Integral of $x e^{cx^3}$

Question

How to evaluate the indefinite integral $\int x e^{cx^3}$. Is there any general form of solution for this integral? some function in terms of hypergeometric function or similar kind of functions? Here c is a positive constant

Answer 1

Start changing variable $$c x^3=-t$$ $$x=-\frac{\sqrt[3]{t}}{\sqrt[3]{c}}$$ $$dx=-\frac{1}{3 \sqrt[3]{c}\, t^{2/3}}\,dt$$ All of that gives $$I=\int x e^{cx^3}\,dx=\frac{1}{3 c^{2/3}}\int t^{-1/3}{e^{-t}}\,dt$$ where you probably recognize the incomplete gamma function.

Answer 2

Hint

We have $$\int xe^{cx^3}dx$$ Let $cx^3=t\implies 3cx^2dx=dt\iff dx=\frac{t^{-2/3}dt}{3\sqrt[3]{c}}$ $$\int \left(\frac{t}{c}\right)^{1/3}e^{t}\frac{t^{-2/3}dt}{3\sqrt[3]{c}}$$ $$=\int \frac{e^{t}t^{-1/3}dt}{3c^{2/3}}$$ $$= \frac{1}{3c^{2/3}}\int t^{-1/3} e^{t}dt$$ I hope you can solve further.