Integrals of Irrational Functions

1.

$\int \dfrac{dx}{\sqrt{a+bx}}=\dfrac{2}{a}\sqrt{ax+b}+C$

2.

$\int \sqrt{ax+b}dx=\dfrac{2}{3a}\left(ax+b\right)^{3/2}+C$

3.

$\int \dfrac{xdx}{\sqrt{ax+b}}=\dfrac{2(ax-2b)}{3a^2}\sqrt{ax+b}+C$

4.

$\int x\sqrt{ax+b}dx=\dfrac{2(3ax-2b)}{15a^2}(ax+b)^{3/2}+C$

5.

$\int \dfrac{dx}{(x+c)\sqrt{ax+b}}=\dfrac{1}{\sqrt{b-ac}} \ln\left|\dfrac{\sqrt{ax+b}-\sqrt{b-ac}}{\sqrt{ax+b}+\sqrt{b-ac}}\right|+C, b-ac>0$

6.

$\int \dfrac{dx}{(x+c)\sqrt{ax+b}}=\dfrac{1}{\sqrt{ac-b}}\arctan\sqrt{\dfrac{ax+b}{ac-b}}+C, b-ac<0$

7.

$\int \sqrt{\dfrac{ax+b}{cx+d}}dx=\dfrac{1}{c}\sqrt{(ax+b)(cx+d)}-\dfrac{ad-bc}{c\sqrt{ac}}\ln\left|\sqrt{a(cx+d)}+\sqrt{c(ax+b)}\right|+C, a>0$

8.

$\int \sqrt{\dfrac{ax+b}{cx+d}}dx=\dfrac{1}{c}\sqrt{(ax+b)(cx+d)}-\dfrac{ad-bc}{c\sqrt{ac}}\arctan\sqrt\frac{a(cx+d)}{c(ax+b)}+C, a<0,c>0$

9.

$\int x^2\sqrt{a+bx}dx=\dfrac{2\left(8a^2-12abx+15b^2x^2\right)}{105b^3}\sqrt{(a+bx)^3}+C$

10.

$\int \dfrac{x^2}{\sqrt{a+bx}} = \dfrac{2\left(8a^2-4abx+3b^2x^2\right)}{15b^3}\sqrt{a+bx}+C$