Optimal. Leaf size=89 \[ -\frac{7 (2-7 x) (2 x+3)^3}{6 \sqrt{3 x^2+2}}-\frac{151}{27} \sqrt{3 x^2+2} (2 x+3)^2-\frac{10}{81} (207 x+185) \sqrt{3 x^2+2}+\frac{880 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0427574, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {819, 833, 780, 215} \[ -\frac{7 (2-7 x) (2 x+3)^3}{6 \sqrt{3 x^2+2}}-\frac{151}{27} \sqrt{3 x^2+2} (2 x+3)^2-\frac{10}{81} (207 x+185) \sqrt{3 x^2+2}+\frac{880 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 819
Rule 833
Rule 780
Rule 215
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx &=-\frac{7 (2-7 x) (3+2 x)^3}{6 \sqrt{2+3 x^2}}+\frac{1}{6} \int \frac{(72-302 x) (3+2 x)^2}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^3}{6 \sqrt{2+3 x^2}}-\frac{151}{27} (3+2 x)^2 \sqrt{2+3 x^2}+\frac{1}{54} \int \frac{(4360-4140 x) (3+2 x)}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^3}{6 \sqrt{2+3 x^2}}-\frac{151}{27} (3+2 x)^2 \sqrt{2+3 x^2}-\frac{10}{81} (185+207 x) \sqrt{2+3 x^2}+\frac{880}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^3}{6 \sqrt{2+3 x^2}}-\frac{151}{27} (3+2 x)^2 \sqrt{2+3 x^2}-\frac{10}{81} (185+207 x) \sqrt{2+3 x^2}+\frac{880 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0550266, size = 58, normalized size = 0.65 \[ -\frac{288 x^4+432 x^3-15024 x^2-15840 \sqrt{9 x^2+6} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+14715 x+33914}{162 \sqrt{3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 79, normalized size = 0.9 \begin{align*} -{\frac{16\,{x}^{4}}{9}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{2504\,{x}^{2}}{27}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{16957}{81}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{8\,{x}^{3}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{545\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{880\,\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 = 1.491, size = 105, normalized size = 1.18 \begin{align*} -\frac{16 \, x^{4}}{9 \, \sqrt{3 \, x^{2} + 2}} - \frac{8 \, x^{3}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{2504 \, x^{2}}{27 \, \sqrt{3 \, x^{2} + 2}} + \frac{880}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{545 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} - \frac{16957}{81 \, \sqrt{3 \, x^{2} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83894, size = 213, normalized size = 2.39 \begin{align*} \frac{7920 \, \sqrt{3}{\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) -{\left (288 \, x^{4} + 432 \, x^{3} - 15024 \, x^{2} + 14715 \, x + 33914\right )} \sqrt{3 \, x^{2} + 2}}{162 \,{\left (3 \, x^{2} + 2\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{999 x}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{864 x^{2}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{264 x^{3}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int \frac{16 x^{4}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int \frac{16 x^{5}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{405}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18165, size = 73, normalized size = 0.82 \begin{align*} -\frac{880}{9} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3 \,{\left (16 \,{\left (3 \,{\left (2 \, x + 3\right )} x - 313\right )} x + 4905\right )} x + 33914}{162 \, \sqrt{3 \, x^{2} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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