Optimal. Leaf size=95 \[ \frac{1}{6} \sqrt{3 x^2-x+2} (2 x+1)^3+\frac{11}{27} \sqrt{3 x^2-x+2} (2 x+1)^2-\frac{143}{324} (3-2 x) \sqrt{3 x^2-x+2}+\frac{4147 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{648 \sqrt{3}} \]
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Rubi [A] time = 0.0989095, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{1}{6} \sqrt{3 x^2-x+2} (2 x+1)^3+\frac{11}{27} \sqrt{3 x^2-x+2} (2 x+1)^2-\frac{143}{324} (3-2 x) \sqrt{3 x^2-x+2}+\frac{4147 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{648 \sqrt{3}} \]
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)^2 \left (1+3 x+4 x^2\right )}{\sqrt{2-x+3 x^2}} \, dx &=\frac{1}{6} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{1}{48} \int \frac{(1+2 x)^2 (-44+176 x)}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{11}{27} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{6} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{1}{432} \int \frac{(1+2 x) (-1716+1144 x)}{\sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{143}{324} (3-2 x) \sqrt{2-x+3 x^2}+\frac{11}{27} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{6} (1+2 x)^3 \sqrt{2-x+3 x^2}-\frac{4147}{648} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{143}{324} (3-2 x) \sqrt{2-x+3 x^2}+\frac{11}{27} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{6} (1+2 x)^3 \sqrt{2-x+3 x^2}-\frac{4147 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{648 \sqrt{69}}\\ &=-\frac{143}{324} (3-2 x) \sqrt{2-x+3 x^2}+\frac{11}{27} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{6} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{4147 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{648 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0327693, size = 55, normalized size = 0.58 \[ \frac{6 \sqrt{3 x^2-x+2} \left (432 x^3+1176 x^2+1138 x-243\right )-4147 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{1944} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 79, normalized size = 0.8 \begin{align*}{\frac{4\,{x}^{3}}{3}\sqrt{3\,{x}^{2}-x+2}}+{\frac{98\,{x}^{2}}{27}\sqrt{3\,{x}^{2}-x+2}}+{\frac{569\,x}{162}\sqrt{3\,{x}^{2}-x+2}}-{\frac{3}{4}\sqrt{3\,{x}^{2}-x+2}}-{\frac{4147\,\sqrt{3}}{1944}{\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 = 1.54187, size = 108, normalized size = 1.14 \begin{align*} \frac{4}{3} \, \sqrt{3 \, x^{2} - x + 2} x^{3} + \frac{98}{27} \, \sqrt{3 \, x^{2} - x + 2} x^{2} + \frac{569}{162} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{4147}{1944} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{3}{4} \, \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 = 1.70268, size = 201, normalized size = 2.12 \begin{align*} \frac{1}{324} \,{\left (432 \, x^{3} + 1176 \, x^{2} + 1138 \, x - 243\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{4147}{3888} \, \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 )^{2} \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 = 1.18767, size = 85, normalized size = 0.89 \begin{align*} \frac{1}{324} \,{\left (2 \,{\left (12 \,{\left (18 \, x + 49\right )} x + 569\right )} x - 243\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{4147}{1944} \, \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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