Optimal. Leaf size=56 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{2}{11}} (2 x+1)}{\sqrt{x^2+x+5}}\right )}{\sqrt{22}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+x+5}}{\sqrt{2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0501775, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1025, 982, 204, 1024, 206} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{2}{11}} (2 x+1)}{\sqrt{x^2+x+5}}\right )}{\sqrt{22}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+x+5}}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1025
Rule 982
Rule 204
Rule 1024
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\left (3+x+x^2\right ) \sqrt{5+x+x^2}} \, dx &=-\left (\frac{1}{2} \int \frac{1}{\left (3+x+x^2\right ) \sqrt{5+x+x^2}} \, dx\right )+\frac{1}{2} \int \frac{1+2 x}{\left (3+x+x^2\right ) \sqrt{5+x+x^2}} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{-11-2 x^2} \, dx,x,\frac{1+2 x}{\sqrt{5+x+x^2}}\right )-\operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{5+x+x^2}\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{2}{11}} (1+2 x)}{\sqrt{5+x+x^2}}\right )}{\sqrt{22}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{5+x+x^2}}{\sqrt{2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0631257, size = 114, normalized size = 2.04 \[ \frac{-\left (\sqrt{11}-i\right ) \tanh ^{-1}\left (\frac{-2 i \sqrt{11} x-i \sqrt{11}+19}{4 \sqrt{2} \sqrt{x^2+x+5}}\right )-\left (\sqrt{11}+i\right ) \tanh ^{-1}\left (\frac{2 i \sqrt{11} x+i \sqrt{11}+19}{4 \sqrt{2} \sqrt{x^2+x+5}}\right )}{2 \sqrt{22}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 45, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{{x}^{2}+x+5}} \right ) }-{\frac{\sqrt{22}}{22}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{22}}{11}{\frac{1}{\sqrt{{x}^{2}+x+5}}}} \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{x}{\sqrt{x^{2} + x + 5}{\left (x^{2} + x + 3\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.47429, size = 1018, normalized size = 18.18 \begin{align*} -\frac{1}{33} \, \sqrt{11} \sqrt{6} \sqrt{3} \arctan \left (\frac{2}{33} \, \sqrt{11} \sqrt{3} \sqrt{\sqrt{6} \sqrt{3}{\left (2 \, x + 1\right )} + 6 \, x^{2} - \sqrt{x^{2} + x + 5}{\left (2 \, \sqrt{6} \sqrt{3} + 6 \, x + 3\right )} + 6 \, x + 30} + \frac{1}{33} \, \sqrt{11}{\left (2 \, \sqrt{6} \sqrt{3} + 6 \, x + 3\right )} - \frac{2}{11} \, \sqrt{11} \sqrt{x^{2} + x + 5}\right ) + \frac{1}{33} \, \sqrt{11} \sqrt{6} \sqrt{3} \arctan \left (-\frac{1}{33} \, \sqrt{11}{\left (2 \, \sqrt{6} \sqrt{3} - 6 \, x - 3\right )} + \frac{1}{33} \, \sqrt{11} \sqrt{-12 \, \sqrt{6} \sqrt{3}{\left (2 \, x + 1\right )} + 72 \, x^{2} + 12 \, \sqrt{x^{2} + x + 5}{\left (2 \, \sqrt{6} \sqrt{3} - 6 \, x - 3\right )} + 72 \, x + 360} - \frac{2}{11} \, \sqrt{11} \sqrt{x^{2} + x + 5}\right ) + \frac{1}{12} \, \sqrt{6} \sqrt{3} \log \left (12 \, \sqrt{6} \sqrt{3}{\left (2 \, x + 1\right )} + 72 \, x^{2} - 12 \, \sqrt{x^{2} + x + 5}{\left (2 \, \sqrt{6} \sqrt{3} + 6 \, x + 3\right )} + 72 \, x + 360\right ) - \frac{1}{12} \, \sqrt{6} \sqrt{3} \log \left (-12 \, \sqrt{6} \sqrt{3}{\left (2 \, x + 1\right )} + 72 \, x^{2} + 12 \, \sqrt{x^{2} + x + 5}{\left (2 \, \sqrt{6} \sqrt{3} - 6 \, x - 3\right )} + 72 \, x + 360\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{x}{\left (x^{2} + x + 3\right ) \sqrt{x^{2} + x + 5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.24447, size = 217, normalized size = 3.88 \begin{align*} -\frac{1}{44} \, \sqrt{2}{\left (-i \, \sqrt{11} - 11\right )} \log \left (9 \, \sqrt{2}{\left (i \, \sqrt{22} - 4\right )} - 36 \, x + 36 \, \sqrt{x^{2} + x + 5} - 18\right ) + \frac{1}{44} \, \sqrt{2}{\left (i \, \sqrt{11} - 11\right )} \log \left (-9 \, \sqrt{2}{\left (i \, \sqrt{22} - 4\right )} - 36 \, x + 36 \, \sqrt{x^{2} + x + 5} - 18\right ) - \frac{1}{44} \, \sqrt{2}{\left (i \, \sqrt{11} - 11\right )} \log \left (9 \, \sqrt{2}{\left (-i \, \sqrt{22} - 4\right )} - 36 \, x + 36 \, \sqrt{x^{2} + x + 5} - 18\right ) + \frac{1}{44} \, \sqrt{2}{\left (-i \, \sqrt{11} - 11\right )} \log \left (-9 \, \sqrt{2}{\left (-i \, \sqrt{22} - 4\right )} - 36 \, x + 36 \, \sqrt{x^{2} + x + 5} - 18\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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