diff --git a/src/sage/sets/disjoint_set.pxd b/src/sage/sets/disjoint_set.pxd
index e8795776fc2..de7f9bf5890 100644
--- a/src/sage/sets/disjoint_set.pxd
+++ b/src/sage/sets/disjoint_set.pxd
@@ -22,6 +22,8 @@ cdef class DisjointSet_class(SageObject):
 cdef class DisjointSet_of_integers(DisjointSet_class):
     cpdef int find(self, int i)
     cpdef void union(self, int i, int j)
+    cdef inline int _find(self, int i)
+    cdef inline void _union(self, int i, int j)
     cpdef root_to_elements_dict(self)
     cpdef element_to_root_dict(self)
     cpdef to_digraph(self)
diff --git a/src/sage/sets/disjoint_set.pyx b/src/sage/sets/disjoint_set.pyx
index 9f95afd5eb2..fa1bc17af2c 100644
--- a/src/sage/sets/disjoint_set.pyx
+++ b/src/sage/sets/disjoint_set.pyx
@@ -14,6 +14,7 @@ AUTHORS:
 - Sébastien Labbé (2008) - Initial version.
 - Sébastien Labbé (2009-11-24) - Pickling support
 - Sébastien Labbé (2010-01) - Inclusion into sage (:issue:`6775`).
+- Giorgos Mousa (2024-04-22): Optimize
 
 EXAMPLES:
 
@@ -447,7 +448,7 @@ cdef class DisjointSet_of_integers(DisjointSet_class):
 
         INPUT:
 
-        - ``i`` -- element in ``self`` (no input checking)
+        - ``i`` -- element in ``self``
 
         EXAMPLES::
 
@@ -468,8 +469,31 @@ cdef class DisjointSet_of_integers(DisjointSet_class):
             {{0}, {1, 2, 3, 4}}
             sage: [e.find(i) for i in range(5)]
             [0, 1, 1, 1, 1]
-            sage: e.find(5)  # no input checking
-            0
+            sage: e.find(2**10)
+            ValueError: i(=1024) must be between 0 and 4
+            ...
+        """
+        card = self.cardinality()
+        if i < 0 or i>= card:
+            raise ValueError('i(=%s) must be between 0 and %s' % (i, card - 1))
+        return OP_find(self._nodes, i)
+
+    cdef inline int _find(self, int i):
+        r"""
+        Return the representative of the set that ``i`` currently belongs to.
+
+        INPUT:
+
+        - ``i`` -- element in ``self``
+
+        EXAMPLES::
+
+            sage: e = DisjointSet(5)
+            sage: e._find(5)  # only C-callable
+            Traceback (most recent call last):
+            ...
+            AttributeError: 'sage.sets.disjoint_set.DisjointSet_of_integers'
+             object has no attribute '_find'. Did you mean: 'find'?
         """
         return OP_find(self._nodes, i)
 
@@ -495,8 +519,38 @@ cdef class DisjointSet_of_integers(DisjointSet_class):
             {{0, 1}, {2, 4}, {3}}
             sage: d.union(1, 4); d
             {{0, 1, 2, 4}, {3}}
-            sage: d.union(1, 5); d  # no input checking
-            {{0, 1, 2, 4}, {3}}
+            sage: d.union(1, 5)
+            ValueError: j(=5) must be between 0 and 4
+            ...
+        """
+        cdef int card = self._nodes.degree
+        if i < 0 or i >= card:
+            raise ValueError('i(=%s) must be between 0 and %s' % (i, card - 1))
+        if j < 0 or j >= card:
+            raise ValueError('j(=%s) must be between 0 and %s' % (j, card - 1))
+        OP_join(self._nodes, i, j)
+
+    cdef inline void _union(self, int i, int j):
+        r"""
+        Combine the set of ``i`` and the set of ``j`` into one.
+
+        All elements in those two sets will share the same representative
+        that can be gotten using find.
+
+        INPUT:
+
+        - ``i`` -- element in ``self``
+        - ``j`` -- element in ``self``
+
+        EXAMPLES::
+
+            sage: d = DisjointSet(5); d
+            {{0}, {1}, {2}, {3}, {4}}
+            sage: d._union(0, 1)  # only C-callable
+            Traceback (most recent call last):
+            ...
+            AttributeError: 'sage.sets.disjoint_set.DisjointSet_of_integers'
+             object has no attribute '_union'. Did you mean: 'union'?
         """
         OP_join(self._nodes, i, j)
 
@@ -727,7 +781,7 @@ cdef class DisjointSet_of_hashables(DisjointSet_class):
 
         INPUT:
 
-        - ``e`` -- element in ``self`` (no input checking)
+        - ``e`` -- element in ``self``
 
         EXAMPLES::
 
@@ -754,7 +808,7 @@ cdef class DisjointSet_of_hashables(DisjointSet_class):
             KeyError: 5
         """
         cdef int i = <int> self._el_to_int[e]
-        cdef int r = <int> self._d.find(i)
+        cdef int r = <int> self._d._find(i)
         return self._int_to_el[r]
 
     cpdef void union(self, e, f):
@@ -766,8 +820,8 @@ cdef class DisjointSet_of_hashables(DisjointSet_class):
 
         INPUT:
 
-        - ``e`` -- element in ``self`` (no input checking)
-        - ``f`` -- element in ``self`` (no input checking)
+        - ``e`` -- element in ``self``
+        - ``f`` -- element in ``self``
 
         EXAMPLES::
 
@@ -779,10 +833,13 @@ cdef class DisjointSet_of_hashables(DisjointSet_class):
             {{'a', 'b'}, {'c', 'e'}, {'d'}}
             sage: e.union('b', 'e'); e
             {{'a', 'b', 'c', 'e'}, {'d'}}
+            sage: e.union('a', 2**10)
+            KeyError: 1024
+            ...
         """
         cdef int i = <int> self._el_to_int[e]
         cdef int j = <int> self._el_to_int[f]
-        self._d.union(i, j)
+        self._d._union(i, j)
 
     cpdef root_to_elements_dict(self):
         r"""
@@ -845,7 +902,7 @@ cdef class DisjointSet_of_hashables(DisjointSet_class):
             sage: g.edges(sort=True)                                                    # needs sage.graphs
             [(0, 0, None), (1, 2, None), (2, 2, None), (3, 2, None), (4, 2, None)]
 
-        The result depends on the ordering of the join::
+        The result depends on the ordering of the union::
 
             sage: d = DisjointSet(range(5))
             sage: d.union(1, 2)