A five-foot-long section of the redesigned shuttle booster rocket cracked after a test firing, but the trial run was still "totally successful," a National Aeronautics and Space Administration official said. The crack was caused by a defective cooling system that is used only in ground tests, said Gerald Smith, manager of the booster program at the Marshall Space Flight Center in Huntsville, Ala. "It occurs after the test," he said. "It has nothing to do with the actual design or the performance of the motor." Discovery of the crack will not alter NASA's plans to launch another shuttle flight in June, 1988, a Marshall center spokesman said. The shuttle booster rocket was test-fired Aug. 30 at Morton Thiokol's test site, 25 miles west of Brigham City, Utah.