Optimal. Leaf size=120 \[ \frac{2}{15} \sqrt{3 x^2-x+2} (2 x+1)^4+\frac{19}{60} \sqrt{3 x^2-x+2} (2 x+1)^3+\frac{44}{135} \sqrt{3 x^2-x+2} (2 x+1)^2-\frac{(6298 x+24897) \sqrt{3 x^2-x+2}}{3240}+\frac{9211 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{1296 \sqrt{3}} \]
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Rubi [A] time = 0.134877, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1653, 832, 779, 619, 215} \[ \frac{2}{15} \sqrt{3 x^2-x+2} (2 x+1)^4+\frac{19}{60} \sqrt{3 x^2-x+2} (2 x+1)^3+\frac{44}{135} \sqrt{3 x^2-x+2} (2 x+1)^2-\frac{(6298 x+24897) \sqrt{3 x^2-x+2}}{3240}+\frac{9211 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{1296 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 832
Rule 779
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt{2-x+3 x^2}} \, dx &=\frac{2}{15} (1+2 x)^4 \sqrt{2-x+3 x^2}+\frac{1}{60} \int \frac{(1+2 x)^3 (-64+228 x)}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{19}{60} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{2}{15} (1+2 x)^4 \sqrt{2-x+3 x^2}+\frac{1}{720} \int \frac{(1+2 x)^2 (-3390+2112 x)}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{44}{135} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{19}{60} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{2}{15} (1+2 x)^4 \sqrt{2-x+3 x^2}+\frac{\int \frac{(-46350-37788 x) (1+2 x)}{\sqrt{2-x+3 x^2}} \, dx}{6480}\\ &=\frac{44}{135} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{19}{60} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{2}{15} (1+2 x)^4 \sqrt{2-x+3 x^2}-\frac{(24897+6298 x) \sqrt{2-x+3 x^2}}{3240}-\frac{9211 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{1296}\\ &=\frac{44}{135} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{19}{60} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{2}{15} (1+2 x)^4 \sqrt{2-x+3 x^2}-\frac{(24897+6298 x) \sqrt{2-x+3 x^2}}{3240}-\frac{9211 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{1296 \sqrt{69}}\\ &=\frac{44}{135} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{19}{60} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{2}{15} (1+2 x)^4 \sqrt{2-x+3 x^2}-\frac{(24897+6298 x) \sqrt{2-x+3 x^2}}{3240}+\frac{9211 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{1296 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.046816, size = 60, normalized size = 0.5 \[ \frac{6 \sqrt{3 x^2-x+2} \left (6912 x^4+22032 x^3+26904 x^2+7538 x-22383\right )-46055 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{19440} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 96, normalized size = 0.8 \begin{align*}{\frac{32\,{x}^{4}}{15}\sqrt{3\,{x}^{2}-x+2}}+{\frac{34\,{x}^{3}}{5}\sqrt{3\,{x}^{2}-x+2}}+{\frac{1121\,{x}^{2}}{135}\sqrt{3\,{x}^{2}-x+2}}+{\frac{3769\,x}{1620}\sqrt{3\,{x}^{2}-x+2}}-{\frac{829}{120}\sqrt{3\,{x}^{2}-x+2}}-{\frac{9211\,\sqrt{3}}{3888}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51555, size = 131, normalized size = 1.09 \begin{align*} \frac{32}{15} \, \sqrt{3 \, x^{2} - x + 2} x^{4} + \frac{34}{5} \, \sqrt{3 \, x^{2} - x + 2} x^{3} + \frac{1121}{135} \, \sqrt{3 \, x^{2} - x + 2} x^{2} + \frac{3769}{1620} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{9211}{3888} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{829}{120} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62303, size = 224, normalized size = 1.87 \begin{align*} \frac{1}{3240} \,{\left (6912 \, x^{4} + 22032 \, x^{3} + 26904 \, x^{2} + 7538 \, x - 22383\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{9211}{7776} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\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 )}{\sqrt{3 x^{2} - x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18289, size = 92, normalized size = 0.77 \begin{align*} \frac{1}{3240} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (16 \, x + 51\right )} x + 1121\right )} x + 3769\right )} x - 22383\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{9211}{3888} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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