Optimal. Leaf size=62 \[ -\frac{2 (101-77 x)}{299 \sqrt{3 x^2-x+2}}-\frac{2 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{13 \sqrt{13}} \]
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Rubi [A] time = 0.0736133, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1646, 12, 724, 206} \[ -\frac{2 (101-77 x)}{299 \sqrt{3 x^2-x+2}}-\frac{2 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{13 \sqrt{13}} \]
Antiderivative was successfully verified.
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Rule 1646
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 )^{3/2}} \, dx &=-\frac{2 (101-77 x)}{299 \sqrt{2-x+3 x^2}}+\frac{2}{23} \int \frac{23}{13 (1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{2 (101-77 x)}{299 \sqrt{2-x+3 x^2}}+\frac{2}{13} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{2 (101-77 x)}{299 \sqrt{2-x+3 x^2}}-\frac{4}{13} \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)}{299 \sqrt{2-x+3 x^2}}-\frac{2 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )}{13 \sqrt{13}}\\ \end{align*}
Mathematica [A] time = 0.0218867, size = 73, normalized size = 1.18 \[ -\frac{2 \left (23 \sqrt{13} \sqrt{3 x^2-x+2} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-1001 x+1313\right )}{3887 \sqrt{3 x^2-x+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 102, normalized size = 1.7 \begin{align*} -{\frac{2}{3}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{-5+30\,x}{69}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{1}{13}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}+{\frac{-4+24\,x}{299}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}-{\frac{2\,\sqrt{13}}{169}{\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 = 1.93413, size = 86, normalized size = 1.39 \begin{align*} \frac{2}{169} \, \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{154 \, x}{299 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{202}{299 \, \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 = 1.0099, size = 247, normalized size = 3.98 \begin{align*} \frac{23 \, \sqrt{13}{\left (3 \, x^{2} - x + 2\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 ) + 26 \, \sqrt{3 \, x^{2} - x + 2}{\left (77 \, x - 101\right )}}{3887 \,{\left (3 \, x^{2} - x + 2\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{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21485, size = 123, normalized size = 1.98 \begin{align*} \frac{2}{169} \, \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 (77 \, x - 101\right )}}{299 \, \sqrt{3 \, x^{2} - x + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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