Optimal. Leaf size=63 \[ -\frac{2 (367 x+73)}{207 \sqrt{3 x^2-x+2}}+\frac{8}{9} \sqrt{3 x^2-x+2}-\frac{14 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0596095, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1660, 640, 619, 215} \[ -\frac{2 (367 x+73)}{207 \sqrt{3 x^2-x+2}}+\frac{8}{9} \sqrt{3 x^2-x+2}-\frac{14 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 640
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx &=-\frac{2 (73+367 x)}{207 \sqrt{2-x+3 x^2}}+\frac{2}{23} \int \frac{\frac{437}{9}+\frac{92 x}{3}}{\sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{2 (73+367 x)}{207 \sqrt{2-x+3 x^2}}+\frac{8}{9} \sqrt{2-x+3 x^2}+\frac{14}{3} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{2 (73+367 x)}{207 \sqrt{2-x+3 x^2}}+\frac{8}{9} \sqrt{2-x+3 x^2}+\frac{14 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{3 \sqrt{69}}\\ &=-\frac{2 (73+367 x)}{207 \sqrt{2-x+3 x^2}}+\frac{8}{9} \sqrt{2-x+3 x^2}-\frac{14 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.130231, size = 50, normalized size = 0.79 \[ \frac{2 \left (92 x^2-153 x+37\right )}{69 \sqrt{3 x^2-x+2}}+\frac{14 \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 81, normalized size = 1.3 \begin{align*}{\frac{8\,{x}^{2}}{3}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{14\,x}{3}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{10}{9}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{-8+48\,x}{207}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{14\,\sqrt{3}}{9}{\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 = 1.5263, size = 85, normalized size = 1.35 \begin{align*} \frac{8 \, x^{2}}{3 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{14}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{102 \, x}{23 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{74}{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 = 0.940043, size = 224, normalized size = 3.56 \begin{align*} \frac{161 \, \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 ) + 6 \,{\left (92 \, x^{2} - 153 \, x + 37\right )} \sqrt{3 \, x^{2} - x + 2}}{207 \,{\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 ) \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 = 1.19603, size = 77, normalized size = 1.22 \begin{align*} -\frac{14}{9} \, \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 \, x - 153\right )} x + 37\right )}}{69 \, \sqrt{3 \, x^{2} - x + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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