fix: Pass extra_body to provider in Responses API requests (#16320)

## Problem
The `extra_body` parameter in `litellm.responses()` and `litellm.aresponses()`
was being accepted but never passed to the HTTP request sent to the LLM provider.
This prevented users from sending custom/experimental parameters to provider APIs.

## Changes
- Added `data.update(extra_body)` in `async_response_api_handler` (line 2138)
- Added `data.update(extra_body)` in `response_api_handler` (line 2012)
- Added tests to `test_openai_responses_api.py` for extra_body functionality

## Testing
- Tests verify extra_body params are passed in both sync and async modes
- Existing Responses API tests continue to pass
- Manually verified with OpenAI API that custom params are sent correctly

## Impact
Users can now pass custom/experimental parameters via extra_body:
```python
litellm.aresponses(
    model="gpt-4o",
    input="hello",
    extra_body={"custom_param": "value"}  # Now works!
)
```

This aligns with the OpenAI SDK pattern and matches behavior in other
LiteLLM endpoints (completion, embedding, etc.) that already support extra_body.

litellm/llms/custom_httpx/llm_http_handler.py

@@ -2009,6 +2009,9 @@ class BaseLLMHTTPHandler:
             headers=headers,
         )
 
+        if extra_body:
+            data.update(extra_body)
+
         ## LOGGING
         logging_obj.pre_call(
             input=input,
@@ -2135,6 +2138,9 @@ class BaseLLMHTTPHandler:
             headers=headers,
         )
 
+        if extra_body:
+            data.update(extra_body)
+
         ## LOGGING
         logging_obj.pre_call(
             input=input,

tests/llm_responses_api_testing/test_openai_responses_api.py

@@ -1676,3 +1676,133 @@ async def test_openai_streaming_logging():
 
     await asyncio.sleep(2)
     assert tcl.validate_usage, "Usage should be validated"
+
+
+# Tests for extra_body parameter passing
+class MockResponse:
+    def __init__(self, json_data, status_code):
+        self._json_data = json_data
+        self.status_code = status_code
+        self.text = str(json_data)
+        self.headers = httpx.Headers({})
+
+    def json(self):
+        return self._json_data
+
+
+@pytest.fixture
+def extra_body_mock_response_data():
+    return {
+        "id": "resp_test123",
+        "object": "response",
+        "created_at": 1234567890,
+        "status": "completed",
+        "model": "gpt-4o",
+        "output": [
+            {
+                "type": "message",
+                "id": "msg_123",
+                "status": "completed",
+                "role": "assistant",
+                "content": [
+                    {"type": "output_text", "text": "Hello!", "annotations": []}
+                ],
+            }
+        ],
+        "usage": {"input_tokens": 10, "output_tokens": 5, "total_tokens": 15},
+        "parallel_tool_calls": True,
+        "text": {"format": {"type": "text"}},
+        "error": None,
+        "metadata": {},
+        "temperature": 1.0,
+        "reasoning": {"effort": None, "summary": None},
+    }
+
+
+@pytest.mark.asyncio
+async def test_aresponses_extra_body_params_passed(extra_body_mock_response_data):
+    """Test that extra_body parameters are passed in async mode."""
+    with patch(
+        "litellm.llms.custom_httpx.http_handler.AsyncHTTPHandler.post",
+        new_callable=AsyncMock,
+    ) as mock_post:
+        mock_post.return_value = MockResponse(extra_body_mock_response_data, 200)
+
+        response = await litellm.aresponses(
+            model="gpt-4o",
+            input="Test input",
+            max_output_tokens=20,
+            extra_body={
+                "custom_param_1": "value1",
+                "custom_param_2": {"nested": "value2"},
+                "experimental_feature": True,
+            },
+        )
+
+        assert response is not None
+        assert response.id is not None
+
+        request_body = mock_post.call_args.kwargs["json"]
+
+        assert "custom_param_1" in request_body
+        assert request_body["custom_param_1"] == "value1"
+        assert "custom_param_2" in request_body
+        assert request_body["custom_param_2"]["nested"] == "value2"
+        assert "experimental_feature" in request_body
+        assert request_body["experimental_feature"] is True
+        assert request_body["model"] == "gpt-4o"
+        assert request_body["input"] == "Test input"
+
+
+def test_responses_extra_body_params_passed_sync(extra_body_mock_response_data):
+    """Test that extra_body parameters are passed in sync mode."""
+    with patch(
+        "litellm.llms.custom_httpx.http_handler.HTTPHandler.post",
+        return_value=MockResponse(extra_body_mock_response_data, 200),
+    ) as mock_post:
+        response = litellm.responses(
+            model="gpt-4o",
+            input="Sync test",
+            max_output_tokens=20,
+            extra_body={
+                "sync_custom_param": "sync_value",
+                "another_param": 42,
+            },
+        )
+
+        assert response is not None
+        assert response.id is not None
+
+        request_body = mock_post.call_args.kwargs["json"]
+
+        assert "sync_custom_param" in request_body
+        assert request_body["sync_custom_param"] == "sync_value"
+        assert "another_param" in request_body
+        assert request_body["another_param"] == 42
+        assert request_body["model"] == "gpt-4o"
+
+
+@pytest.mark.asyncio
+async def test_extra_body_merges_with_request_data(extra_body_mock_response_data):
+    """Test that extra_body is merged into the request data."""
+    with patch(
+        "litellm.llms.custom_httpx.http_handler.AsyncHTTPHandler.post",
+        new_callable=AsyncMock,
+    ) as mock_post:
+        mock_post.return_value = MockResponse(extra_body_mock_response_data, 200)
+
+        await litellm.aresponses(
+            model="gpt-4o",
+            input="Test",
+            temperature=0.7,
+            max_output_tokens=20,
+            extra_body={
+                "custom_field": "custom_value",
+            },
+        )
+
+        request_body = mock_post.call_args.kwargs["json"]
+
+        assert "temperature" in request_body
+        assert "custom_field" in request_body
+        assert request_body["custom_field"] == "custom_value"