Optimal. Leaf size=101 \[ \frac{2}{9} \left (3 x^2-x+2\right )^{3/2}+\frac{1}{72} (30 x+13) \sqrt{3 x^2-x+2}-\frac{1}{8} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-\frac{43 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{144 \sqrt{3}} \]
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Rubi [A] time = 0.116899, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {1653, 814, 843, 619, 215, 724, 206} \[ \frac{2}{9} \left (3 x^2-x+2\right )^{3/2}+\frac{1}{72} (30 x+13) \sqrt{3 x^2-x+2}-\frac{1}{8} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-\frac{43 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{144 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 814
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{2-x+3 x^2} \left (1+3 x+4 x^2\right )}{1+2 x} \, dx &=\frac{2}{9} \left (2-x+3 x^2\right )^{3/2}+\frac{1}{36} \int \frac{(48+60 x) \sqrt{2-x+3 x^2}}{1+2 x} \, dx\\ &=\frac{1}{72} (13+30 x) \sqrt{2-x+3 x^2}+\frac{2}{9} \left (2-x+3 x^2\right )^{3/2}-\frac{\int \frac{-3324-1032 x}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{1728}\\ &=\frac{1}{72} (13+30 x) \sqrt{2-x+3 x^2}+\frac{2}{9} \left (2-x+3 x^2\right )^{3/2}+\frac{43}{144} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx+\frac{13}{8} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=\frac{1}{72} (13+30 x) \sqrt{2-x+3 x^2}+\frac{2}{9} \left (2-x+3 x^2\right )^{3/2}-\frac{13}{4} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )+\frac{43 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{144 \sqrt{69}}\\ &=\frac{1}{72} (13+30 x) \sqrt{2-x+3 x^2}+\frac{2}{9} \left (2-x+3 x^2\right )^{3/2}-\frac{43 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{144 \sqrt{3}}-\frac{1}{8} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.055795, size = 86, normalized size = 0.85 \[ \frac{1}{432} \left (6 \sqrt{3 x^2-x+2} \left (48 x^2+14 x+45\right )-54 \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+43 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 95, normalized size = 0.9 \begin{align*}{\frac{2}{9} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{-5+30\,x}{72}\sqrt{3\,{x}^{2}-x+2}}+{\frac{43\,\sqrt{3}}{432}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{1}{8}\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}-{\frac{\sqrt{13}}{8}{\it Artanh} \left ({\frac{2\,\sqrt{13}}{13} \left ({\frac{9}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52865, size = 130, normalized size = 1.29 \begin{align*} \frac{2}{9} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} + \frac{5}{12} \, \sqrt{3 \, x^{2} - x + 2} x + \frac{43}{432} \, \sqrt{3} \operatorname{arsinh}\left (\frac{6}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{1}{8} \, \sqrt{13} \operatorname{arsinh}\left (\frac{8 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 1 \right |}} - \frac{9 \, \sqrt{23}}{23 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{13}{72} \, \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 = 1.6215, size = 321, normalized size = 3.18 \begin{align*} \frac{1}{72} \,{\left (48 \, x^{2} + 14 \, x + 45\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{43}{864} \, \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 ) + \frac{1}{16} \, \sqrt{13} \log \left (-\frac{4 \, \sqrt{13} \sqrt{3 \, x^{2} - x + 2}{\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\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{\sqrt{3 x^{2} - x + 2} \left (4 x^{2} + 3 x + 1\right )}{2 x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.261, size = 170, normalized size = 1.68 \begin{align*} \frac{1}{72} \,{\left (2 \,{\left (24 \, x + 7\right )} x + 45\right )} \sqrt{3 \, x^{2} - x + 2} - \frac{43}{432} \, \sqrt{3} \log \left (-6 \, \sqrt{3} x + \sqrt{3} + 6 \, \sqrt{3 \, x^{2} - x + 2}\right ) + \frac{1}{8} \, \sqrt{13} \log \left (-\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{13} - 2 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} - x + 2} \right |}}{2 \,{\left (2 \, \sqrt{3} x - \sqrt{13} + \sqrt{3} - 2 \, \sqrt{3 \, x^{2} - x + 2}\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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