Optimal. Leaf size=47 \[ -\frac{4 (3889-4290 x)}{14283 \sqrt{3 x^2-x+2}}-\frac{2 (367 x+73)}{621 \left (3 x^2-x+2\right )^{3/2}} \]
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Rubi [A] time = 0.0476253, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1660, 636} \[ -\frac{4 (3889-4290 x)}{14283 \sqrt{3 x^2-x+2}}-\frac{2 (367 x+73)}{621 \left (3 x^2-x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 636
Rubi steps
\begin{align*} \int \frac{(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (73+367 x)}{621 \left (2-x+3 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{577}{9}+92 x}{\left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (73+367 x)}{621 \left (2-x+3 x^2\right )^{3/2}}-\frac{4 (3889-4290 x)}{14283 \sqrt{2-x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0602662, size = 33, normalized size = 0.7 \[ \frac{2 \left (2860 x^3-3546 x^2+1833 x-1915\right )}{1587 \left (3 x^2-x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 30, normalized size = 0.6 \begin{align*}{\frac{5720\,{x}^{3}-7092\,{x}^{2}+3666\,x-3830}{1587} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00424, size = 103, normalized size = 2.19 \begin{align*} \frac{5720 \, x}{4761 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{8 \, x^{2}}{3 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{2860}{14283 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{182 \, x}{621 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{1250}{621 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.00179, size = 136, normalized size = 2.89 \begin{align*} \frac{2 \,{\left (2860 \, x^{3} - 3546 \, x^{2} + 1833 \, x - 1915\right )} \sqrt{3 \, x^{2} - x + 2}}{1587 \,{\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\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{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18293, size = 38, normalized size = 0.81 \begin{align*} \frac{2 \,{\left ({\left (2 \,{\left (1430 \, x - 1773\right )} x + 1833\right )} x - 1915\right )}}{1587 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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