Table of Integrals, Series, and Products, 8th Edition

3.18 Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions

3.181

1. 

$\begin{array}{rr}\hfill {\displaystyle {\int }_b^u\frac{\text{d}x}{\sqrt[4]{\left(a-x\right)\left(x-b\right)}}}& \hfill =\sqrt{a-b}\{2\left[E\left(\frac{1}{\sqrt{2}}\right)+E\left(\arccos \sqrt[4]{\frac{4\left(a-u\right)\left(u-b\right)}{{\left(a-b\right)}^2}},\frac{1}{\sqrt{2}}\right)\right]\\ \hfill -\lceil k\left(\frac{1}{\sqrt{2}}\right)+F\left(\arccos \sqrt[4]{\frac{4\left(a-u\right)\left(u-b\right)}{{\left(a-b\right)}^2}},\frac{1}{\sqrt{2}}\right)\rceil \}\end{array}$

  BY (271.05)

$\left[a\ge u>b\right]$

2. 

$\begin{array}{r}\hfill {\displaystyle {\int }_b^u\frac{\text{d}x}{\sqrt[4]{\left(x-a\right)\left(x-b\right)}}\sqrt{\frac{a-b}{2}}F[\left(\arccos \frac{a-b-2\sqrt{\left(u-a\right)\left(u-b\right)}}{a-b+2\sqrt{\left(u-a\right)\left(u-b\right)}},\frac{1}{\sqrt{2}}\right)}\\ \hfill -2E\left(\arccos \frac{a-b-2\sqrt{\left(u-a\right)\left(u-b\right)}}{a-b+2\sqrt{\left(u-a\right)\left(u-b\right)}},\frac{1}{\sqrt{2}}\right)]\\ \hfill +\frac{2\left(2u-a-b\right)\sqrt[4]{\left(u-a\right)\left(u-b\right)}}{a-b+2\sqrt{\left(u-a\right)\left(u-b\right)}}\end{array}$

  BY (272.05)

$\left[u>a>b\right]$

3.182

1. 

${\displaystyle {\int }_b^u\frac{\text{d}x}{\text{exception 25:}}}$