Optimal. Leaf size=73 \[ -\frac{38-21 x}{198 \left (3 x^2+2\right )^{3/2}}+\frac{95 x+24}{726 \sqrt{3 x^2+2}}-\frac{8 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{121 \sqrt{11}} \]
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Rubi [A] time = 0.0850621, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1647, 823, 12, 725, 206} \[ -\frac{38-21 x}{198 \left (3 x^2+2\right )^{3/2}}+\frac{95 x+24}{726 \sqrt{3 x^2+2}}-\frac{8 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{121 \sqrt{11}} \]
Antiderivative was successfully verified.
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Rule 1647
Rule 823
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 )^{5/2}} \, dx &=-\frac{38-21 x}{198 \left (2+3 x^2\right )^{3/2}}-\frac{1}{18} \int \frac{-\frac{78}{11}-\frac{84 x}{11}}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac{24+95 x}{726 \sqrt{2+3 x^2}}+\frac{\int \frac{864}{11 (1+2 x) \sqrt{2+3 x^2}} \, dx}{1188}\\ &=-\frac{38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac{24+95 x}{726 \sqrt{2+3 x^2}}+\frac{8}{121} \int \frac{1}{(1+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{38-21 x}{198 \left (2+3 x^2\right )^{3/2}}+\frac{24+95 x}{726 \sqrt{2+3 x^2}}-\frac{8}{121} \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}{198 \left (2+3 x^2\right )^{3/2}}+\frac{24+95 x}{726 \sqrt{2+3 x^2}}-\frac{8 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{2+3 x^2}}\right )}{121 \sqrt{11}}\\ \end{align*}
Mathematica [A] time = 0.0514794, size = 58, normalized size = 0.79 \[ \frac{855 x^3+216 x^2+801 x-274}{2178 \left (3 x^2+2\right )^{3/2}}-\frac{8 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{33 x^2+22}}\right )}{121 \sqrt{11}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 133, normalized size = 1.8 \begin{align*} -{\frac{2}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{x}{12} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{x}{12}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{1}{33} \left ( 3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{x}{44} \left ( 3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{23\,x}{484}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}+{\frac{4}{121}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}-{\frac{8\,\sqrt{11}}{1331}{\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 = 1.49518, size = 109, normalized size = 1.49 \begin{align*} \frac{8}{1331} \, \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{95 \, x}{726 \, \sqrt{3 \, x^{2} + 2}} + \frac{4}{121 \, \sqrt{3 \, x^{2} + 2}} + \frac{7 \, x}{66 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{19}{99 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56274, size = 273, normalized size = 3.74 \begin{align*} \frac{72 \, \sqrt{11}{\left (9 \, x^{4} + 12 \, x^{2} + 4\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 \,{\left (855 \, x^{3} + 216 \, x^{2} + 801 \, x - 274\right )} \sqrt{3 \, x^{2} + 2}}{23958 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27288, size = 123, normalized size = 1.68 \begin{align*} \frac{8}{1331} \, \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{9 \,{\left ({\left (95 \, x + 24\right )} x + 89\right )} x - 274}{2178 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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