@@ -2128,6 +2128,24 @@ here that writes one does: left inside, a rule below differentiated it, and
 | `"sqrt(a^2+2*a*b*x+b^2*x^2)*sqrt(c+pe*x+d*x^2)".Integrate("x")` | left unevaluated | the antiderivative |
 | `"(a^2+2*a*b*x+b^2*x^2)^(5/2)".Integrate("x")` | the antiderivative, through the substitution search | `sgn(a + b x) (a + b x)^6 |b|^5/(6 b)` up to the form |
 
+### The root of a quadratic that is a square is the modulus of a linear, and was read as any quadratic
+
+**Wrong answers, silent.** The table's entry for `k/sqrt(Q)` and `sqrt(Q)` reads a quadratic by the
+sign of its leading coefficient: an arcsine over the root of the discriminant, or the logarithm of
+`2 a x + b + 2 sqrt(a) sqrt(Q)`. Where `Q` is a square the discriminant is zero, and that logarithm
+is of zero beyond the root. So `1/sqrt(x^2 + 2x + 1)` had no value for any `x < -1`, where it is
+`1/|x + 1|`, and with a negative leading coefficient there was no value anywhere. The root of a
+square is now the modulus of its linear, `sqrt(a (x + h)^2) = sqrt(a) |x + h|`, for a leading
+coefficient of either sign ([#1670](https://github.com/asc-community/AngouriMath/issues/1670)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/sqrt(x^2+2*x+1)".Integrate("x")` | `ln(2 x + 2 + 2 sqrt(x^2 + 2 x + 1))`, `ln(0)` for every `x < -1` | `sgn(x + 1) ln(x + 1)` |
+| `"sqrt(x^2+2*x+1)".Integrate("x")` | `(2 x + 2) sqrt(x^2 + 2 x + 1)/4 provided not 2 x + 2 + 2 sqrt(x^2 + 2 x + 1) = 0`, which says nothing for `x <= -1` | `sgn(x + 1) (x^2/2 + x)` |
+| `"1/sqrt(-4-4*x-x^2)".Integrate("x")`, and `sqrt(-4 - 4 x - x^2)` | `NaN` | `sgn(x + 2) ln(i x + 2 i)/i`, and `i sgn(x + 2) (x^2/2 + 2 x)` |
+| `"1/sqrt(-a^2-2*a*b*x-b^2*x^2)".Integrate("x")` | a piecewise whose arcsine divides by `sqrt(0)`, `NaN` for every real `b` but zero | a logarithm over `sqrt(-b^2)`, `provided b^2 > 0` |
+| `"1/(x*sqrt(a^2+2*a*b*x+b^2*x^2))".Integrate("x")`, the same with `-a^2 - 2 a b x - b^2 x^2`, or with `d + e x` for `x` | left unevaluated | the antiderivative, `provided b^2 > 0` |
+
 ### A power of the variable times a sine or cosine of a logarithm, and a power of a monomial
 
 `x^2 sin(a + b ln(c x^n))` and `(c x^n)^b` were both left as written. Two rules, each exact:
 
@@ -13697,14 +13697,11 @@ private static bool IsAWholeOrHalfOddMultiple(Entity timesN)
         /// Symbolically as well as numerically: for <c>a^2 + 2 a b x + b^2 x^2</c> it is
         /// <c>4 a^2 b^2 - 4 a^2 b^2</c>, which <see cref="Entity.InnerSimplified"/> does not
         /// collect, so a guard that asks only what it evaluates to reads the square as an
-        /// ordinary quadratic.
+        /// ordinary quadratic. Asked of every quadratic under a root the table meets, so the
+        /// simplification is only for a discriminant that is zero with its symbols pinned.
         /// </summary>
-        private static bool IsAPerfectSquareDiscriminant(Entity a, Entity b, Entity c)
-        {
-            var discriminant = (b * b - Number.Integer.Create(4) * a * c).InnerSimplified;
-            return discriminant.Evaled is Number.Complex { IsZero: true }
-                || discriminant.Vars.Any() && Functions.PartialFractions.Bare(discriminant.Simplify()).Evaled is Number.Complex { IsZero: true };
-        }
+        internal static bool IsAPerfectSquareDiscriminant(Entity a, Entity b, Entity c)
+            => VanishesIdentically(b * b - Number.Integer.Create(4) * a * c);
 
         /// <summary>
         /// A square root of a perfect square in <paramref name="x"/> is the modulus:
@@ -13798,15 +13795,17 @@ bool MayBeARootOfASquare(Entity node, Entity.Variable x)
                 var b = monomials.TryGetValue(half, out var b1) ? b1 : Number.Integer.Zero;
                 var c = monomials.TryGetValue(EInteger.Zero, out var c0) ? c0 : Number.Integer.Zero;
                 Entity w = half.Equals(EInteger.One) ? x : MathS.Pow(x, Number.Integer.Create(half));
-                // A positive leading coefficient, since `sqrt(a (x + h)^2)` is `sqrt(a) |x + h|`:
-                // a positive number, or one positive for a real parameter -- `b^2` is, and
-                // `a^2 + 2 a b x + b^2 x^2` is the square Rubi writes -- whose condition travels
+                // A leading coefficient of known sign, since `sqrt(a (x + h)^2)` is
+                // `sqrt(a) |x + h|` for either: a number, or one positive or negative for a real
+                // parameter -- `b^2` is, and `a^2 + 2 a b x + b^2 x^2` is the square Rubi writes,
+                // and `-b^2` is, whose square has an imaginary root -- whose condition travels
                 // with the answer. The other two coefficients are real or symbolic, not a
-                // number off the line, for the same reason.
+                // number off the line, so that `(x + h)^2` is not negative.
                 // A whole power needs no sign of anything: (a (w + h)^2)^n is a^n (w + h)^(2n).
-                if (!whole && a.Evaled is not Number.Real { IsPositive: true })
+                if (!whole && a.Evaled is not Number.Real { IsZero: false })
                 {
-                    if (!IsPositiveForARealParameter(a, x, out var leadingAssumed, assumed))
+                    if (!IsPositiveForARealParameter(a, x, out var leadingAssumed, assumed)
+                        && !IsPositiveForARealParameter((-a).InnerSimplified, x, out leadingAssumed, assumed))
                         return node;
                     assumed = leadingAssumed;
                 }
 
@@ -480,10 +480,11 @@ _ when GaussianMoment(expr, x) is { } moment => moment,
                         / (rate * rate + frequency * frequency),
 
             // ∫ sqrt(ax^2 + bx + c) dx, which is one integration by parts away from the
-            // reciprocal form below and is written in terms of it.
+            // reciprocal form below and is written in terms of it. Not of a square, as below.
             Entity.Powf(var radicand, Entity.Number.Rational(Entity.Number.Integer(1), Entity.Number.Integer(2))) when
                 TreeAnalyzer.TryGetPolyQuadratic(radicand, x, out var qa, out var qb, out var qc)
                 && qa.Evaled is Entity.Number.Complex { IsZero: false }
+                && !IndefiniteIntegralSolver.IsAPerfectSquareDiscriminant(qa, qb, qc)
                     => IntegrateRootOfQuadratic(qa, qb, qc, radicand, x),
 
             // ∫ k / (x^2 * sqrt(ax^2 + c)) dx, the shape a trigonometric substitution is
@@ -496,10 +497,17 @@ _ when TryReadOverSquareTimesRoot(expr, x, out var overFactor, out var overRadic
 
             // ∫ k / sqrt(ax^2 + bx + c) dx -- the arcsine and logarithm forms. Without
             // these, 1/sqrt(1 - x^2) had no antiderivative at all.
+            // Not of a square: its root is the modulus of a linear, `sqrt(a (x + h)^2)` being
+            // `sqrt(a) |x + h|`, which the rule for a root of a perfect square writes. Here the
+            // arcsine divides by the root of the discriminant, zero, and the logarithm is of
+            // `2a (x + h) + 2a |x + h|`, zero beyond the root: `1/sqrt(x^2 + 2x + 1)` was
+            // `ln(0)` for every x below -1, and `1/sqrt(-a^2 - 2abx - b^2 x^2)` NaN everywhere.
+            // https://github.com/asc-community/AngouriMath/issues/1670
             Entity.Divf(var numerator,
                         Entity.Powf(var radicand, Entity.Number.Rational(Entity.Number.Integer(1), Entity.Number.Integer(2)))) when
                 !numerator.ContainsNode(x)
                 && TreeAnalyzer.TryGetPolyQuadratic(radicand, x, out var ra, out var rb, out var rc)
+                && !IndefiniteIntegralSolver.IsAPerfectSquareDiscriminant(ra, rb, rc)
                     => IntegrateOverRootOfQuadratic(numerator, ra, rb, rc, radicand, x),
 
             // ∫ (px + q)/(ax^2 + bx + c) dx. Only the constant numerator was covered, so
 
@@ -162,6 +162,72 @@ public void AWholePowerOfASquareInAPowerOfTheVariable(string integrand)
             Assert.True(compared >= 5, $"only {compared} points were comparable for {integrand}");
         }
 
+        /// <summary>
+        /// The table's arcsine and logarithm for a root of a quadratic are not for a square: the
+        /// arcsine divides by the root of the discriminant, and the logarithm is of zero beyond
+        /// the root, so <c>1/sqrt(x^2 + 2x + 1)</c> was <c>ln(0)</c> for every x below -1 and
+        /// <c>1/sqrt(-a^2 - 2 a b x - b^2 x^2)</c> NaN everywhere. Compared at every point where
+        /// the integrand has a value, on both sides of each root -- an answer with none there is
+        /// wrong, not incomparable -- with the leading coefficient of either sign: <c>-b^2</c> is
+        /// negative for a real <c>b</c>, and the root of its square is imaginary. Rubi's
+        /// 1.2.1.2 #2737 among them.
+        /// <a href="https://github.com/asc-community/AngouriMath/issues/1670">#1670</a>
+        /// </summary>
+        [Theory]
+        [InlineData("1/sqrt(x^2 + 2*x + 1)")]
+        [InlineData("sqrt(x^2 + 2*x + 1)")]
+        [InlineData("3/sqrt(9*x^2 - 6*x + 1)")]
+        [InlineData("1/sqrt(-4 - 4*x - x^2)")]
+        [InlineData("sqrt(-4 - 4*x - x^2)")]
+        [InlineData("1/sqrt(-a^2 - 2*a*b*x - b^2*x^2)")]
+        [InlineData("x/sqrt(-a^2 - 2*a*b*x - b^2*x^2)")]
+        [InlineData("1/(x*sqrt(-a^2 - 2*a*b*x - b^2*x^2))")]
+        [InlineData("1/(x*sqrt(a^2 + 2*a*b*x + b^2*x^2))")]
+        [InlineData("1/((d + h*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2))")]
+        [InlineData("1/((d + h*x)*sqrt(-a^2 - 2*a*b*x - b^2*x^2))")]
+        public void TheTableDoesNotReadASquareAsAQuadratic(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x").Substitute("C", 0);
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            HoldsWhereverTheIntegrandHasAValue(integrand, integral);
+        }
+
+        /// <summary>
+        /// A square with a symbol of no known sign in front, <c>c (a + b x)^2</c>, is not read as
+        /// one, and the table's forms gave no value between its roots: declined, or answered
+        /// everywhere the integrand has a value.
+        /// </summary>
+        [Fact]
+        public void ASquareOfUnknownSignIsNotAnsweredWrongly()
+        {
+            var integrand = "1/(x*sqrt(c*(a + b*x)^2))";
+            var integral = integrand.ToEntity().Integrate("x").Substitute("C", 0);
+            if (!integral.Stringize().Contains("integral("))
+                HoldsWhereverTheIntegrandHasAValue(integrand, integral);
+        }
+
+        private static void HoldsWhereverTheIntegrandHasAValue(string integrand, Entity integral)
+        {
+            Entity Pin(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.6)
+                .Substitute("d", 1.9).Substitute("h", 1.1);
+            var derivative = Pin(integral).Differentiate("x");
+            var original = Pin(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -3.7, -2.6, -1.8, -1.2, -0.6, 0.4, 1.9 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN)
+                    continue;
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                compared++;
+                var difference = Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart);
+                var scale = Math.Max(1.0, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart));
+                Assert.True(difference / scale < 1e-9,
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}: {integral}");
+            }
+            Assert.True(compared >= 6, $"only {compared} points were comparable for {integrand}");
+        }
+
         [Fact]
         public void TheSignIsTheSignOfTheLinearFactor()
         {