Function Expression :
\[f(x)=\frac{x+3}{(x^3+27 )(1-3x^2 )} \]Domain
\[\left]-\infty, -3\right[ \cup \left]-3, - \frac{\sqrt{3}}{3}\right[ \cup \left]- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}\right[ \cup \left]\frac{\sqrt{3}}{3}, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = 0 \]\[\lim_{x \overset{<}{\rightarrow-3} }f(x) = - \frac{1}{702} \]
\[\lim_{x \overset{>}{\rightarrow-3} }f(x) = - \frac{1}{702} \]
\[\lim_{x \overset{<}{\rightarrow- \frac{\sqrt{3}}{3}} }f(x) = -\infty \]
\[\lim_{x \overset{>}{\rightarrow- \frac{\sqrt{3}}{3}} }f(x) = +\infty \]
\[\lim_{x \overset{<}{\rightarrow\frac{\sqrt{3}}{3}} }f(x) = +\infty \]
\[\lim_{x \overset{>}{\rightarrow\frac{\sqrt{3}}{3}} }f(x) = -\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = 0 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{\left(x + 3\right) \left(- 3 x^{2} \left(\left(-1\right) 3 x^{2} + 1\right) + 6 x \left(x^{3} + 27\right)\right)}{\left(x^{3} + 27\right)^{2} \left(\left(-1\right) 3 x^{2} + 1\right)^{2}} + \frac{1}{\left(x^{3} + 27\right) \left(\left(-1\right) 3 x^{2} + 1\right)} \]\[f^{\,\prime}(x)=\frac{3 x \left(x + 3\right) \left(2 x^{3} + x \left(3 x^{2} - 1\right) + 54\right) - \left(3 x^{2} - 1\right) \left(x^{3} + 27\right)}{\left(3 x^{2} - 1\right)^{2} \left(x^{3} + 27\right)^{2}} \]
\[ \]
Integral
\[F(x) = \frac{9 \log{\left(x^{2} - 3 x + 9 \right)}}{1514} - \left(\frac{9}{1514} + \frac{14 \sqrt{3}}{757}\right) \log{\left(x - \frac{3988649812604 \sqrt{3}}{271479557247} - \frac{6919485755589}{633452300243} - \frac{1708376544045 \left(\frac{9}{1514} + \frac{14 \sqrt{3}}{757}\right)^{2}}{836792999} + \frac{591822677772666 \left(\frac{9}{1514} + \frac{14 \sqrt{3}}{757}\right)^{3}}{836792999} \right)} - \left(\frac{9}{1514} - \frac{14 \sqrt{3}}{757}\right) \log{\left(x + \frac{591822677772666 \left(\frac{9}{1514} - \frac{14 \sqrt{3}}{757}\right)^{3}}{836792999} - \frac{6919485755589}{633452300243} - \frac{1708376544045 \left(\frac{9}{1514} - \frac{14 \sqrt{3}}{757}\right)^{2}}{836792999} + \frac{3988649812604 \sqrt{3}}{271479557247} \right)} + \frac{29 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right)}}{6813} \]Sign Table
Variation Table
Plot
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