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\( \int \frac{3 x+4}{\sqrt{5 x^{2}+8}} \)

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\(\int\frac{3x+4}{\sqrt{5x^2+8}}dx\)

 = \(\int(\frac{3x}{\sqrt{5x^2+8}}+\frac4{\sqrt{5x^2+8}})dx\) 

 = \(\frac3{10}\int\frac{10x}{\sqrt{5x^2+8}}dx+\frac4{\sqrt 5}\int\frac{dx}{\sqrt{x^2+(\sqrt{\frac85})^2}}\)

 = \(\frac3{10}(2\sqrt{5x^2+8})\) + \(\frac4{\sqrt 5}log|x + \sqrt{x^2+\frac85}|+k\) 

 = \(\frac35\sqrt{{5x^2+8}}\)  + \(\frac4{\sqrt 5}log(\frac{\sqrt5x+\sqrt{5x^2+8}}{\sqrt5})+k\) 

 = \(\frac35\sqrt{{5x^2+8}}\) + \(\frac4{\sqrt 5}log(\sqrt 5x+\sqrt{5x^2+8})\) - \(\frac4{\sqrt5}log\sqrt5+k\) 

 = \(\frac35\sqrt{{5x^2+8}}\) + \(\frac4{\sqrt 5}log(\sqrt5x + \sqrt{5x^2+8})+c\)

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