Optimal. Leaf size=71 \[ -\frac{\sqrt{3 x^2+2}}{11 (2 x+1)}+\frac{4 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{11 \sqrt{11}}+\frac{\sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.069962, antiderivative size = 71, 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 = {1651, 844, 215, 725, 206} \[ -\frac{\sqrt{3 x^2+2}}{11 (2 x+1)}+\frac{4 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right )}{11 \sqrt{11}}+\frac{\sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1651
Rule 844
Rule 215
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x)^2 \sqrt{2+3 x^2}} \, dx &=-\frac{\sqrt{2+3 x^2}}{11 (1+2 x)}-\frac{1}{11} \int \frac{-7-22 x}{(1+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{\sqrt{2+3 x^2}}{11 (1+2 x)}-\frac{4}{11} \int \frac{1}{(1+2 x) \sqrt{2+3 x^2}} \, dx+\int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{\sqrt{2+3 x^2}}{11 (1+2 x)}+\frac{\sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}}+\frac{4}{11} \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}}{11 (1+2 x)}+\frac{\sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}}+\frac{4 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{11} \sqrt{2+3 x^2}}\right )}{11 \sqrt{11}}\\ \end{align*}
Mathematica [A] time = 0.106519, size = 64, normalized size = 0.9 \[ -\frac{\sqrt{3 x^2+2}}{22 x+11}+\frac{4 \tanh ^{-1}\left (\frac{4-3 x}{\sqrt{33 x^2+22}}\right )}{11 \sqrt{11}}+\frac{\sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 65, normalized size = 0.9 \begin{align*}{\frac{\sqrt{3}}{3}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{1}{22}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-3\,x+{\frac{5}{4}}} \left ( x+{\frac{1}{2}} \right ) ^{-1}}+{\frac{4\,\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 = 1.49284, size = 88, normalized size = 1.24 \begin{align*} \frac{1}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{4}{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{\sqrt{3 \, x^{2} + 2}}{11 \,{\left (2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65382, size = 285, normalized size = 4.01 \begin{align*} \frac{121 \, \sqrt{3}{\left (2 \, x + 1\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 12 \, \sqrt{11}{\left (2 \, 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 ) - 66 \, \sqrt{3 \, x^{2} + 2}}{726 \,{\left (2 \, x + 1\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 )^{2} \sqrt{3 x^{2} + 2}}\, dx \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}{\sqrt{3 \, x^{2} + 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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