Preheat the oven to 350˚ F. Butter a 9-by-13-inch baking pan (for a thinner cake, use a 13-by-18-inch sheet pan). Dust the bottom and sides of the pan with flour.
Whisk 3 cups flour, 1 1/2 teaspoons baking powder, 1 teaspoon salt and 1/4 teaspoon baking soda in a bowl.
Beat 1 1/2 sticks softened butter and 1/4 cup vegetable oil in a separate bowl with a mixer on medium-high speed until combined. Add 1 1/2 cups granulated sugar and beat until fluffy, about 4 minutes. Beat in 3 room-temperature eggs one at a time, then 1 tablespoon vanilla. Beat in the flour mixture in three batches on low speed, alternating with 1 1/4 cups buttermilk. Beat on medium-high speed until combined.
Spread the batter in the prepared pan and tap on the counter to release any air bubbles. Bake until a toothpick inserted into the center comes out clean, 35 to 40 minutes (if using a sheet pan, bake only 30 to 35 minutes). Transfer to a rack and let cool completely in the pan before frosting.