Optimal. Leaf size=53 \[ -\frac{38-21 x}{66 \sqrt{3 x^2+2}}-\frac{2 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{11 \sqrt{11}} \]
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Rubi [A] time = 0.0599846, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1647, 12, 725, 206} \[ -\frac{38-21 x}{66 \sqrt{3 x^2+2}}-\frac{2 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{11 \sqrt{11}} \]
Antiderivative was successfully verified.
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Rule 1647
Rule 12
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx &=-\frac{38-21 x}{66 \sqrt{2+3 x^2}}-\frac{1}{6} \int -\frac{12}{11 (1+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{38-21 x}{66 \sqrt{2+3 x^2}}+\frac{2}{11} \int \frac{1}{(1+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{38-21 x}{66 \sqrt{2+3 x^2}}-\frac{2}{11} \operatorname{Subst}\left (\int \frac{1}{11-x^2} \, dx,x,\frac{4-3 x}{\sqrt{2+3 x^2}}\right )\\ &=-\frac{38-21 x}{66 \sqrt{2+3 x^2}}-\frac{2 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{2+3 x^2}}\right )}{11 \sqrt{11}}\\ \end{align*}
Mathematica [A] time = 0.0238863, size = 51, normalized size = 0.96 \[ \frac{-12 \sqrt{33 x^2+22} \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{33 x^2+22}}\right )+231 x-418}{726 \sqrt{3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 88, normalized size = 1.7 \begin{align*} -{\frac{2}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{x}{4}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{1}{11}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}+{\frac{3\,x}{44}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}-{\frac{2\,\sqrt{11}}{121}{\it Artanh} \left ({\frac{ \left ( 8-6\,x \right ) \sqrt{11}}{11}{\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-12\,x+5}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50389, size = 78, normalized size = 1.47 \begin{align*} \frac{2}{121} \, \sqrt{11} \operatorname{arsinh}\left (\frac{\sqrt{6} x}{2 \,{\left | 2 \, x + 1 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{7 \, x}{22 \, \sqrt{3 \, x^{2} + 2}} - \frac{19}{33 \, \sqrt{3 \, x^{2} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81235, size = 215, normalized size = 4.06 \begin{align*} \frac{6 \, \sqrt{11}{\left (3 \, x^{2} + 2\right )} \log \left (-\frac{\sqrt{11} \sqrt{3 \, x^{2} + 2}{\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \, \sqrt{3 \, x^{2} + 2}{\left (21 \, x - 38\right )}}{726 \,{\left (3 \, x^{2} + 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} + 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 = 1.20817, size = 111, normalized size = 2.09 \begin{align*} \frac{2}{121} \, \sqrt{11} \log \left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{11} - \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{11} + \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) + \frac{21 \, x - 38}{66 \, \sqrt{3 \, x^{2} + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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