Optimal. Leaf size=85 \[ -\frac{4 (691-13668 x)}{268203 \sqrt{3 x^2-x+2}}-\frac{2 (101-77 x)}{897 \left (3 x^2-x+2\right )^{3/2}}-\frac{8 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{169 \sqrt{13}} \]
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Rubi [A] time = 0.0944011, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1646, 822, 12, 724, 206} \[ -\frac{4 (691-13668 x)}{268203 \sqrt{3 x^2-x+2}}-\frac{2 (101-77 x)}{897 \left (3 x^2-x+2\right )^{3/2}}-\frac{8 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{169 \sqrt{13}} \]
Antiderivative was successfully verified.
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Rule 1646
Rule 822
Rule 12
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x) \left (2-x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (101-77 x)}{897 \left (2-x+3 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{223}{13}+\frac{308 x}{13}}{(1+2 x) \left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (101-77 x)}{897 \left (2-x+3 x^2\right )^{3/2}}-\frac{4 (691-13668 x)}{268203 \sqrt{2-x+3 x^2}}+\frac{4 \int \frac{3174}{13 (1+2 x) \sqrt{2-x+3 x^2}} \, dx}{20631}\\ &=-\frac{2 (101-77 x)}{897 \left (2-x+3 x^2\right )^{3/2}}-\frac{4 (691-13668 x)}{268203 \sqrt{2-x+3 x^2}}+\frac{8}{169} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{2 (101-77 x)}{897 \left (2-x+3 x^2\right )^{3/2}}-\frac{4 (691-13668 x)}{268203 \sqrt{2-x+3 x^2}}-\frac{16}{169} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )\\ &=-\frac{2 (101-77 x)}{897 \left (2-x+3 x^2\right )^{3/2}}-\frac{4 (691-13668 x)}{268203 \sqrt{2-x+3 x^2}}-\frac{8 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )}{169 \sqrt{13}}\\ \end{align*}
Mathematica [A] time = 0.0482146, size = 72, normalized size = 0.85 \[ \frac{2 \left (82008 x^3-31482 x^2+79077 x-32963\right )}{268203 \left (3 x^2-x+2\right )^{3/2}}-\frac{8 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{169 \sqrt{13}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 158, normalized size = 1.9 \begin{align*} -{\frac{2}{9} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{-5+30\,x}{207} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{-40+240\,x}{1587}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{1}{39} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-4+24\,x}{897} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-784+4704\,x}{89401}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}+{\frac{4}{169}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}-{\frac{8\,\sqrt{13}}{2197}{\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 = 1.47288, size = 126, normalized size = 1.48 \begin{align*} \frac{8}{2197} \, \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{18224 \, x}{89401 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{2764}{268203 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{154 \, x}{897 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{202}{897 \,{\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 = 1.10805, size = 344, normalized size = 4.05 \begin{align*} \frac{2 \,{\left (3174 \, \sqrt{13}{\left (9 \, x^{4} - 6 \, x^{3} + 13 \, 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 (82008 \, x^{3} - 31482 \, x^{2} + 79077 \, x - 32963\right )} \sqrt{3 \, x^{2} - x + 2}\right )}}{3486639 \,{\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{4 x^{2} + 3 x + 1}{\left (2 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 = 1.20905, size = 136, normalized size = 1.6 \begin{align*} \frac{8}{2197} \, \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 ) + \frac{2 \,{\left (3 \,{\left (6 \,{\left (4556 \, x - 1749\right )} x + 26359\right )} x - 32963\right )}}{268203 \,{\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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