{2}+2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50345, size = 86, normalized size = 1.02 \begin{align*} -\frac{2}{3} \, \sqrt{3 \, x^{2} + 2} x^{3} + \frac{4}{9} \, \sqrt{3 \, x^{2} + 2} x^{2} + \frac{65}{3} \, \sqrt{3 \, x^{2} + 2} x + \frac{275}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{2171}{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 = 1.7754, size = 158, normalized size = 1.88 \begin{align*} -\frac{1}{27} \,{\left (18 \, x^{3} - 12 \, x^{2} - 585 \, x - 2171\right )} \sqrt{3 \, x^{2} + 2} + \frac{275}{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 = 1.17184, size = 80, normalized size = 0.95 \begin{align*} - \frac{2 x^{3} \sqrt{3 x^{2} + 2}}{3} + \frac{4 x^{2} \sqrt{3 x^{2} + 2}}{9} + \frac{65 x \sqrt{3 x^{2} + 2}}{3} + \frac{2171 \sqrt{3 x^{2} + 2}}{27} + \frac{275 \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 = 1.13243, size = 66, normalized size = 0.79 \begin{align*} -\frac{1}{27} \,{\left (3 \,{\left (2 \,{\left (3 \, x - 2\right )} x - 195\right )} x - 2171\right )} \sqrt{3 \, x^{2} + 2} - \frac{275}{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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