Optimal. Leaf size=82 \[ \frac{2 (1249-2273 x)}{621 \sqrt{3 x^2-x+2}}+\frac{8}{9} x \sqrt{3 x^2-x+2}+\frac{112}{27} \sqrt{3 x^2-x+2}-\frac{64 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{9 \sqrt{3}} \]
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Rubi [A] time = 0.108118, antiderivative size = 82, 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 = {1660, 1661, 640, 619, 215} \[ \frac{2 (1249-2273 x)}{621 \sqrt{3 x^2-x+2}}+\frac{8}{9} x \sqrt{3 x^2-x+2}+\frac{112}{27} \sqrt{3 x^2-x+2}-\frac{64 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{9 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 1661
Rule 640
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx &=\frac{2 (1249-2273 x)}{621 \sqrt{2-x+3 x^2}}+\frac{2}{23} \int \frac{\frac{2116}{27}+\frac{1150 x}{9}+\frac{184 x^2}{3}}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (1249-2273 x)}{621 \sqrt{2-x+3 x^2}}+\frac{8}{9} x \sqrt{2-x+3 x^2}+\frac{1}{69} \int \frac{\frac{3128}{9}+\frac{2576 x}{3}}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (1249-2273 x)}{621 \sqrt{2-x+3 x^2}}+\frac{112}{27} \sqrt{2-x+3 x^2}+\frac{8}{9} x \sqrt{2-x+3 x^2}+\frac{64}{9} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (1249-2273 x)}{621 \sqrt{2-x+3 x^2}}+\frac{112}{27} \sqrt{2-x+3 x^2}+\frac{8}{9} x \sqrt{2-x+3 x^2}+\frac{64 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{9 \sqrt{69}}\\ &=\frac{2 (1249-2273 x)}{621 \sqrt{2-x+3 x^2}}+\frac{112}{27} \sqrt{2-x+3 x^2}+\frac{8}{9} x \sqrt{2-x+3 x^2}-\frac{64 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{9 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0318988, size = 61, normalized size = 0.74 \[ \frac{2 \left (828 x^3+3588 x^2+736 \sqrt{9 x^2-3 x+6} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )-3009 x+3825\right )}{621 \sqrt{3 x^2-x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 98, normalized size = 1.2 \begin{align*}{\frac{8\,{x}^{3}}{3}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{104\,{x}^{2}}{9}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{64\,x}{9}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{107}{9}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{-89+534\,x}{207}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{64\,\sqrt{3}}{27}{\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 = 1.55439, size = 108, normalized size = 1.32 \begin{align*} \frac{8 \, x^{3}}{3 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{104 \, x^{2}}{9 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{64}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{2006 \, x}{207 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{850}{69 \, \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 = 1.12729, size = 244, normalized size = 2.98 \begin{align*} \frac{2 \,{\left (368 \, \sqrt{3}{\left (3 \, x^{2} - x + 2\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 3 \,{\left (276 \, x^{3} + 1196 \, x^{2} - 1003 \, x + 1275\right )} \sqrt{3 \, x^{2} - x + 2}\right )}}{621 \,{\left (3 \, x^{2} - x + 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{\left (2 x + 1\right )^{2} \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} - x + 2\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15107, size = 84, normalized size = 1.02 \begin{align*} -\frac{64}{27} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) + \frac{2 \,{\left ({\left (92 \,{\left (3 \, x + 13\right )} x - 1003\right )} x + 1275\right )}}{207 \, \sqrt{3 \, x^{2} - x + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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