diff --git a/include/boost/math/special_functions/detail/ibeta_inverse.hpp b/include/boost/math/special_functions/detail/ibeta_inverse.hpp
index 4ec944a9c..7de5da5c5 100644
--- a/include/boost/math/special_functions/detail/ibeta_inverse.hpp
+++ b/include/boost/math/special_functions/detail/ibeta_inverse.hpp
@@ -673,7 +673,28 @@ T ibeta_inv_imp(T a, T b, T p, T q, const Policy& pol, T* py)
          invert = !invert;
          xs = 1 - xs;
       }
-      T xg = pow(a * p * boost::math::beta(a, b, pol), 1/a);
+      if (a < tools::min_value<T>())
+         return invert ? 1 : 0; // nothing interesting going on here.
+      //
+      // The call to beta may overflow, plus the alternative using lgamma may do the same
+      // if T is a type where 1/T is infinite for small values (denorms for example).
+      //
+      T bet = 0;
+      T xg;
+      bool overflow = false;
+      try {
+         bet = boost::math::beta(a, b, pol);
+      }
+      catch (const std::runtime_error&)
+      {
+         overflow = true;
+      }
+      if (overflow || !(boost::math::isfinite)(bet))
+      {
+         xg = exp((boost::math::lgamma(a + 1, pol) + boost::math::lgamma(b + 1) - boost::math::lgamma(a + b + 1) + log(p) - log(b) + log(a + b)) / a);
+      }
+      else
+         xg = pow(a * p * bet, 1/a);
       x = xg / (1 + xg);
       y = 1 / (1 + xg);
       //
diff --git a/test/test_students_t.cpp b/test/test_students_t.cpp
index 9a133835e..47e9c4916 100644
--- a/test/test_students_t.cpp
+++ b/test/test_students_t.cpp
@@ -381,6 +381,13 @@ void test_spots(RealType)
                boost::math::quantile(
                   students_t_distribution<RealType>(static_cast<RealType>(0x0fffffff)), static_cast<RealType>(0.25f))), 
             static_cast<RealType>(0.25f), tolerance);
+         //
+         // Bug cases:
+         //
+         if (std::numeric_limits<RealType>::is_specialized && std::numeric_limits<RealType>::has_denorm)
+         {
+            BOOST_CHECK_THROW(boost::math::quantile(students_t_distribution<RealType>((std::numeric_limits<RealType>::min)() / 2), static_cast<RealType>(0.0025f)), std::overflow_error);
+         }
       }
 
   // Student's t pdf tests.