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

3.181

1.

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

BY (271.05)

$[a≥u>b]$

2.

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

BY (272.05)

$[u>a>b]$

3.182

1.

${\int }_{b}^{u}\frac{\text{d}x}{\text{exception 25:}}$

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