Optimal. Leaf size=77 \[ \frac{13 \sqrt{3 x^2+2}}{242 (2 x+1)}-\frac{\sqrt{3 x^2+2}}{22 (2 x+1)^2}-\frac{103 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{121 \sqrt{11}} \]
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Rubi [A] time = 0.066702, antiderivative size = 77, 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 = {1651, 807, 725, 206} \[ \frac{13 \sqrt{3 x^2+2}}{242 (2 x+1)}-\frac{\sqrt{3 x^2+2}}{22 (2 x+1)^2}-\frac{103 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{121 \sqrt{11}} \]
Antiderivative was successfully verified.
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Rule 1651
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x)^3 \sqrt{2+3 x^2}} \, dx &=-\frac{\sqrt{2+3 x^2}}{22 (1+2 x)^2}-\frac{1}{22} \int \frac{-14-41 x}{(1+2 x)^2 \sqrt{2+3 x^2}} \, dx\\ &=-\frac{\sqrt{2+3 x^2}}{22 (1+2 x)^2}+\frac{13 \sqrt{2+3 x^2}}{242 (1+2 x)}+\frac{103}{121} \int \frac{1}{(1+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{\sqrt{2+3 x^2}}{22 (1+2 x)^2}+\frac{13 \sqrt{2+3 x^2}}{242 (1+2 x)}-\frac{103}{121} \operatorname{Subst}\left (\int \frac{1}{11-x^2} \, dx,x,\frac{4-3 x}{\sqrt{2+3 x^2}}\right )\\ &=-\frac{\sqrt{2+3 x^2}}{22 (1+2 x)^2}+\frac{13 \sqrt{2+3 x^2}}{242 (1+2 x)}-\frac{103 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{2+3 x^2}}\right )}{121 \sqrt{11}}\\ \end{align*}
Mathematica [A] time = 0.0677386, size = 55, normalized size = 0.71 \[ \frac{\frac{11 (13 x+1) \sqrt{3 x^2+2}}{(2 x+1)^2}-103 \sqrt{11} \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{33 x^2+22}}\right )}{1331} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 74, normalized size = 1. \begin{align*}{\frac{13}{484}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}} \left ( x+{\frac{1}{2}} \right ) ^{-1}}-{\frac{103\,\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 ) }-{\frac{1}{88}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}} \left ( x+{\frac{1}{2}} \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47774, size = 103, normalized size = 1.34 \begin{align*} \frac{103}{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{\sqrt{3 \, x^{2} + 2}}{22 \,{\left (4 \, x^{2} + 4 \, x + 1\right )}} + \frac{13 \, \sqrt{3 \, x^{2} + 2}}{242 \,{\left (2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57685, size = 234, normalized size = 3.04 \begin{align*} \frac{103 \, \sqrt{11}{\left (4 \, x^{2} + 4 \, x + 1\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 ) + 22 \, \sqrt{3 \, x^{2} + 2}{\left (13 \, x + 1\right )}}{2662 \,{\left (4 \, x^{2} + 4 \, x + 1\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 [B] time = 1.26592, size = 243, normalized size = 3.16 \begin{align*} \frac{103}{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{72 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 13 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 168 \, \sqrt{3} x + 104 \, \sqrt{3} + 168 \, \sqrt{3 \, x^{2} + 2}}{484 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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