Optimal. Leaf size=110 \[ -\frac{24 (841-6633 x)}{1162213 \sqrt{3 x^2-x+2}}-\frac{16 \sqrt{3 x^2-x+2}}{2197 (2 x+1)}-\frac{2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}-\frac{56 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{2197 \sqrt{13}} \]
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Rubi [A] time = 0.150562, antiderivative size = 110, 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 = {1646, 806, 724, 206} \[ -\frac{24 (841-6633 x)}{1162213 \sqrt{3 x^2-x+2}}-\frac{16 \sqrt{3 x^2-x+2}}{2197 (2 x+1)}-\frac{2 (197-837 x)}{11661 \left (3 x^2-x+2\right )^{3/2}}-\frac{56 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{2197 \sqrt{13}} \]
Antiderivative was successfully verified.
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Rule 1646
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{2226}{169}+\frac{462 x}{13}+\frac{6696 x^2}{169}}{(1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}-\frac{24 (841-6633 x)}{1162213 \sqrt{2-x+3 x^2}}+\frac{4 \int \frac{\frac{50784}{2197}+\frac{19044 x}{2197}}{(1+2 x)^2 \sqrt{2-x+3 x^2}} \, dx}{1587}\\ &=-\frac{2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}-\frac{24 (841-6633 x)}{1162213 \sqrt{2-x+3 x^2}}-\frac{16 \sqrt{2-x+3 x^2}}{2197 (1+2 x)}+\frac{56 \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{2197}\\ &=-\frac{2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}-\frac{24 (841-6633 x)}{1162213 \sqrt{2-x+3 x^2}}-\frac{16 \sqrt{2-x+3 x^2}}{2197 (1+2 x)}-\frac{112 \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )}{2197}\\ &=-\frac{2 (197-837 x)}{11661 \left (2-x+3 x^2\right )^{3/2}}-\frac{24 (841-6633 x)}{1162213 \sqrt{2-x+3 x^2}}-\frac{16 \sqrt{2-x+3 x^2}}{2197 (1+2 x)}-\frac{56 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )}{2197 \sqrt{13}}\\ \end{align*}
Mathematica [A] time = 0.0616853, size = 111, normalized size = 1.01 \[ \frac{26 \left (1318464 x^4+133308 x^3+1021566 x^2+569989 x-170239\right )-88872 \sqrt{13} \sqrt{3 x^2-x+2} \left (6 x^3+x^2+3 x+2\right ) \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{45326307 (2 x+1) \left (3 x^2-x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 165, normalized size = 1.5 \begin{align*}{\frac{-2+12\,x}{69} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{-16+96\,x}{529}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{7}{507} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{-128+768\,x}{11661} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{-10736+64416\,x}{1162213}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}+{\frac{28}{2197}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}-{\frac{56\,\sqrt{13}}{28561}{\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} \left ( x+{\frac{1}{2}} \right ) ^{-1} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48197, size = 169, normalized size = 1.54 \begin{align*} \frac{56}{28561} \, \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{146496 \, x}{1162213 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{9604}{1162213 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{420 \, x}{3887 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{1}{13 \,{\left (2 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x +{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{49}{11661 \,{\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 = 1.08474, size = 397, normalized size = 3.61 \begin{align*} \frac{2 \,{\left (22218 \, \sqrt{13}{\left (18 \, x^{5} - 3 \, x^{4} + 20 \, x^{3} + 5 \, x^{2} + 4 \, x + 4\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 ) + 13 \,{\left (1318464 \, x^{4} + 133308 \, x^{3} + 1021566 \, x^{2} + 569989 \, x - 170239\right )} \sqrt{3 \, x^{2} - x + 2}\right )}}{45326307 \,{\left (18 \, x^{5} - 3 \, x^{4} + 20 \, x^{3} + 5 \, 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{4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{2} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4 \, x^{2} + 3 \, x + 1}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{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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