If `a, b, c` are non-zero than minimumm value of expression
`(((a^(4)+3a^(2)+1)(b^(4)+5b^(2)+1)(c^(4)+7c^(2)+1))/(a^(2)b^(2)c^(2)))` equals `pqr` find `p + q+ r`.

Explanation

To find the minimum value of the expression

\[ \frac{(a^4 + 3a^2 + 1)(b^4 + 5b^2 + 1)(c^4 + 7c^2 + 1)}{a^2b^2c^2}, \]

we will analyze each factor in the numerator separately.

### Step 1: Analyze the first term \( a^4 + 3a^2 + 1 \)

We can rewrite this expression in terms of \( x = a^2 \):

\[ a^4 + 3a^2 + 1 = x^2 + 3x + 1. \]

To find the minimum value of this quadratic expression, we can use the vertex formula \( x = -\frac{b}{2a} \):

\[ x = -\frac{3}{2 \cdot 1} = -\frac{3}{2}. \]

However, since \( a \) is non-zero, \( a^2 \) (or \( x \)) must be positive. Thus, we evaluate the expression at \( x = 1 \):

\[ 1^2 + 3 \cdot 1 + 1 = 1 + 3 + 1 = 5. \]

### Step 2: Analyze the second term \( b^4 + 5b^2 + 1 \)

Similarly, we set \( y = b^2 \):

\[ b^4 + 5b^2 + 1 = y^2 + 5y + 1. \]

Using the vertex formula again:

\[ y = -\frac{5}{2 \cdot 1} = -\frac{5}{2}. \]

Again, since \( b \) is non-zero, we evaluate at \( y = 1 \):

\[ 1^2 + 5 \cdot 1 + 1 = 1 + 5 + 1 = 7. \]

### Step 3: Analyze the third term \( c^4 + 7c^2 + 1 \)

Let \( z = c^2 \):

\[ c^4 + 7c^2 + 1 = z^2 + 7z + 1. \]

Using the vertex formula:

\[ z = -\frac{7}{2 \cdot 1} = -\frac{7}{2}. \]

Evaluating at \( z = 1 \):

\[ 1^2 + 7 \cdot 1 + 1 = 1 + 7 + 1 = 9. \]

### Step 4: Combine the results

Now we can combine the minimum values of each term:

\[ \text{Minimum value} = \frac{(5)(7)(9)}{(1)(1)(1)} = 5 \cdot 7 \cdot 9. \]

### Step 5: Calculate \( pqr \)

Let \( p = 5 \), \( q = 7 \), and \( r = 9 \). Thus:

\[ pqr = 5 \cdot 7 \cdot 9. \]

### Step 6: Find \( p + q + r \)

Now, we calculate:

\[ p + q + r = 5 + 7 + 9 = 21. \]

### Final Answer

The final answer is:

\[ \boxed{21}. \]