Integration of functions with radicals
\(\circ\,\,\,\,\large{F\Big(x,\,\,\sqrt[n]{\frac{\alpha x+\beta}{\gamma x+\delta}}\Big)\,\,\,_{\large{n\in \mathbb{N}}}}\)
$$\text{The substitution is}\,\,t=\sqrt[n]{\frac{\alpha x+\beta}{\gamma x+\delta}}$$
\(\circ\,\,\,\,\Large{\frac{R(x)}{\sqrt{ax^2+bx+c}}}\,\,_{\normalsize{R(x)-\text{rational function}}}\)
$$R(x)=P(x)+\frac{K(x)}{Q(x)}\,\,\,,\,^{P(x)-\text{integer part}}_{\frac{K(x)}{Q(x)}-\text{fractional part}}$$
$$\frac{K(x)}{Q(x)}\,\text{is decomposable into a sum of simple fractions}\Rightarrow$$
$$\Rightarrow \int\frac{R(x)}{\sqrt{ax^2+bx+c}}=\int\frac{P(x)dx}{\sqrt{ax^2+bx+c}}+\sum_{i=1}^{n}\int\frac{A_idx}{(x-\alpha_i)^{k_i}{\sqrt{ax^2+bx+c}}}+\sum_{i=1}^{m}\int\frac{(M_ix+N_i)dx}{(x^2+p_ix+q_i)^{k_i}{\sqrt{ax^2+bx+c}}}$$
$$\text{I.}\int\frac{P(x)dx}{\sqrt{ax^2+bx+c}}= Q(x)\sqrt{ax^2+bx+c}+\gamma\int\frac{dx}{\sqrt{ax^2+bx+c}}\,\,\,\,_{\normalsize{Q(x)'s\,\text{multiplicity}=n-1}}^{\normalsize{P(x)'s\,\text{multiplicity}=n}}\,\,\,(1)$$
$$\gamma\,\,\text{and}\,\,Q(x)\text{'s coffecients are being found by differentiating }\large{(1)}$$
$$\text{II.}\int\frac{Adx}{(x-\alpha)^k\sqrt{ax^2+bx+c}}$$
$$\text{The substitution is}\,\,\,t=\frac{1}{x-\alpha}$$
$$\text{III.}\int\frac{Mx+N}{(x^2+px+q)^m\sqrt{ax^2+bx+c})}dx$$
$$\longrightarrow\text{if }\,\,\,ax^2+bx+c=a(x^2+px+q):$$
$$\int\frac{Mx+N}{(x^2+px+q)^m\sqrt{ax^2+bx+c})}=\int\frac{M_1x+N_1}{(x^2+px+q)^{m+\frac{1}{2}}}dx=\int\frac{\frac{M_1}{2}(2x+p)+N_1-\frac{M_1p}{2}}{(x^2+px+q)^{m+\frac{1}{2}}}dx=$$
$$=C_1\int\frac{d(x^2+px+q)}{(x^2+px+q)^{m+\frac{1}{2}}}+\underbrace{B_1\int\frac{dx}{(x^2+px+q)^{m+\frac{1}{2}}}}_{\text{substitute}\,\,\Large{t=\frac{d}{dx}\sqrt{x^2+px+q}}}$$
$$\longrightarrow\text{if not}:\text{all }x^1\text{ from below have to be removed:}$$
$$\text{(How) The substitution: }\left[ \begin{gathered}
x=\frac{\gamma t+\lambda}{t+1}\,,\,p\ne \frac{b}{a} \\
x=t-\frac{p}{2}\,,\,p=\frac{b}{a}\\
\end{gathered}\right.$$
$$\text{Result: }$$
$$\int\frac{At+B}{(t^2+\lambda)^m\sqrt{\delta t^2+r}}dt=\underbrace{A\int\frac{tdt}{(t^2+\lambda)^m\sqrt{\delta t^2+r}}}_{\text{substitute }\Large{u=\sqrt{\delta t^2+r}} }+\underbrace{B\int\frac{dt}{(t^2+\lambda)^m\sqrt{\delta t^2+r}}}_{\text{substitute }\Large{v=\frac{d}{dx}\sqrt{
\delta t^2+r}}}$$
1
$$\int\frac{dx}{1+\sqrt{x}}$$ Answer 2
$$\int\frac{dx}{x\Big(1+2\sqrt{x}+\sqrt[3]{x}\Big)}$$ Answer 3
$$\int\frac{dx}{(3-x)\sqrt{1-x}}$$ Answer 4
$$\int\frac{dx}{(1+\sqrt[4]{x})^3\sqrt{x}}$$ Answer 5
$$\int\frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}dx$$ Answer 6
$$\int\frac{x^4dx}{\sqrt{1-x^2}}$$ Answer 7
$$\int\frac{dx}{(1+x^2)\sqrt{1-x^2}}$$ Answer 8
$$\int\frac{dx}{(x+1)\sqrt{x^2+x+1}}$$ Answer 9
$$\int\frac{xdx}{(1+x)\sqrt{1-x-x^2}}$$ Answer 10
$$\int\frac{dx}{(1-x)^2\sqrt{1-x^2}}$$ Answer 11
$$\int\frac{dx}{(1-x^4)\sqrt{1+x^2}}$$ Answer 12
$$\int\frac{dx}{x^3\sqrt{x^2+1}}$$ Answer 13
$$\int\frac{dx}{x^4\sqrt{x^2-1}}$$ Answer 14
$$\int\frac{dx}{(x-1)^3\sqrt{x^2+3x+1}}$$ Answer 15
$$\int\frac{dx}{(x+1)^3\sqrt{x^2+2x}}$$ Answer 16
$$\int\frac{\sqrt{x^2+2}}{x^2+1}dx$$ Answer 17
$$\int\frac{\sqrt{x^2+5}}{x^2+2}dx$$ Answer 18
$$\int\frac{\sqrt{x^2+2x+2}}{x}dx$$ Answer 19
$$\int\frac{\sqrt{x^2+x+1}}{(x+1)^2}dx$$ Answer 20
$$\int\frac{dx}{(x^2+x)^\frac{3}{2}}$$ Answer 21
$$\int\frac{1-\sqrt{x+1}}{1+\sqrt[3]{x+1}}dx$$ Answer 22
$$\int\frac{x^2}{\sqrt{1+x+x^2}}dx$$ Answer 23
$$\int\frac{1-x+x^2}{\sqrt{1+x-x^2}}dx$$ Answer 24
$$\int\frac{x^3}{\sqrt{1+2x-x^2}}dx$$ Answer 25
$$\int\frac{x^3-6x^2+11x-6}{\sqrt{x^2+4x+3}}dx$$ Answer 26
$$\int\frac{x^4-5x^3+6x-7}{\sqrt{x^2+2x+3}}dx$$ Answer 27
$$\int\frac{x^{10}dx}{\sqrt{1+x^2}}$$ Answer 28
$$\int x^4\sqrt{a^2-x^2}dx$$ Answer 29
$$\int\frac{xdx}{(x-1)^2\sqrt{1+2x-x^2}}$$ Answer 30
$$\int\frac{xdx}{(x^2-3x+2)\sqrt{x^2-4x+3}}$$ Answer 31
$$\int\frac{xdx}{(x^2-1)\sqrt{x^2-x-1}}$$ Answer 32
$$\int\frac{dx}{(x^2+2)\sqrt{2x^2-2x+5}}$$ Answer 33
$$\int\frac{x^2+1}{(x^2-1)\sqrt{x^2-2x-1}}dx$$ Answer 34
$$\int\frac{x^3}{(1+x)\sqrt{x+2x-x^2}}dx$$ Answer 35
$$\int\frac{dx}{x-\sqrt{x^2-x-2}}$$ Answer 36
$$\int\frac{dx}{(x^2+x+1)\sqrt{x^2+x-1}}$$ Answer 37
$$\int\frac{x^2dx}{(4-2x+x^2)\sqrt{x+2x-x^2}}$$ Answer 38
$$\int\frac{(x+1)dx}{(x^2+x+1)\sqrt{x^2+x+1}}$$ Answer 39
$$\int\frac{dx}{(x^2-x+1)\sqrt{x^2+x+1}}$$ Answer 40
$$\int(x+1)\sqrt{x^2+4x+1}dx$$ Answer 41
$$\int\frac{x^2-1}{x\sqrt{1+3x^2+x^4}}dx$$ Answer 42
$$\int\frac{x^2+3x+1}{(x^2+2x-1)^\frac{5}{2}}dx$$ Answer 43
$$\int\frac{dx}{\sqrt{x}+\sqrt[3]{x}}$$ Answer 44
$$\int\frac{dx}{\sqrt[3]{x(x+1)^2}}$$ Answer 45
$$\int\sqrt[3]{\frac{x+1}{x-1}}dx$$ Answer 46
$$\int\frac{dx}{\sqrt[3]{(x+1)^2(x-1)^4}}$$ Answer 47
$$\int\frac{dx}{\sqrt[3]{(x+1)^2(x-1)^7}}$$ Answer 48
$$\int\frac{xdx}{\sqrt[4]{x^3(a-x)}}\,\,\,\small{(a>0)}$$ Answer 49
$$\int\frac{dx}{\sqrt[n]{(x-a)^{n+1}(x-b)^{n-1})}}\,\,\small{n\in \mathbb{N}}$$ Answer 50
$$\int\frac{dx}{(x-1)\sqrt[3]{x}}$$ Answer 51
$$\int\frac{x\sqrt[3]{2+x}}{x+\sqrt[3]{2+x}}dx$$ Answer 52
$$\int\frac{dx}{1+\sqrt{x}+\sqrt{1+x}}$$ Answer 53
$$\int\frac{dx}{x+\sqrt{x^2+x+1}}$$ Answer 54
$$\int\frac{dx}{1+\sqrt{1-2x-x^2}}$$ Answer 55
$$\int x\sqrt{x^2-2x+2}dx$$ Answer 56
$$\int\frac{dx}{\sqrt{2x-1}-\sqrt[4]{2x-1}}$$ Answer 57
$$\int x\sqrt{\frac{x+2}{x-3}}dx$$ Answer 58
$$\int\frac{1+\sqrt[6]{x}}{(\sqrt[3]{x}-\sqrt[4]{x})\sqrt[4]{x^3}}dx$$ Answer 59
$$\int\frac{x-\sqrt{x^2+3x+2}}{x+\sqrt{x^2+3x+2}}dx$$ Answer 60
$$\int\frac{dx}{(1+\sqrt{x^2+x}\big)^2}$$ Answer 61
$$\int\frac{dx}{\sqrt{x^2+1}-\sqrt{x^2-1}}$$ Answer 62
$$\int\frac{xdx}{(1-x^3)\sqrt{1-x^2}}$$ Answer 63
$$\int\frac{dx}{\sqrt2+\sqrt{1-x}+\sqrt{1+x}}$$ Answer 64
$$\int\frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}dx$$ Answer 65
$$\int\frac{\sqrt{x(x+1)}}{\sqrt{x}+\sqrt{x+1}}dx$$ Answer 66
$$\int\frac{(x^2-1)dx}{(x^2+1)\sqrt{x^4+1}}$$ Answer 67
$$\int\frac{(x^2+1)dx}{(x^2-1)\sqrt{x^4+1}}$$ Answer 68
$$\int\frac{dx}{x\sqrt{x^4+2x^2-1}}$$ Answer 69
$$\int\frac{(x^2+1)dx}{x\sqrt{x^4+x^2+1}}$$ Answer 70
$$\int\sqrt{x^3+x^4}dx$$ Answer 71
$$\int\frac{\sqrt{x}}{(1+\sqrt[3]{x})^2}dx$$ Answer 72
$$\int\frac{xdx}{\sqrt{1+\sqrt[3]{x^2}}}$$ Answer 73
$$\int\frac{x^5dx}{\sqrt{1-x^2}}$$ Answer 74
$$\int\frac{dx}{\sqrt[3]{1+x^3}}$$ Answer 75
$$\int\frac{dx}{\sqrt[4]{1+x^4}}$$ Answer 76
$$\int\frac{dx}{x\sqrt[6]{1+x^6}}$$ Answer 77
$$\int\frac{dx}{x^3\sqrt[5]{1+\frac{1}{x}}}$$ Answer 78
$$\int\sqrt[3]{3x-x^3}dx$$ Answer
