My question is about symbolic integration.
Which simple general integration rules valid for all integrable complex-valued functions $f$ of one complex variable are there for the integrals that are representable by the integral expression below?
$a_0\in\mathbb{Z}$
$a_1,..,a_4,n_1,...,n_4\in\mathbb{N}_0$
$n_1\neq n_2\neq n_3\neq n_4$
$n_1<n_2$
$n_3<n_4$
$$\int x^{a_0}\frac{\left(f^{(n_1)}(x)\right)^{a_1}\cdot\left(f^{(n_2)}(x)\right)^{a_2}}{\left(f^{(n_3)}(x)\right)^{a_3}\cdot\left(f^{(n_4)}(x)\right)^{a_4}}\ dx$$
I myself give an answer below.
The question and its answer are inspired e.g. by the following questions.
Does $\int{f(x)\cdot f'(x)}dx=\frac{f^2(x)}{2}+c$ work?
Calculate $\int f(x) f''(x)dx$
Find the integral $\int f(x)f'''(x) dx$.
Is there a general solution to the integral $\int \frac{f(x)}{f'(x)}dx$?
Integration of $\frac{f'(x)}{f(x)}$?
How to integrate $\int \frac{f'(x)}{f^2(x)}$?
Solve $\int \frac{1}{f(x)f'(x)}\,dx$
Find the Antiderivative $\int \frac{1}{f(x)f''(x)}\mathrm{d}x$
Integrate $\int x \frac{f'(x)}{f(x)} dx$
the integration of $\int \frac{f''(x)f'(x)}{f(x)}\mathrm{d}x$
Regarding rules for simple antiderivatives for the integrals from the question for all functions $f$, most integral tables contain only the rules $\int f'(x)f(x)\ dx=\frac{1}{2}\left(f(x)\right)^2$ and $\int\frac{f'(x)}{f(x)}\ dx=\ln(f(x))$ (logarithmic integration). Clearly, replacing $f(x)$ by $f^{(n)}(x)$ yields the corresponding more general rules for $n\in\mathbb{N}_0$ (equations 3 and 5 below respectively).
To get a general integration rule, we can i.a.
a) take a simple expression, differentiate it and see if the result is simple enough to choose it as integrand,
b) use the known general integration rules/techniques (some of them collected e.g. in [Will 2017]),
c) use integration algorithms to find simple antiderivatives.
Application of partial integration and substitution rule for the general case seem to yield mostly integrands that are more complicated. Only in some special cases, simplified integrands or antiderivatives result.
Even using the integration algorithms implemented in computer algebra software MAPLE and MATHEMATICA, I was able to find only the simple antiderivatives below. They are the result of a systematic search.
The following integration rules can be verified by differentiating both sides of the equations.
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$a,m,n\in\mathbb{N}_0$
$$n>0:\ \ \int f^{(n)}(x)\ dx=f^{(n-1)}(x)$$
$$n>a:\ \ \int x^af^{(n)}(x)\ dx=a!\sum_{i=0}^a\frac{(-1)^{a+i}}{i!}x^if^{(n+i-a-1)}(x)$$
$$\int \left(f^{(n)}(x)\right)^af^{(n+1)}(x)\ dx=\frac{\left(f^{(n)}(x)\right)^{a+1}}{a+1}$$
$$\int f^{(m)}(x)f^{(m+2n+1)}(x)\ dx=\left(\sum_{i=m}^{m+n-1}(-1)^{m-i}f^{(i)}(x)f^{(2(m+n)-i)}(x)\right)+\frac{(-1)^n}{2}\left(f^{(m+n)}(x)\right)^2$$ $\ $
$$\int\frac{f^{(n+1)}(x)}{f^{(n)}(x)}\ dx=\ln(f^{(n)}(x))$$
$$a\neq 1:\ \ \ \int\frac{f^{(n+1)}(x)}{\left(f^{(n)}(x)\right)^a}\ dx=-\frac{1}{(a-1)\left(f^{(n)}(x)\right)^{a-1}}$$
$$\int\frac{f^{(n)}(x)f^{(n+2)}(x)}{\left(f^{(n+1)}(x)\right)^2}\ dx=x-\frac{f^{(n)}(x)}{f^{(n+1)}(x)}$$ $\ $
$$\int\left(\frac{f^{(n+1)}(x)}{f^{(n)}(x)}\right)^2\ dx=-\frac{f^{(n+1)}(x)}{f^{(n)}(x)}+\int\frac{f^{(n+2)}(x)}{f^{(n)}(x)}\ dx$$
$$\int\frac{f^{(n+2)}(x)}{f^{(n)}(x)}\ dx=\frac{f^{(n+1)}(x)}{f^{(n)}(x)}+\int\left(\frac{f^{(n+1)}(x)}{f^{(n)}(x)}\right)^2\ dx$$
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The existence of closed-form antiderivatives of closed-form integrands is answered by the mathematical subject area symbolic integration / integration in finite terms. For the elementary functions, Risch algorithm with Liouville's theorem does exist. Extensions of Risch algorithm for the elementary functions + some special functions are also known ([Raab 2013]). Maybe the types of the transcendental and differential equations of these algorithms can help to answer the general question above further at least for some large classes of functions.
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[Raab 2013] Raab, C. G.: Generalization of Risch's Algorithm to Special Functions. 2013
[Will 2017]: Will, J.: Product rule, quotient rule, reciprocal rule, chain rule and inverse rule for integration. May 2017, Will, J.: Produktregel, Quotientenregel, Reziprokenregel, Kettenregel und Umkehrregel für die Integration. May 2017