Optimal. Leaf size=68 \[ \frac{2 (1249-2273 x)}{1863 \left (3 x^2-x+2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{3 x^2-x+2}}-\frac{16 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{9 \sqrt{3}} \]
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Rubi [A] time = 0.0946948, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1660, 12, 619, 215} \[ \frac{2 (1249-2273 x)}{1863 \left (3 x^2-x+2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{3 x^2-x+2}}-\frac{16 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{9 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
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 )^{5/2}} \, dx &=\frac{2 (1249-2273 x)}{1863 \left (2-x+3 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{1802}{27}+\frac{1150 x}{3}+184 x^2}{\left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (1249-2273 x)}{1863 \left (2-x+3 x^2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{2-x+3 x^2}}+\frac{4 \int \frac{2116}{3 \sqrt{2-x+3 x^2}} \, dx}{1587}\\ &=\frac{2 (1249-2273 x)}{1863 \left (2-x+3 x^2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{2-x+3 x^2}}+\frac{16}{9} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (1249-2273 x)}{1863 \left (2-x+3 x^2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{2-x+3 x^2}}+\frac{16 \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)}{1863 \left (2-x+3 x^2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{2-x+3 x^2}}-\frac{16 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{9 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.148035, size = 66, normalized size = 0.97 \[ \frac{2 \left (5892 x^3-94992 x^2+4232 \sqrt{3} \left (3 x^2-x+2\right )^{3/2} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )+17511 x-52443\right )}{14283 \left (3 x^2-x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 146, normalized size = 2.2 \begin{align*} -{\frac{16\,{x}^{3}}{9} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{92\,{x}^{2}}{9} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{67\,x}{27} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{2653}{486} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{-4585+27510\,x}{11178} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{-18892+113352\,x}{42849}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{16\,x}{9}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{8}{27}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{16\,\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 [B] time = 1.4618, size = 250, normalized size = 3.68 \begin{align*} \frac{16}{14283} \, x{\left (\frac{426 \, x}{\sqrt{3 \, x^{2} - x + 2}} - \frac{4761 \, x^{2}}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{71}{\sqrt{3 \, x^{2} - x + 2}} + \frac{805 \, x}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{2162}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}}\right )} + \frac{16}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{2272}{14283} \, \sqrt{3 \, x^{2} - x + 2} + \frac{28184 \, x}{14283 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{28 \, x^{2}}{3 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{2956}{4761 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{106 \, x}{621 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{3394}{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 [B] time = 1.0629, size = 304, normalized size = 4.47 \begin{align*} \frac{2 \,{\left (2116 \, \sqrt{3}{\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\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 (1964 \, x^{3} - 31664 \, x^{2} + 5837 \, x - 17481\right )} \sqrt{3 \, x^{2} - x + 2}\right )}}{14283 \,{\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 )^{2} \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 = 1.15909, size = 84, normalized size = 1.24 \begin{align*} -\frac{16}{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 (4 \,{\left (491 \, x - 7916\right )} x + 5837\right )} x - 17481\right )}}{4761 \,{\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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