A best solution for \frac using mtpro2 fonts

Question

Starting from this old this question How to make fractions in powers look good, I'm searching a optimal solution to aesthetically define a fraction of the type \frac{...}{...}. For my book I've used the 2nd formula as in the image (green line),

but I'm not very happy about the results.

(a) Meanwhile I would like to understand if there is any minimal difference between the second and fifth formula.

(b) After, I'd like to know how to build the smallest power ^{-\frac{3}{2}} in size and aesthetically more beautiful and that it's higher left near the round bracket ). In fact if you look carefully this fraction is almost close to the next x.

Here there is my MWE. Thank you.

\documentclass[12pt]{article}
\usepackage{newtxtext,mathtools}
%\usepackage{newtxmath}
\usepackage[lite]{mtpro2}
\usepackage{xfrac}
\usepackage[ugly]{nicefrac}
\begin{document}

\begin{equation}
f(x)=(1+x)^{-3/2} x+\cdots 
\end{equation}

\begin{equation}
f(x)=(1+x)^{-\frac32} x+\cdots 
\end{equation}

\begin{equation}
f(x)=(1+x)^{-\tfrac 32} x+\cdots 
\end{equation}

\begin{equation}
f(x)=(1+x)^{-\sfrac 32} x+\cdots %https://tex.stackexchange.com/questions/202325/how-to-make-fractions-in-powers-look-good
\end{equation}


\begin{equation}
f(x)=(1+x)^{-\nicefrac{3}{2}} x+\cdots %https://tex.stackexchange.com/questions/202325/how-to-make-fractions-in-powers-look-good
\end{equation}

\end{document}

Answer 1

At the OP's request, I have turned my comment into an answer. Since the OP seemed to be leaning toward choice 2, I tried a variation on that choice, attempting to both raise the superscript and bring it further left, as indicated by the OP as desirable.

The macro \supfrac[-]{a}{b} is how I propose to do this. It is intended purely for superscript fractions in displaystyle, as it will mess with the line spacing in textstyle.

Because it moves the fraction leftward with an \mkern, it has to know whether there is a leading negative sign. Thus, the negative is introduced to \supfrac as an optional argument, so that it, too, may be shifted leftward.

The raising of the superscript is accomplished by adding a downward \rule to the argument. The value -6pt can be tweaked to suit.

In the MWE, the original choice 2 is on the top line, with \supfrac on the next two lines (one with a negative fraction, the next with a positive fraction. The last line shows that, by using a \rule rather than a \raisebox, one can add other math to the \supfrac and retain a proper baseline in the exponent.

\documentclass[12pt]{article}
\usepackage{newtxtext,mathtools}
\usepackage{newtxmath}
%\usepackage[lite]{mtpro2}
\usepackage{xfrac}
\usepackage[ugly]{nicefrac}
\newcommand\supfrac[3][]{\mkern-2mu#1\frac{#2}{#3}\rule[-6pt]{0pt}{0pt}}
\begin{document}
\begin{equation}
f(x)=(1+x)^{-\frac32} x+\cdots 
\end{equation}
\[
f(x)=(1+x)^{\supfrac[-]{3}{2}} x+\cdots
\]
\[
f(x)=(1+x)^{\supfrac{3}{2}} x+\cdots
\]
\[
f(x)=(1+x)^{\supfrac{3}{2}x+1} x+\cdots
\]
\end{document}