$\int {[\frac{1}{{(x + a)}^{3} {(x + b)}^{5}}]}^{1 / 4} d x$

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Solution

$I = \int \frac{1}{\sqrt[4]{(x + a)^{3}} (x + b)^{5}} d x$
$I = \int \frac{1}{(x + a)^{3 / 4} (x + b)^{5 / 4}} d x$
Let $t = \sqrt[4]{x + a}$
$d x = 4 (x + a)^{3 / 4} d t$
$I = 4 \int \frac{d t}{(t^{4} + b - a)^{5 / 4}}$
Let $P = \frac{4 \sqrt{t^{4} + b - a}}{t}$
$d t = \frac{1}{\frac{t^{2}}{(t^{2} + b - a)^{3 / 4}} - \frac{\sqrt[4]{T^{2} + b - a}}{t^{2}}} d p$
Therefore, $I = - \int \frac{1}{(b - a) p^{2}} d p$
$I = \frac{1}{(a - b)} \int \frac{d p}{p^{2}}$
$I = - \frac{1}{(a - b)} [\frac{1}{p}] + c$
$I = \frac{1}{(b - a)} \frac{1}{p} + c$
$I = \frac{1}{(b - a)} [\frac{t}{\sqrt[4]{t^{4} + b - a}}] + c$
$I = \frac{1}{(b - a)} [\frac{\sqrt[4]{x + a}}{\sqrt[4]{(x + a) + b - a}}] + c$
$I = \frac{1}{(b - a)} [\frac{4 \sqrt[4]{x + a}}{\sqrt[4]{x + b}}] + c$

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