Optimal. Leaf size=76 \[ -\frac{\sqrt{x^2+2 x+4} (3-x)}{4 \left (x^2+2 x+3\right )}-\frac{\tan ^{-1}\left (\frac{x+1}{\sqrt{2} \sqrt{x^2+2 x+4}}\right )}{4 \sqrt{2}}+\tanh ^{-1}\left (\sqrt{x^2+2 x+4}\right ) \]
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Rubi [A] time = 0.0677463, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {1016, 1025, 982, 204, 1024, 206} \[ -\frac{\sqrt{x^2+2 x+4} (3-x)}{4 \left (x^2+2 x+3\right )}-\frac{\tan ^{-1}\left (\frac{x+1}{\sqrt{2} \sqrt{x^2+2 x+4}}\right )}{4 \sqrt{2}}+\tanh ^{-1}\left (\sqrt{x^2+2 x+4}\right ) \]
Antiderivative was successfully verified.
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Rule 1016
Rule 1025
Rule 982
Rule 204
Rule 1024
Rule 206
Rubi steps
\begin{align*} \int \frac{3+2 x}{\left (3+2 x+x^2\right )^2 \sqrt{4+2 x+x^2}} \, dx &=-\frac{(3-x) \sqrt{4+2 x+x^2}}{4 \left (3+2 x+x^2\right )}+\frac{1}{8} \int \frac{-10-8 x}{\left (3+2 x+x^2\right ) \sqrt{4+2 x+x^2}} \, dx\\ &=-\frac{(3-x) \sqrt{4+2 x+x^2}}{4 \left (3+2 x+x^2\right )}-\frac{1}{4} \int \frac{1}{\left (3+2 x+x^2\right ) \sqrt{4+2 x+x^2}} \, dx-\frac{1}{2} \int \frac{2+2 x}{\left (3+2 x+x^2\right ) \sqrt{4+2 x+x^2}} \, dx\\ &=-\frac{(3-x) \sqrt{4+2 x+x^2}}{4 \left (3+2 x+x^2\right )}+2 \operatorname{Subst}\left (\int \frac{1}{2-2 x^2} \, dx,x,\sqrt{4+2 x+x^2}\right )+\operatorname{Subst}\left (\int \frac{1}{-16-2 x^2} \, dx,x,\frac{2+2 x}{\sqrt{4+2 x+x^2}}\right )\\ &=-\frac{(3-x) \sqrt{4+2 x+x^2}}{4 \left (3+2 x+x^2\right )}-\frac{\tan ^{-1}\left (\frac{2+2 x}{2 \sqrt{2} \sqrt{4+2 x+x^2}}\right )}{4 \sqrt{2}}+\tanh ^{-1}\left (\sqrt{4+2 x+x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.368974, size = 146, normalized size = 1.92 \[ \frac{1}{32} \left (8 \left (\frac{\sqrt{x^2+2 x+4} (x-3)}{x^2+2 x+3}-2 \log \left (\left (x^2+2 x+3\right )^2\right )+2 \log \left (\left (x^2+2 x+3\right ) \left (x^2+2 \sqrt{x^2+2 x+4}+2 x+5\right )\right )\right )-4 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \left (5 x^2+10 x+4\right )}{4 x^2+\left (11 \sqrt{x^2+2 x+4}+8\right ) x+11 \sqrt{x^2+2 x+4}+12}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 123, normalized size = 1.6 \begin{align*} -{\frac{1}{2} \left ( 1+\sqrt{{x}^{2}+2\,x+4} \right ) ^{-1}}+{\frac{1}{2}\ln \left ( 1+\sqrt{{x}^{2}+2\,x+4} \right ) }-{\frac{1}{2} \left ( -1+\sqrt{{x}^{2}+2\,x+4} \right ) ^{-1}}-{\frac{1}{2}\ln \left ( -1+\sqrt{{x}^{2}+2\,x+4} \right ) }+{\frac{3+3\,x}{4}{\frac{1}{\sqrt{{x}^{2}+2\,x+4}}} \left ({\frac{ \left ( 1+x \right ) ^{2}}{{x}^{2}+2\,x+4}}+2 \right ) ^{-1}}-{\frac{\sqrt{2}}{8}\arctan \left ({\frac{ \left ( 1+x \right ) \sqrt{2}}{2}{\frac{1}{\sqrt{{x}^{2}+2\,x+4}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x + 3}{\sqrt{x^{2} + 2 \, x + 4}{\left (x^{2} + 2 \, x + 3\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88254, size = 504, normalized size = 6.63 \begin{align*} \frac{\sqrt{2}{\left (x^{2} + 2 \, x + 3\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (x + 2\right )} + \frac{1}{2} \, \sqrt{2} \sqrt{x^{2} + 2 \, x + 4}\right ) - \sqrt{2}{\left (x^{2} + 2 \, x + 3\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2} x + \frac{1}{2} \, \sqrt{2} \sqrt{x^{2} + 2 \, x + 4}\right ) + 2 \, x^{2} - 4 \,{\left (x^{2} + 2 \, x + 3\right )} \log \left (x^{2} - \sqrt{x^{2} + 2 \, x + 4}{\left (x + 2\right )} + 3 \, x + 5\right ) + 4 \,{\left (x^{2} + 2 \, x + 3\right )} \log \left (x^{2} - \sqrt{x^{2} + 2 \, x + 4} x + x + 3\right ) + 2 \, \sqrt{x^{2} + 2 \, x + 4}{\left (x - 3\right )} + 4 \, x + 6}{8 \,{\left (x^{2} + 2 \, x + 3\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{2 x + 3}{\left (x^{2} + 2 x + 3\right )^{2} \sqrt{x^{2} + 2 x + 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09552, size = 317, normalized size = 4.17 \begin{align*} \frac{1}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (x - \sqrt{x^{2} + 2 \, x + 4} + 2\right )}\right ) - \frac{1}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (x - \sqrt{x^{2} + 2 \, x + 4}\right )}\right ) + \frac{4 \,{\left (x - \sqrt{x^{2} + 2 \, x + 4}\right )}^{3} + 13 \,{\left (x - \sqrt{x^{2} + 2 \, x + 4}\right )}^{2} + 26 \, x - 26 \, \sqrt{x^{2} + 2 \, x + 4} + 26}{2 \,{\left ({\left (x - \sqrt{x^{2} + 2 \, x + 4}\right )}^{4} + 4 \,{\left (x - \sqrt{x^{2} + 2 \, x + 4}\right )}^{3} + 8 \,{\left (x - \sqrt{x^{2} + 2 \, x + 4}\right )}^{2} + 8 \, x - 8 \, \sqrt{x^{2} + 2 \, x + 4} + 12\right )}} - \frac{1}{2} \, \log \left ({\left (x - \sqrt{x^{2} + 2 \, x + 4}\right )}^{2} + 4 \, x - 4 \, \sqrt{x^{2} + 2 \, x + 4} + 6\right ) + \frac{1}{2} \, \log \left ({\left (x - \sqrt{x^{2} + 2 \, x + 4}\right )}^{2} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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