Optimal. Leaf size=75 \[ -\frac{10-97 x}{242 \sqrt{3 x^2+2}}-\frac{4 \sqrt{3 x^2+2}}{121 (2 x+1)}+\frac{4 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{121 \sqrt{11}} \]
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Rubi [A] time = 0.0772505, antiderivative size = 75, 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, 807, 725, 206} \[ -\frac{10-97 x}{242 \sqrt{3 x^2+2}}-\frac{4 \sqrt{3 x^2+2}}{121 (2 x+1)}+\frac{4 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{121 \sqrt{11}} \]
Antiderivative was successfully verified.
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Rule 1647
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx &=-\frac{10-97 x}{242 \sqrt{2+3 x^2}}-\frac{1}{6} \int \frac{-\frac{72}{121}+\frac{120 x}{121}}{(1+2 x)^2 \sqrt{2+3 x^2}} \, dx\\ &=-\frac{10-97 x}{242 \sqrt{2+3 x^2}}-\frac{4 \sqrt{2+3 x^2}}{121 (1+2 x)}-\frac{4}{121} \int \frac{1}{(1+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{10-97 x}{242 \sqrt{2+3 x^2}}-\frac{4 \sqrt{2+3 x^2}}{121 (1+2 x)}+\frac{4}{121} \operatorname{Subst}\left (\int \frac{1}{11-x^2} \, dx,x,\frac{4-3 x}{\sqrt{2+3 x^2}}\right )\\ &=-\frac{10-97 x}{242 \sqrt{2+3 x^2}}-\frac{4 \sqrt{2+3 x^2}}{121 (1+2 x)}+\frac{4 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{2+3 x^2}}\right )}{121 \sqrt{11}}\\ \end{align*}
Mathematica [A] time = 0.0419811, size = 71, normalized size = 0.95 \[ \frac{11 \left (170 x^2+77 x-26\right )+8 (2 x+1) \sqrt{33 x^2+22} \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{33 x^2+22}}\right )}{2662 (2 x+1) \sqrt{3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 98, normalized size = 1.3 \begin{align*}{\frac{x}{2}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{1}{22} \left ( x+{\frac{1}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}-{\frac{2}{121}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}-{\frac{18\,x}{121}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}+{\frac{4\,\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 = 1.48201, size = 113, normalized size = 1.51 \begin{align*} -\frac{4}{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{85 \, x}{242 \, \sqrt{3 \, x^{2} + 2}} - \frac{2}{121 \, \sqrt{3 \, x^{2} + 2}} - \frac{1}{11 \,{\left (2 \, \sqrt{3 \, x^{2} + 2} x + \sqrt{3 \, x^{2} + 2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70027, size = 266, normalized size = 3.55 \begin{align*} \frac{4 \, \sqrt{11}{\left (6 \, x^{3} + 3 \, x^{2} + 4 \, x + 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 \,{\left (170 \, x^{2} + 77 \, x - 26\right )} \sqrt{3 \, x^{2} + 2}}{2662 \,{\left (6 \, x^{3} + 3 \, x^{2} + 4 \, x + 2\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4 \, x^{2} + 3 \, x + 1}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{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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