Integral of the exponential function

Question

I am searching the indefinite integral of this function: $\dfrac{\exp(x)}{(1+x)^{5/3}}$. Thank you alot.

Answer 1

This integral is not elementary:
First rewrite your integral: $$\int{\frac{e^x}{(x+1)^{5/3}}}{dx} = e^{-1}\int{\frac{e^{x+1}}{(x+1)^{5/3}}}{dx}$$ Make a substitution $u = x+1$ and $du = dx$ $$e^{-1}\int{\frac{e^{x+1}}{(x+1)^{5/3}}}{dx} = e^{-1}\int{\frac{e^{u}}{u^{5/3}}}{du}$$ Make another substitution: $s = u^{1/3}$ and $du = 3s^2ds$ $$e^{-1}\int{\frac{e^{u}}{u^{5/3}}}{du} = 3e^{-1}\int{\frac{e^{s^3}}{s^3}}{ds}$$ The last integral is not elementary (which can be proven by the Risch Algorithm). Thus you can conclude that your initial integral is not an elementary function.

However, your integral has a closed form in terms of special functions (using Mathematica): $$\int_{-\infty}^{\infty}{s(\omega)e^{\tau*I*\omega}}{d\omega} = \frac{\left(\frac{2}{3}\right)^{2/3} \pi \text{c1} |r|^{2/3} \left(-i \text{sgn}(r) \left(3 \left(\sqrt[3]{-\text{c2}}-\sqrt[3]{\text{c2}}\right) \cos \left(\frac{2 |r|}{3 \text{c2}}\right)-\sqrt{3} \left(\sqrt[3]{-\text{c2}}+\sqrt[3]{\text{c2}}\right) \sin \left(\frac{2 |r|}{3 \text{c2}}\right)\right)+3 \left(\sqrt[3]{\text{c2}}-\sqrt[3]{-\text{c2}}\right) \sin \left(\frac{2 |r|}{3 \text{c2}}\right)-\sqrt{3} \left(\sqrt[3]{-\text{c2}}+\sqrt[3]{\text{c2}}\right) \cos \left(\frac{2 |r|}{3 \text{c2}}\right)\right)}{3 \text{c2}^2 \Gamma \left(\frac{2}{3}\right)}$$

with the restrictions: $r \in \mathbb{R} \land c2 \in \mathbb{C}\backslash\mathbb{R} \lor c2 = 0$