Find a polynomial with integer coefficients which has $\sqrt2 + \sqrt3$ and $\sqrt2 + \sqrt[3]{3}$ as roots.
1. **Finding a polynomial with $\sqrt{2} + \sqrt{3}$ as a root:**

   Let \( x = \sqrt{2} + \sqrt{3} \). We need to find a polynomial with integer coefficients that has \( x \) as a root.

   \[
   x^2 = (\sqrt{2} + \sqrt{3})^2 = 2 + 2\sqrt{6} + 3 = 5 + 2\sqrt{6}
   \]

   Rearrange to isolate the radical:

   \[
   x^2 - 5 = 2\sqrt{6}
   \]

   Square both sides to eliminate the square root:

   \[
   (x^2 - 5)^2 = (2\sqrt{6})^2
   \]

   \[
   x^4 - 10x^2 + 25 = 24
   \]

   Simplify:

   \[
   x^4 - 10x^2 + 1 = 0
   \]

   Therefore, the polynomial \( x^4 - 10x^2 + 1 \) has \( \sqrt{2} + \sqrt{3} \) as a root.

2. **Finding a polynomial with $\sqrt{2} + \sqrt[3]{3}$ as a root:**

   Let \( y = \sqrt{2} + \sqrt[3]{3} \). We need to find a polynomial with integer coefficients that has \( y \) as a root.

   Consider the expression:

   \[
   (y - \sqrt{2})^3 = y^3 - 3y^2\sqrt{2} + 3y(\sqrt{2})^2 - (\sqrt{2})^3
   \]

   Simplify the terms:

   \[
   (y - \sqrt{2})^3 = y^3 - 3y^2\sqrt{2} + 6y - 2\sqrt{2}
   \]

   Since \( y = \sqrt{2} + \sqrt[3]{3} \), we know:

   \[
   (y - \sqrt{2})^3 = (\sqrt[3]{3})^3 = 3
   \]

   Therefore:

   \[
   y^3 - 3y^2\sqrt{2} + 6y - 2\sqrt{2} = 3
   \]

   Rearrange to isolate the radicals:

   \[
   y^3 + 6y - 3 = 3y^2\sqrt{2} + 2\sqrt{2}
   \]

   Square both sides to eliminate the square roots:

   \[
   (y^3 + 6y - 3)^2 = (3y^2\sqrt{2} + 2\sqrt{2})^2
   \]

   Expand both sides:

   \[
   (y^3 + 6y - 3)^2 = 9y^4 \cdot 2 + 12y^2 \cdot 2 + 4 \cdot 2
   \]

   \[
   (y^3 + 6y - 3)^2 = 18y^4 + 24y^2 + 8
   \]

   Simplify:

   \[
   y^6 + 12y^4 - 6y^3 + 36y^2 - 36y + 9 = 18y^4 + 24y^2 + 8
   \]

   Rearrange to form a polynomial:

   \[
   y^6 - 6y^4 - 6y^3 + 12y^2 - 36y + 1 = 0
   \]

   Therefore, the polynomial \( y^6 - 6y^4 - 6y^3 + 12y^2 - 36y + 1 \) has \( \sqrt{2} + \sqrt[3]{3} \) as a root.

3. **Combining the polynomials:**

   The requested polynomial that has both \( \sqrt{2} + \sqrt{3} \) and \( \sqrt{2} + \sqrt[3]{3} \) as roots is the product of the two polynomials found:

   \[
   (x^4 - 10x^2 + 1)(x^6 - 6x^4 - 6x^3 + 12x^2 - 36x + 1)
   \]

   The polynomial with integer coefficients is:

   \[
   \boxed{(x^4 - 10x^2 + 1)(x^6 - 6x^4 - 6x^3 + 12x^2 - 36x + 1)}
   \]