  We show that the energy of a perturbed system can be fully recovered from the
unperturbed system's electron density. We derive an alchemical integral
transform by parametrizing space in terms of transmutations, the chain rule and
integration by parts. Within the radius of convergence, the zeroth order yields
the energy expansion at all orders, restricting the textbook statement by
Wigner that the $p$-th order wave function derivative is necessary to describe
the $(2p+1)$-th energy derivative. Without the need for derivatives of the
electron density, this allows to cover entire chemical neighborhoods from just
one quantum calculation instead of single systems one-by-one. Numerical
evidence presented indicates that predictive accuracy is achieved in the range
of mHa for the harmonic oscillator or the Morse potential, and in the range of
machine accuracy for hydrogen-like atoms. Considering iso-electronic nuclear
charge variations by one proton in all multi-electron atoms from He to Ne,
alchemical integral transform based estimates of the relative energy deviate by
only few mHa from corresponding Hartree-Fock reference numbers.
