Solution:

The distance is $|(1 + 2i)z^3 - z^5| = |z|^3 \cdot |1 + 2i - z^2| = 125\,|1 + 2i - z^2|.$ As $z$ runs over the circle $|z| = 5$ with $b \gt 0,$ the square $z^2$ attains every point of the circle $|w| = 25$ (the condition $b \gt 0$ merely selects one of the two square roots). The point of that circle farthest from $1 + 2i$ is diametrically opposite in direction: $$z^2 = -25 \cdot \frac{1 + 2i}{|1 + 2i|} = -5\sqrt{5}\,(1 + 2i).$$

Squaring, $z^4 = 125\,(1 + 2i)^2 = 125\,(-3 + 4i) = -375 + 500i,$ so $c + d = -375 + 500 = 125.$