The problem is: The integral $\int(a+bt)^{p}t^{q}\text{d}t$ can be expressed in terms of elementary functions if one of the numbers $p$, $q$ or $p+q$ is an integer.
The first two cases (either $p$ or $q$ is an integer) are easy to use substitution. Assume $p$ is an integer, we can denote $q=\frac{m}{n}$ so that $p$, $m$ and $n$ are all integers, then let $s=t^{\frac{1}{n}}$ so that $t=s^{n}$ and $\text{d}t=ns^{n-1}\text{d}s$, we have: $$\int(a+bt)^{p}t^{q}=\int(a+bs^{n})^{p}s^{m}ns^{n-1}\text{d}s$$
Since $p$, $m$ and $n$ are all integers, this is obviously a rational integration regarding $s$. The case of $q$ is an integer can be deduced in a similar way.
However I am stuck with the case $p+q$ being an integer. Intuitively I let $p=\frac{m}{n}$ and $q=z-p=z-\frac{m}{n}=\frac{nz-m}{n}$. Let $s=t^{\frac{1}{n}}$ so that $t=s^{n}$ and $\text{d}t=ns^{n-1}\text{d}s$. We have $$\int(a+bt)^{p}t^{q}\text{d}t=\int(a+bs^{n})^{\frac{m}{n}}s^{nz-m}ns^{n-1}\text{d}s=n\int(a+bs^{n})^{\frac{m}{n}}s^{n(z+1)-m-1}\text{d}s$$ All good except the $(a+bs^{n})^{\frac{m}{n}}$ part. Intuitively this is just some amplification and translation from $s^{n}$ and should be a rational expression of $s$, but how to reason it in mathematics?
Thanks in advance!