Optimal. Leaf size=97 \[ \frac{351 x+358}{2662 \sqrt{3 x^2+2}}+\frac{2 \sqrt{3 x^2+2}}{1331 (2 x+1)}-\frac{2 \sqrt{3 x^2+2}}{121 (2 x+1)^2}-\frac{322 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{1331 \sqrt{11}} \]
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Rubi [A] time = 0.122966, antiderivative size = 97, 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, 1651, 807, 725, 206} \[ \frac{351 x+358}{2662 \sqrt{3 x^2+2}}+\frac{2 \sqrt{3 x^2+2}}{1331 (2 x+1)}-\frac{2 \sqrt{3 x^2+2}}{121 (2 x+1)^2}-\frac{322 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{1331 \sqrt{11}} \]
Antiderivative was successfully verified.
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Rule 1647
Rule 1651
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx &=\frac{358+351 x}{2662 \sqrt{2+3 x^2}}-\frac{1}{6} \int \frac{-\frac{2940}{1331}-\frac{7272 x}{1331}-\frac{8592 x^2}{1331}}{(1+2 x)^3 \sqrt{2+3 x^2}} \, dx\\ &=\frac{358+351 x}{2662 \sqrt{2+3 x^2}}-\frac{2 \sqrt{2+3 x^2}}{121 (1+2 x)^2}+\frac{1}{132} \int \frac{\frac{3768}{121}+\frac{7800 x}{121}}{(1+2 x)^2 \sqrt{2+3 x^2}} \, dx\\ &=\frac{358+351 x}{2662 \sqrt{2+3 x^2}}-\frac{2 \sqrt{2+3 x^2}}{121 (1+2 x)^2}+\frac{2 \sqrt{2+3 x^2}}{1331 (1+2 x)}+\frac{322 \int \frac{1}{(1+2 x) \sqrt{2+3 x^2}} \, dx}{1331}\\ &=\frac{358+351 x}{2662 \sqrt{2+3 x^2}}-\frac{2 \sqrt{2+3 x^2}}{121 (1+2 x)^2}+\frac{2 \sqrt{2+3 x^2}}{1331 (1+2 x)}-\frac{322 \operatorname{Subst}\left (\int \frac{1}{11-x^2} \, dx,x,\frac{4-3 x}{\sqrt{2+3 x^2}}\right )}{1331}\\ &=\frac{358+351 x}{2662 \sqrt{2+3 x^2}}-\frac{2 \sqrt{2+3 x^2}}{121 (1+2 x)^2}+\frac{2 \sqrt{2+3 x^2}}{1331 (1+2 x)}-\frac{322 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{2+3 x^2}}\right )}{1331 \sqrt{11}}\\ \end{align*}
Mathematica [A] time = 0.088082, size = 78, normalized size = 0.8 \[ \frac{11 \left (1428 x^3+2716 x^2+1799 x+278\right )-644 (2 x+1)^2 \sqrt{33 x^2+22} \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{33 x^2+22}}\right )}{29282 (2 x+1)^2 \sqrt{3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 107, normalized size = 1.1 \begin{align*}{\frac{7}{484} \left ( x+{\frac{1}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}+{\frac{161}{1331}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}+{\frac{357\,x}{2662}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}}-{\frac{322\,\sqrt{11}}{14641}{\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} \left ( x+{\frac{1}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53963, size = 167, normalized size = 1.72 \begin{align*} \frac{322}{14641} \, \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{357 \, x}{2662 \, \sqrt{3 \, x^{2} + 2}} + \frac{161}{1331 \, \sqrt{3 \, x^{2} + 2}} - \frac{1}{22 \,{\left (4 \, \sqrt{3 \, x^{2} + 2} x^{2} + 4 \, \sqrt{3 \, x^{2} + 2} x + \sqrt{3 \, x^{2} + 2}\right )}} + \frac{7}{242 \,{\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 = 1.65576, size = 321, normalized size = 3.31 \begin{align*} \frac{322 \, \sqrt{11}{\left (12 \, x^{4} + 12 \, x^{3} + 11 \, x^{2} + 8 \, 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 (1428 \, x^{3} + 2716 \, x^{2} + 1799 \, x + 278\right )} \sqrt{3 \, x^{2} + 2}}{29282 \,{\left (12 \, x^{4} + 12 \, x^{3} + 11 \, x^{2} + 8 \, 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 [B] time = 1.30109, size = 265, normalized size = 2.73 \begin{align*} \frac{322}{14641} \, \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{351 \, x + 358}{2662 \, \sqrt{3 \, x^{2} + 2}} + \frac{36 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 48 \, \sqrt{3} x + 8 \, \sqrt{3} - 48 \, \sqrt{3 \, x^{2} + 2}}{1331 \,{\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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