Table of Integrals of Irrational Functions

$$\int \frac{\sqrt{x} \, dx}{X} = \pm \frac{2\sqrt{x}}{b^2} \mp \frac{2a}{b^3} \, Y$$

$$\int \frac{\sqrt{x^3} \, dx}{X} = \pm \frac{2}{3} \cdot \frac{\sqrt{x^3}}{b^2} - \frac{2a^2\sqrt{x}}{b^4} + \frac{2a^3}{b^5} \, Y$$

$$\int \frac{\sqrt{x} \, dx}{X^2} = \pm \frac{\sqrt{x}}{b^2 X} \pm \frac{1}{a b^3} \, Y$$

$$\int \frac{\sqrt{x^3} \, dx}{X^2} = \pm \frac{2\sqrt{x^3}}{b^2 X} + \frac{3a^2\sqrt{x}}{b^4 X} - \frac{3a^3}{b^5} \, Y$$

$$\int \frac{dx}{X \sqrt{x}} = \frac{2}{a b} \, Y$$

$$\int \frac{dx}{X \sqrt{x^3}} = -\frac{2}{a^3 \sqrt{x}} \mp \frac{2b}{a^3} \, Y$$

$$\int \frac{dx}{X^2 \sqrt{x}} = \frac{\sqrt{x}}{a^2 X} + \frac{b}{a^3} \, Y$$

$$\int \frac{dx}{X^3 \sqrt{x^3}} = -\frac{2}{a^3 X \sqrt{x}} \mp \frac{3b^2 \sqrt{x}}{a^4 X} \mp \frac{3b^3}{a^5} \, Y$$

---

$$\int \frac{\sqrt{x} \, dx}{a^4 + x^2} = -\frac{1}{2a\sqrt{2}} \ln \left( \frac{x + a\sqrt{2x} + a^2}{x - a\sqrt{2x} + a^2} \right) + \frac{1}{a\sqrt{2}} \arctan \left( \frac{a\sqrt{2x}}{a^2 - x} \right)$$

$$\int \frac{dx}{(a^4 + x^2) \sqrt{x}} = \frac{1}{2a^3\sqrt{2}} \ln \left( \frac{x + a\sqrt{2x} + a^2}{x - a\sqrt{2x} + a^2} \right) + \frac{1}{a^3\sqrt{2}} \arctan \left( \frac{a\sqrt{2x}}{a^2 - x} \right)$$

$$\int \frac{\sqrt{x} \, dx}{a^4 - x^2} = \frac{1}{2a} \ln \left| \frac{a + \sqrt{x}}{a - \sqrt{x}} \right| - \frac{1}{a} \arctan \left( \frac{\sqrt{x}}{a} \right)$$

$$\int \frac{dx}{(a^4 - x^2) \sqrt{x}} = \frac{1}{2a^3} \ln \left| \frac{a + \sqrt{x}}{a - \sqrt{x}} \right| + \frac{1}{a^3} \arctan \left( \frac{\sqrt{x}}{a} \right)$$

---

$$\int \sqrt{X} \, dx = \frac{2}{3a} \, X^{3/2}$$

$$\int x \sqrt{X} \, dx = \frac{2(3ax - 2b)}{15a^2} \, X^{3/2}$$

$$\int x^2 \sqrt{X} \, dx = \frac{2(15a^2 x^2 - 12abx + 8b^2)}{105a^3} \, X^{3/2}$$

$$\int \frac{dx}{\sqrt{X}} = \frac{2}{a} \sqrt{X}$$

$$\int \frac{x \, dx}{\sqrt{X}} = \frac{2(ax - 2b)}{3a^2} \sqrt{X}$$

$$\int \frac{x^2 \, dx}{\sqrt{X}} = \frac{2(3a^2 x^2 - 4abx + 8b^2)}{15a^3} \sqrt{X}$$

$$\int \frac{dx}{x \sqrt{X}} = \begin{cases} \displaystyle \frac{1}{\sqrt{b}} \ln \left| \frac{\sqrt{X} - \sqrt{b}}{\sqrt{X} + \sqrt{b}} \right|, & b > 0, \\[2ex] \displaystyle \frac{2}{\sqrt{-b}} \arctan \sqrt{\frac{X}{-b}}, & b < 0. \end{cases}$$

$$\int \frac{\sqrt{X}}{x} \, dx = 2 \sqrt{X} + b \int \frac{dx}{x \sqrt{X}} \qquad \text{(see 127)}$$

$$\int \frac{dx}{x^2 \sqrt{X}} = -\frac{\sqrt{X}}{bx} - \frac{a}{2b} \int \frac{dx}{x \sqrt{X}} \qquad \text{(see 127)}$$

$$\int \frac{\sqrt{X}}{x^2} \, dx = -\frac{\sqrt{X}}{x} + \frac{a}{2} \int \frac{dx}{x \sqrt{X}} \qquad \text{(see 127)}$$

$$\int \frac{dx}{x^n \sqrt{X}} = -\frac{\sqrt{X}}{(n-1) b x^{n-1}} - \frac{(2n-3)a}{(2n-2)b} \int \frac{dx}{x^{n-1} \sqrt{X}}, \qquad n>1$$

$$\int X^{3/2} \, dx = \frac{2}{5a} \, X^{5/2}$$

$$\int x X^{3/2} \, dx = \frac{2}{35a^2} (5a x - 2b) \, X^{5/2}$$

$$\int x^2 X^{3/2} \, dx = \frac{2}{315a^3} \bigl( 35a^2 x^2 - 20ab x + 8b^2 \bigr) \, X^{5/2}$$

$$\int \frac{X^{3/2}}{x} \, dx = \frac{2}{3} X^{3/2} + 2b \sqrt{X} + b^2 \int \frac{dx}{x \sqrt{X}} \qquad \text{(see 127)}$$

$$\int \frac{x \, dx}{X^{3/2}} = \frac{2}{a^2} \left( \sqrt{X} + \frac{b}{\sqrt{X}} \right)$$

$$\int \frac{x^2 \, dx}{X^{3/2}} = \frac{2}{a^3} \left( \frac{X^{3/2}}{3} - 2b \sqrt{X} - \frac{b^2}{\sqrt{X}} \right)$$

$$\int \frac{dx}{x X^{3/2}} = \frac{2}{b\sqrt{X}} + \frac{1}{b} \int \frac{dx}{x \sqrt{X}} \qquad \text{(see 127)}$$

$$\int \frac{dx}{x^2 X^{3/2}} = -\frac{1}{b x \sqrt{X}} - \frac{3a}{2b} \left( \frac{2}{b\sqrt{X}} + \frac{1}{b} \int \frac{dx}{x \sqrt{X}} \right)$$

$$\int X^{n/2} \, dx = \frac{2}{a(n+2)} \, X^{(n+2)/2}, \qquad n \neq -2$$

$$\int x X^{n/2} \, dx = \frac{2}{a^2} \left( \frac{X^{(n+4)/2}}{n+4} - \frac{b X^{(n+2)/2}}{n+2} \right), \qquad n \neq -2, -4$$

$$\int x^2 X^{n/2} \, dx = \frac{2}{a^3} \left( \frac{X^{(n+6)/2}}{n+6} - \frac{2b X^{(n+4)/2}}{n+4} + \frac{b^2 X^{(n+2)/2}}{n+2} \right), \qquad n \neq -2, -4, -6$$

$$\int \frac{X^{n/2}}{x} \, dx = \frac{2}{n} X^{n/2} + b \int \frac{X^{(n-2)/2}}{x} \, dx, \qquad n \neq 0$$

$$\int \frac{dx}{x X^{n/2}} = \frac{2}{(2-n)b X^{(n-2)/2}} + \frac{1}{b} \int \frac{dx}{x X^{(n-2)/2}}, \qquad n \neq 2$$

$$\int \frac{dx}{x^2 X^{n/2}} = -\frac{1}{b x X^{(n-2)/2}} - \frac{na}{2b} \int \frac{dx}{x X^{n/2}}$$

---

$$\int \frac{dx}{\sqrt{XY}} = \begin{cases} \displaystyle \frac{2}{\sqrt{-af}} \, \arctan \sqrt{\frac{-fX}{aY}}, & \text{if } af < 0, \\[2ex] \displaystyle \frac{1}{\sqrt{af}} \, \ln \left| \sqrt{aY} + \sqrt{fX} \right|, & \text{if } af > 0. \end{cases}$$

$$\int \frac{x \, dx}{\sqrt{XY}} = \frac{\sqrt{XY}}{af} - \frac{ag + bf}{2af} \int \frac{dx}{\sqrt{XY}} \qquad \text{(see 146)}$$

$$\int \frac{dx}{\sqrt{X} \, Y^{3/2}} = -\frac{2\sqrt{X}}{\Delta \sqrt{Y}}$$

$$\int \frac{dx}{Y \sqrt{X}} = \begin{cases} \displaystyle \frac{2}{\sqrt{-\Delta f}} \, \arctan \sqrt{ \frac{fX}{-\Delta} }, & \text{if } \Delta f < 0, \\[2ex] \displaystyle \frac{1}{\sqrt{\Delta f}} \, \ln \left| \frac{\sqrt{fX} - \sqrt{\Delta f}}{\sqrt{fX} + \sqrt{\Delta f}} \right|, & \text{if } \Delta f > 0. \end{cases}$$

$$\int \sqrt{XY} \, dx = \frac{\Delta + 2aY}{4af} \sqrt{XY} - \frac{\Delta^2}{8af} \int \frac{dx}{\sqrt{XY}} \qquad \text{(see 146)}$$

$$\int \sqrt{\frac{Y}{X}} \, dx = \frac{1}{a} \sqrt{XY} - \frac{\Delta}{2a} \int \frac{dx}{\sqrt{XY}} \qquad \text{(see 146)}$$

$$\int \frac{\sqrt{X}}{Y} \, dx = \frac{2\sqrt{X}}{f} + \frac{\Delta}{f} \int \frac{dx}{Y\sqrt{X}} \qquad \text{(see 149)}$$

$$\int \frac{Y^n \, dx}{\sqrt{X}} = \frac{2}{(2n+1)a} \left( \sqrt{X} \, Y^n - n\Delta \int \frac{Y^{n-1} \, dx}{\sqrt{X}} \right)$$

$$\int \frac{dx}{\sqrt{X} \, Y^n} = -\frac{1}{(n-1)\Delta} \left( \frac{\sqrt{X}}{Y^{n-1}} + \frac{(2n-3)a}{2} \int \frac{dx}{\sqrt{X} \, Y^{n-1}} \right)$$

$$\int \sqrt{X} \, Y^n \, dx = \frac{2}{(2n+3)f} \left( \sqrt{X} \, Y^{n+1} - \frac{\Delta}{2} \int \frac{Y^n \, dx}{\sqrt{X}} \right) \qquad \text{(see 153)}$$

$$\int \frac{\sqrt{X} \, dx}{Y^n} = -\frac{1}{(n-1)f} \left( \frac{\sqrt{X}}{Y^{n-1}} + \frac{a}{2} \int \frac{dx}{\sqrt{X} \, Y^{n-1}} \right)$$

---

$$\int \sqrt{X} \, dx = \frac{1}{2} \left( x\sqrt{X} + a^2 \arcsin\frac{x}{a} \right)$$

$$\int x\sqrt{X} \, dx = -\frac{1}{3} X^{3/2}$$

$$\int x^2 \sqrt{X} \, dx = -\frac{x}{4} X^{3/2} + \frac{a^2}{8} \left( x\sqrt{X} + a^2 \arcsin\frac{x}{a} \right)$$

$$\int x^3 \sqrt{X} \, dx = \left( \frac{X^{5/2}}{5} - \frac{a^2 X^{3/2}}{3} \right)$$

$$\int \frac{\sqrt{X}}{x} \, dx = \sqrt{X} - a \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{\sqrt{X}}{x^2} \, dx = -\frac{\sqrt{X}}{x} - \arcsin\frac{x}{a}$$

$$\int \frac{\sqrt{X}}{x^3} \, dx = -\frac{\sqrt{X}}{2x^2} + \frac{1}{2a} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{dx}{\sqrt{X}} = \arcsin\frac{x}{a}$$

$$\int \frac{x\,dx}{\sqrt{X}} = -\sqrt{X}$$

$$\int \frac{x^2\,dx}{\sqrt{X}} = -\frac{x}{2}\sqrt{X} + \frac{a^2}{2} \arcsin\frac{x}{a}$$

$$\int \frac{x^3\,dx}{\sqrt{X}} = \frac{X^{3/2}}{3} - a^2 \sqrt{X}$$

$$\int \frac{dx}{x\sqrt{X}} = -\frac{1}{a} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{dx}{x^2\sqrt{X}} = -\frac{\sqrt{X}}{a^2 x}$$

$$\int \frac{dx}{x^3\sqrt{X}} = -\frac{\sqrt{X}}{2a^2 x^2} - \frac{1}{2a^3} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int X^{3/2} \, dx = \frac{1}{4} \left( x X^{3/2} + \frac{3a^2 x}{2} \sqrt{X} + \frac{3a^4}{2} \arcsin\frac{x}{a} \right)$$

$$\int x X^{3/2} \, dx = -\frac{1}{5} X^{5/2}$$

$$\int x^2 X^{3/2} \, dx = -\frac{x}{6} X^{5/2} + \frac{a^2}{24} \left( x X^{3/2} + \frac{3a^2 x}{2} \sqrt{X} + \frac{3a^4}{2} \arcsin\frac{x}{a} \right)$$

$$\int x^3 X^{3/2} \, dx = \frac{X^{7/2}}{7} - \frac{a^2 X^{5/2}}{5}$$

$$\int \frac{X^{3/2}}{x} \, dx = \frac{X^{3/2}}{3} + a^2 \sqrt{X} - a^3 \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{X^{3/2}}{x^2} \, dx = -\frac{X^{3/2}}{x} - \frac{3}{2} x \sqrt{X} - \frac{3a^2}{2} \arcsin\frac{x}{a}$$

$$\int \frac{X^{3/2}}{x^3} \, dx = -\frac{X^{3/2}}{2x^2} - \frac{3\sqrt{X}}{2} + \frac{3a}{2} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{dx}{X^{3/2}} = \frac{x}{a^2 \sqrt{X}}$$

$$\int \frac{x\,dx}{X^{3/2}} = \frac{1}{\sqrt{X}}$$

$$\int \frac{x^2\,dx}{X^{3/2}} = \frac{x}{\sqrt{X}} - \arcsin\frac{x}{a}$$

$$\int \frac{x^3\,dx}{X^{3/2}} = \sqrt{X} + \frac{a^2}{\sqrt{X}}$$

$$\int \frac{dx}{x X^{3/2}} = \frac{1}{a^2 \sqrt{X}} - \frac{1}{a^3} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{dx}{x^2 X^{3/2}} = -\frac{\sqrt{X}}{a^4 x} + \frac{x}{a^4 \sqrt{X}}$$

$$\int \frac{dx}{x^3 X^{3/2}} = -\frac{1}{2a^2 x^2 \sqrt{X}} + \frac{3}{2a^4 \sqrt{X}} - \frac{3}{2a^5} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

---

$$\int \sqrt{X} \, dx = \frac{1}{2} \left( x\sqrt{X} + a^2 \ln\bigl(x + \sqrt{X}\bigr) \right) + C$$

$$\int x\sqrt{X} \, dx = \frac{1}{3} X^{3/2}$$

$$\int x^2 \sqrt{X} \, dx = \frac{x}{4} X^{3/2} - \frac{a^2}{8} \left( x\sqrt{X} + a^2 \ln\bigl(x + \sqrt{X}\bigr) \right) + C$$

$$\int x^3 \sqrt{X} \, dx = \frac{X^{5/2}}{5} - \frac{a^2 X^{3/2}}{3}$$

$$\int \frac{\sqrt{X}}{x} \, dx = \sqrt{X} - a \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{\sqrt{X}}{x^2} \, dx = -\frac{\sqrt{X}}{x} + \ln\bigl(x + \sqrt{X}\bigr) + C$$

$$\int \frac{\sqrt{X}}{x^3} \, dx = -\frac{\sqrt{X}}{2x^2} - \frac{1}{2a} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{dx}{\sqrt{X}} = \ln\bigl(x + \sqrt{X}\bigr) + C$$

$$\int \frac{x\,dx}{\sqrt{X}} = \sqrt{X}$$

$$\int \frac{x^2\,dx}{\sqrt{X}} = \frac{x}{2} \sqrt{X} - \frac{a^2}{2} \ln\bigl(x + \sqrt{X}\bigr) + C$$

$$\int \frac{x^3\,dx}{\sqrt{X}} = \frac{X^{3/2}}{3} - a^2 \sqrt{X}$$

$$\int \frac{dx}{x\sqrt{X}} = -\frac{1}{a} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{dx}{x^2\sqrt{X}} = -\frac{\sqrt{X}}{a^2 x}$$

$$\int \frac{dx}{x^3\sqrt{X}} = -\frac{\sqrt{X}}{2a^2 x^2} + \frac{1}{2a^3} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int X^{3/2} \, dx = \frac{1}{4} \left( x X^{3/2} + \frac{3a^2 x}{2} \sqrt{X} + \frac{3a^4}{2} \ln\bigl(x + \sqrt{X}\bigr) \right) + C$$

$$\int x X^{3/2} \, dx = \frac{1}{5} X^{5/2}$$

$$\int x^2 X^{3/2} \, dx = \frac{x}{6} X^{5/2} - \frac{a^2}{24} \left( x X^{3/2} + \frac{3a^2 x}{2} \sqrt{X} + \frac{3a^4}{2} \ln\bigl(x + \sqrt{X}\bigr) \right) + C$$

$$\int x^3 X^{3/2} \, dx = \frac{X^{7/2}}{7} - \frac{a^2 X^{5/2}}{5}$$

$$\int \frac{X^{3/2}}{x} \, dx = \frac{X^{3/2}}{3} + a^2 \sqrt{X} - a^3 \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{X^{3/2}}{x^2} \, dx = -\frac{X^{3/2}}{x} + \frac{3}{2} x \sqrt{X} + \frac{3a^2}{2} \ln\bigl(x + \sqrt{X}\bigr) + C$$

$$\int \frac{X^{3/2}}{x^3} \, dx = -\frac{X^{3/2}}{2x^2} + \frac{3}{2} \sqrt{X} - \frac{3a}{2} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{dx}{X^{3/2}} = \frac{x}{a^2 \sqrt{X}}$$

$$\int \frac{x\,dx}{X^{3/2}} = -\frac{1}{\sqrt{X}}$$

$$\int \frac{x^2\,dx}{X^{3/2}} = -\frac{x}{\sqrt{X}} + \ln\bigl(x + \sqrt{X}\bigr) + C$$

$$\int \frac{x^3\,dx}{X^{3/2}} = \sqrt{X} + \frac{a^2}{\sqrt{X}}$$

$$\int \frac{dx}{x X^{3/2}} = \frac{1}{a^2 \sqrt{X}} - \frac{1}{a^3} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

$$\int \frac{dx}{x^2 X^{3/2}} = -\frac{\sqrt{X}}{a^4 x} - \frac{x}{a^4 \sqrt{X}}$$

$$\int \frac{dx}{x^3 X^{3/2}} = -\frac{1}{2a^2 x^2 \sqrt{X}} - \frac{3}{2a^4 \sqrt{X}} + \frac{3}{2a^6} \ln\left| \frac{a + \sqrt{X}}{x} \right|$$

---

$$\int \sqrt{X} \, dx = \frac{1}{2} \left( x\sqrt{X} - a^2 \ln\bigl|x + \sqrt{X}\bigr| \right) + C$$

$$\int x\sqrt{X} \, dx = \frac{1}{3} X^{3/2}$$

$$\int x^2 \sqrt{X} \, dx = \frac{x}{4} X^{3/2} + \frac{a^2}{8} \left( x\sqrt{X} - a^2 \ln\bigl|x + \sqrt{X}\bigr| \right) + C$$

$$\int x^3 \sqrt{X} \, dx = \frac{X^{5/2}}{5} + \frac{a^2 X^{3/2}}{3}$$

$$\int \frac{\sqrt{X}}{x} \, dx = \sqrt{X} - a \arccos\frac{a}{|x|}$$

$$\int \frac{\sqrt{X}}{x^2} \, dx = -\frac{\sqrt{X}}{x} + \ln\bigl|x + \sqrt{X}\bigr| + C$$

$$\int \frac{\sqrt{X}}{x^3} \, dx = -\frac{\sqrt{X}}{2x^2} + \frac{1}{2a} \arccos\frac{a}{|x|}$$

$$\int \frac{dx}{\sqrt{X}} = \ln\bigl|x + \sqrt{X}\bigr| + C$$

$$\int \frac{x\,dx}{\sqrt{X}} = \sqrt{X}$$

$$\int \frac{x^2\,dx}{\sqrt{X}} = \frac{x}{2} \sqrt{X} + \frac{a^2}{2} \ln\bigl|x + \sqrt{X}\bigr| + C$$

$$\int \frac{x^3\,dx}{\sqrt{X}} = \frac{X^{3/2}}{3} + a^2 \sqrt{X}$$

$$\int \frac{dx}{x\sqrt{X}} = \frac{1}{a} \arccos\frac{a}{|x|}$$

$$\int \frac{dx}{x^2\sqrt{X}} = \frac{\sqrt{X}}{a^2 x}$$

$$\int \frac{dx}{x^3\sqrt{X}} = \frac{\sqrt{X}}{2a^2 x^2} + \frac{1}{2a^3} \arccos\frac{a}{|x|}$$

$$\int X^{3/2} \, dx = \frac{1}{4} \left( x X^{3/2} + \frac{3a^2 x}{2} \sqrt{X} - \frac{3a^4}{2} \ln\bigl|x + \sqrt{X}\bigr| \right) + C$$

$$\int x X^{3/2} \, dx = \frac{1}{5} X^{5/2}$$

$$\int x^2 X^{3/2} \, dx = \frac{x}{6} X^{5/2} + \frac{a^2}{24} \left( x X^{3/2} + \frac{3a^2 x}{2} \sqrt{X} - \frac{3a^4}{2} \ln\bigl|x + \sqrt{X}\bigr| \right) + C$$

$$\int x^3 X^{3/2} \, dx = \frac{X^{7/2}}{7} + \frac{a^2 X^{5/2}}{5}$$

$$\int \frac{X^{3/2}}{x} \, dx = \frac{X^{3/2}}{3} - a^2 \sqrt{X} + a^3 \arccos\frac{a}{|x|}$$

$$\int \frac{X^{3/2}}{x^2} \, dx = -\frac{X^{3/2}}{x} + \frac{3}{2} x \sqrt{X} + \frac{3a^2}{2} \ln\bigl|x + \sqrt{X}\bigr| + C$$

$$\int \frac{X^{3/2}}{x^3} \, dx = -\frac{X^{3/2}}{2x^2} - \frac{3}{2} \sqrt{X} - \frac{3a}{2} \arccos\frac{a}{|x|}$$

$$\int \frac{dx}{X^{3/2}} = -\frac{x}{a^2 \sqrt{X}}$$

$$\int \frac{x\,dx}{X^{3/2}} = -\frac{1}{\sqrt{X}}$$

$$\int \frac{x^2\,dx}{X^{3/2}} = -\frac{x}{\sqrt{X}} - \ln\bigl|x + \sqrt{X}\bigr| + C$$

$$\int \frac{x^3\,dx}{X^{3/2}} = \sqrt{X} - \frac{a^2}{\sqrt{X}}$$

$$\int \frac{dx}{x X^{3/2}} = -\frac{1}{a^2 \sqrt{X}} - \frac{1}{a^3} \arccos\frac{a}{|x|}$$

$$\int \frac{dx}{x^2 X^{3/2}} = -\frac{\sqrt{X}}{a^4 x} + \frac{x}{a^4 \sqrt{X}}$$

$$\int \frac{dx}{x^3 X^{3/2}} = -\frac{1}{2a^2 x^2 \sqrt{X}} + \frac{3}{2a^4 \sqrt{X}} - \frac{3}{2a^6} \arccos\frac{a}{|x|}$$

---

$$\int \frac{dx}{\sqrt{X}} = \begin{cases} \displaystyle \frac{1}{\sqrt{a}} \ln\left| 2\sqrt{aX} + 2ax + b \right| + C, & a > 0, \\[2ex] \displaystyle \frac{1}{\sqrt{-a}} \arcsin\frac{2ax+b}{\sqrt{-\Delta}} + C, & a < 0,\ \Delta < 0. \end{cases}$$

$$\int \frac{dx}{x\sqrt{X}} = -\frac{1}{\sqrt{c}} \ln\left| \frac{2\sqrt{cX} + 2c + bx}{x} \right| \quad (c > 0)$$

$$\int \frac{dx}{X^{3/2}} = \frac{2(2ax+b)}{\Delta\sqrt{X}}$$

$$\int \frac{dx}{X^{(2n+1)/2}} = \frac{2(2ax+b)}{(2n-1)\Delta X^{(2n-1)/2}} + \frac{2k(n-1)}{2n-1} \int \frac{dx}{X^{(2n-1)/2}}$$

$$\int \sqrt{X} \, dx = \frac{(2ax+b)\sqrt{X}}{4a} + \frac{\Delta}{8a} \int \frac{dx}{\sqrt{X}} \qquad \text{(see 241)}$$

$$\int X^{3/2} \, dx = \frac{(2ax+b)X\sqrt{X}}{8a} + \frac{3\Delta}{16a} \int \sqrt{X} \, dx \qquad \text{(see 245)}$$

$$\int X^{5/2} \, dx = \frac{(2ax+b)X^2\sqrt{X}}{12a} + \frac{5\Delta}{24a} \int X^{3/2} \, dx \qquad \text{(see 246)}$$

$$\int X^{(2n+1)/2} \, dx = \frac{(2ax+b)X^{(2n-1)/2}\sqrt{X}}{4an} + \frac{(2n-1)\Delta}{8an} \int X^{(2n-3)/2} \, dx$$

$$\int \frac{x\,dx}{\sqrt{X}} = \frac{\sqrt{X}}{a} - \frac{b}{2a} \int \frac{dx}{\sqrt{X}} \qquad \text{(see 241)}$$

$$\int \frac{x\,dx}{X^{3/2}} = -\frac{2(bx+2c)}{\Delta\sqrt{X}}$$

$$\int \frac{x\,dx}{X^{(2n+1)/2}} = -\frac{1}{(2n-1)a X^{(2n-1)/2}} - \frac{b}{2a} \int \frac{dx}{X^{(2n+1)/2}} \qquad \text{(see 244)}$$

$$\int \frac{x^2\,dx}{\sqrt{X}} = \left( \frac{x}{2a} - \frac{3b}{4a^2} \right) \sqrt{X} + \frac{3b^2 - 4ac}{8a^2} \int \frac{dx}{\sqrt{X}} \qquad \text{(see 241)}$$

$$\int \frac{x^2\,dx}{X^{3/2}} = \frac{(2b^2-4ac)x + 2bc}{a\Delta\sqrt{X}} + \frac{1}{a} \int \frac{dx}{\sqrt{X}} \qquad \text{(see 241)}$$

$$\int x\sqrt{X} \, dx = \frac{X^{3/2}}{3a} - \frac{b}{2a} \int \sqrt{X} \, dx \qquad \text{(see 245)}$$

$$\int x X^{3/2} \, dx = \frac{X^{5/2}}{5a} - \frac{b}{2a} \int X^{3/2} \, dx \qquad \text{(see 246)}$$

$$\int x X^{(2n+1)/2} \, dx = \frac{X^{(2n+3)/2}}{(2n+3)a} - \frac{b}{2a} \int X^{(2n+1)/2} \, dx \qquad \text{(see 248)}$$

$$\int x^2 \sqrt{X} \, dx = \frac{x X^{3/2}}{4a} - \frac{5b}{12a} \cdot \frac{X^{3/2}}{a} + \frac{5b^2 - 4ac}{8a^2} \int \sqrt{X} \, dx \qquad \text{(see 245)}$$

$$\int \frac{dx}{x\sqrt{X}} = \begin{cases} \displaystyle -\frac{1}{\sqrt{c}} \ln\left| \frac{2\sqrt{cX} + 2c + bx}{x} \right|, & c > 0, \\[2ex] \displaystyle \frac{1}{\sqrt{-c}} \arcsin\frac{bx+2c}{|x|\sqrt{-\Delta}}, & c < 0. \end{cases}$$

$$\int \frac{dx}{x^2\sqrt{X}} = -\frac{\sqrt{X}}{cx} - \frac{b}{2c} \int \frac{dx}{x\sqrt{X}} \qquad \text{(see 258)}$$

$$\int \frac{\sqrt{X}}{x} \, dx = \sqrt{X} + \frac{b}{2} \int \frac{dx}{\sqrt{X}} + c \int \frac{dx}{x\sqrt{X}} \qquad \text{(see 241, 258)}$$

$$\int \frac{\sqrt{X}}{x^2} \, dx = -\frac{\sqrt{X}}{x} + a \int \frac{dx}{\sqrt{X}} + \frac{b}{2} \int \frac{dx}{x\sqrt{X}} \qquad \text{(see 241, 258)}$$

$$\int \frac{X^{(2n+1)/2}}{x} \, dx = \frac{X^{(2n+1)/2}}{2n+1} + \frac{b}{2} \int X^{(2n-1)/2} \, dx + c \int \frac{X^{(2n-1)/2}}{x} \, dx$$

$$\int \frac{dx}{x\sqrt{ax^4 + bx}} = -\frac{2}{bx} \sqrt{ax^4 + bx}$$

$$\int \frac{dx}{\sqrt{2ax - x^2}} = \arcsin\frac{x-a}{a}$$

$$\int \frac{x\,dx}{\sqrt{2ax - x^2}} = -\sqrt{2ax - x^2} + a \arcsin\frac{x-a}{a}$$

$$\int \sqrt{2ax - x^2} \, dx = \frac{x-a}{2} \sqrt{2ax - x^2} + \frac{a^2}{2} \arcsin\frac{x-a}{a}$$

$$\int \frac{dx}{(ax^3 + b)\sqrt{fx^3 + g}} = \begin{cases} \displaystyle \frac{1}{\sqrt{b}\sqrt{ag-bf}} \arctan\frac{\sqrt{ag-bf}\sqrt{fx^3+g}}{\sqrt{b}\sqrt{fx^3+g} - \sqrt{g}\sqrt{ax^3+b}}, & ag>bf, \\[2ex] \displaystyle \frac{1}{2\sqrt{b}\sqrt{bf-ag}} \ln\left| \frac{\sqrt{b}\sqrt{fx^3+g} + \sqrt{g}\sqrt{ax^3+b}}{\sqrt{b}\sqrt{fx^3+g} - \sqrt{g}\sqrt{ax^3+b}} \right|, & ag<bf. \end{cases}$$

---

$$\int \sqrt[n]{ax + b} \, dx = \frac{n}{(n+1)a} (ax + b)^{\frac{n+1}{n}} + C$$

$$\int \frac{dx}{\sqrt[n]{ax + b}} = \frac{n}{(n-1)a} (ax + b)^{\frac{n-1}{n}} + C \quad (n \neq 1)$$

$$\int \frac{dx}{x \sqrt{x^n + a^n}} = -\frac{2}{na} \ln\left| \frac{a + \sqrt{x^n + a^n}}{x^{n/2}} \right| + C$$

$$\int \frac{dx}{x \sqrt{x^n - a^n}} = \frac{2}{na} \arccos\left( \frac{a}{x^{n/2}} \right) + C \quad (x > a > 0)$$

$$\int \frac{\sqrt{x} \, dx}{\sqrt{a^n - x^n}} = \frac{2}{3} \arcsin\left( \frac{x^{n/2}}{a^{n/2}} \right) + C$$

---

For the binomial integral $\displaystyle \int x^m (ax^n + b)^p \, dx$:

$$\int x^m (ax^n + b)^p \, dx = \frac{1}{m+np+1} \left[ x^{m+1} (ax^n + b)^p + npb \int x^m (ax^n + b)^{p-1} \, dx \right]$$

$$= \frac{1}{bn(p+1)} \left[ -x^{m+1} (ax^n + b)^{p+1} + (m+n+np+1) \int x^m (ax^n + b)^{p+1} \, dx \right]$$

$$= \frac{1}{(m+1)b} \left[ x^{m+1} (ax^n + b)^{p+1} - a(m+n+np+1) \int x^{m+n} (ax^n + b)^p \, dx \right]$$

$$= \frac{1}{a(m+np+1)} \left[ x^{m-n+1} (ax^n + b)^{p+1} - (m-n+1)b \int x^{m-n} (ax^n + b)^p \, dx \right]$$