How to evaluate the integral $\displaystyle\int \frac{x^{15}}{{(1+x^3)^{\frac{2}{5}}}} dx$

Question

We were given this integral in one textbook for our high school calculus class. I tried applying all possible methods, however nothing worked. Can you give me some advice on how to approach this problem. $$\int \frac{x^{15}}{{(1+x^3)^{\frac{2}{5}}}} dx = \int \frac{x^{15}}{\sqrt[5]{(1+x^3)^2}} dx$$

Edit: The task comes from our calculus book, however it asks for solving the definite integral, however we must still solve the indefinite and apply the formula to get the value then.

$$\int_{0}^{\sqrt[4]{2}} \frac{x^{15}}{{(1+x^3)^{\frac{2}{5}}}} dx $$

Answer 1

$$\int\frac{x^{15}}{(1+x^3)^{2/5}}\,\mathrm dx$$

Let $u=1+x^3$, so that $x=(u-1)^{1/3}$ and $\mathrm dx=\frac13(u-1)^{-2/3}\,\mathrm du$. Then the integral becomes

$$\frac13\int\frac{(u-1)^5}{u^{2/5}}(u-1)^{-2/3}\,\mathrm du=\frac13\int(u-1)^{13/3}u^{-2/5}\,\mathrm du$$

which almost resembles the beta function. However, you mention in the comments that the original integral is taken over the set $x\in[0,\sqrt[4]2]$, so the latter integral is taken over $u\in[1,1+\sqrt[4]8]$ and has a value of

$$\frac13\int_1^{1+\sqrt[4]8}(u-1)^{13/3}u^{-2/5}\,\mathrm du=3\,{}_2F_1\left(\frac25,\frac{16}3;\frac{19}3;-\sqrt[4]8\right)\approx2.11381$$

(computation courtesy of Mathematica) where ${}_2F_1$ is the hypergeometric function. I wouldn't expect this to have an elementary form, so perhaps there is an error made by the author or your instructor.

Answer 2

Just for fun, let’s use @user$170231$’s result:

$$3\,{}_2F_1\left(\frac25,\frac{16}3;\frac{19}3;-\sqrt[4]8\right)$$

which is just:

$$3\,{}_2\text F_1\left(\frac25,\frac{16}3;\frac{16}3+1;-\sqrt[4]8\right) $$

and by the use of the following identity:

$$\,_2\text F_1(a,b,b+1,z)=\frac b{z^b}\text B_z(b,1-a)$$

Therefore:

$$\int_{0}^{\sqrt[4]{2}} \frac{x^{15}}{{(1+x^3)^{\frac{2}{5}}}} dx = 3\,_2\text F_1\left(\frac25,\frac{16}3;\frac{16}3+1;-\sqrt[4]8\right) =(-1)^\frac 23 \text B_{-2^\frac34}\left(\frac{16}3,\frac35\right)= 2.11380849407852224338555434770918887014885712827... $$

as shown in this computation. Please correct me and give me feedback!