For desired input range 1V to 5V, and desired DC gain of 5, Max Vo = 25V.
To achieve this, we do the following:
We can set the output dynamic range to be 30V using +15V for Vcc and -15V for Vee supply rail.
We use a non-inverting amplifier with gain = 1 + Rf/R1 = 5. Therefore Rf = 47kΩ and R1 = 12kΩ to give Rf/R1 = 3.91, which is very close to 4.
We set 2.5V as the crossing reference for a 5V input dynamic range.
To achieve this, instead of letting the common reference for input and output be a ground which is usually the case, we let this common reference be -2.5V.
Vo = Non-inverting Gain x Vin, where Vin = V(+) - V(-), such that V(+) is the voltage at the non-inverting input terminal and V(-) is the voltage at the inverting input terminal.
You will need to level shift at the input to bring 0V to -2.5V, and level shift at the output to bring -2.5V to 0V.
To achieve this, use 4 diodes connected in series because 4 x 0.6V = 2.4V which is close to 2.5V.
To level shift the input and output, connect the anode of first diode to 0V and cathode of last diode to -2.5V