<s>[INST] How can I find the reduction formula for $I_n=\int^1_0{x^6 (1+x^3)^{n}}dx$ I need to find the reduction formula for $I_n=\int^1_0{x^6 (1+x^3)^{n}}dx$ so that I may find the relationship between $I_4$ and $I_3$, I have done the following:
Setting up for Integration by parts
$$u=(1+x^3)^n \implies u^{'}=3nx^2(1+x^3)^{n-1}$$ $$v^{'}=x^6 \implies v=\frac{1}{7}x^7$$
Applying Integration by parts
$$I_n=[\frac{1}{7}x^7(1+x^3)^n]^1_0 - \frac{3n}{7}\int^1_0{x^9(1+x^3)^{n-1}dx}$$
$$=\frac{2^n}{7}-\frac{3n}{7}\int^1_0{x^9(1+x^3)^{n-1}dx}$$

However I have no idea how to bring the integral of $\frac{3n}{7}\int^1_0{x^9(1+x^3)^{n-1}dx}$ into the form of $\int^1_0{x^6(1+x^3)^{n-1}dx}$ so that I may substitute it as $I_{n-1}$.

How can I do it, so that I may find a relationship between $I_4$ and $I_3$ ?
 [/INST] Notice that $$
I_n  - I_{n - 1}  = \int_0^1 {x^9 (1 + x^3 )^{n - 1} dx} .
$$
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