# Integral of $\sqrt{x^3 + 8}$?

I have issues solving the following integral:

$$\int\sqrt{x^3+8}~dx$$

I tried substitution and integration by parts, but with no use. I'm guessing I have to use some trigonometric substitution.

Can anybody help solve this integral?

• Circular functions won't cut it. You'll need elliptic integrals. – David H Sep 22 '14 at 20:31
• @DavidH You're probably right. I tried to input that in wolfram alpha. It looks messy, and there is F (elliptical integral of the first kind) – Ant Sep 22 '14 at 20:39
• Gradshteyn & Rhyzhik includes the following related integral (3.139.7): $$\int_u^1\sqrt{1-x^3}\,dx=\frac{1}{5}\left\{\sqrt[4]{27}F(\beta,\sin 75^\circ)-2u\sqrt{1-u^3}\right\}$$ with $F(\psi,k)$ as the incomplete elliptic integral of the first kind. – Semiclassical Sep 23 '14 at 1:52

For any real number $x$:
When $|x|\leq2$ ,
$$\begin{array}\int\sqrt{x^3+8}\,dx &=\int2\sqrt2\sqrt{\dfrac{x^3}{8}+1}\,dx\\ &=\int\sum\limits_{n=0}^\infty\dfrac{2\sqrt2(-1)^n(2n)!x^{3n}}{8^n4^n(n!)^2(1-2n)}\,dx\\ &=\int\sum\limits_{n=0}^\infty\dfrac{2\sqrt2(-1)^n(2n)!x^{3n}}{32^n(n!)^2(1-2n)}\,dx\\ &=\sum\limits_{n=0}^\infty\dfrac{2\sqrt2(-1)^n(2n)!x^{3n+1}}{32^n(n!)^2(1-2n)(3n+1)}+C\end{array}$$
When $|x|\geq2$ ,
$$\begin{array}\int\sqrt{x^3+8}\,dx &=\int x^\frac{3}{2}\sqrt{1+\dfrac{8}{x^3}}\,dx\\ &=\int x^\frac{3}{2}\sum\limits_{n=0}^\infty\dfrac{(-1)^n(2n)!8^n}{4^n(n!)^2(1-2n)x^{3n}}\,dx\\ &=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^n(2n)!2^nx^{\frac{3}{2}-3n}}{(n!)^2(1-2n)}dx\\ &=\sum\limits_{n=0}^\infty\dfrac{(-1)^n(2n)!2^nx^{\frac{5}{2}-3n}}{(n!)^2(1-2n)\left(\dfrac{5}{2}-3n\right)}+C\\ &=\sum\limits_{n=0}^\infty\dfrac{(-1)^n(2n)!2^{n+1}}{(n!)^2(2n-1)(6n-5)x^{3n-\frac{5}{2}}}+C\end{array}$$