Optimal. Leaf size=83 \[ -\frac{\sqrt{3 x^2-x+2}}{13 (2 x+1)}+\frac{9 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{26 \sqrt{13}}-\frac{\sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0950249, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1650, 843, 619, 215, 724, 206} \[ -\frac{\sqrt{3 x^2-x+2}}{13 (2 x+1)}+\frac{9 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{26 \sqrt{13}}-\frac{\sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x)^2 \sqrt{2-x+3 x^2}} \, dx &=-\frac{\sqrt{2-x+3 x^2}}{13 (1+2 x)}-\frac{1}{13} \int \frac{-\frac{17}{2}-26 x}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{\sqrt{2-x+3 x^2}}{13 (1+2 x)}-\frac{9}{26} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx+\int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{\sqrt{2-x+3 x^2}}{13 (1+2 x)}+\frac{9}{13} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{\sqrt{69}}\\ &=-\frac{\sqrt{2-x+3 x^2}}{13 (1+2 x)}-\frac{\sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{\sqrt{3}}+\frac{9 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )}{26 \sqrt{13}}\\ \end{align*}
Mathematica [A] time = 0.0518867, size = 82, normalized size = 0.99 \[ -\frac{\sqrt{3 x^2-x+2}}{13 (2 x+1)}+\frac{9 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{26 \sqrt{13}}+\frac{\sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 67, normalized size = 0.8 \begin{align*}{\frac{\sqrt{3}}{3}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{9\,\sqrt{13}}{338}{\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 ) }-{\frac{1}{26}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}} \left ( x+{\frac{1}{2}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54286, size = 100, normalized size = 1.2 \begin{align*} \frac{1}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{6}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{9}{338} \, \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{\sqrt{3 \, x^{2} - x + 2}}{13 \,{\left (2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70077, size = 336, normalized size = 4.05 \begin{align*} \frac{338 \, \sqrt{3}{\left (2 \, x + 1\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 27 \, \sqrt{13}{\left (2 \, x + 1\right )} \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 ) - 156 \, \sqrt{3 \, x^{2} - x + 2}}{2028 \,{\left (2 \, 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{4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{2} \sqrt{3 x^{2} - x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4 \, x^{2} + 3 \, x + 1}{\sqrt{3 \, x^{2} - x + 2}{\left (2 \, x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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