1,0,0,127,3.8141874,"\int x^3 (A+B x) \sqrt{a+b x^2} \, dx","IntegrateAlgebraic[x^3*(A + B*x)*Sqrt[a + b*x^2],x]","\int x^3 (A+B x) \sqrt{a+b x^2} \, dx","\frac{a^3 B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{5/2}}+\frac{a^2 B x \sqrt{a+b x^2}}{16 b^2}-\frac{a \left(a+b x^2\right)^{3/2} (16 A+15 B x)}{120 b^2}+\frac{A x^2 \left(a+b x^2\right)^{3/2}}{5 b}+\frac{B x^3 \left(a+b x^2\right)^{3/2}}{6 b}",1,"Could not integrate","F",-1
2,0,0,104,3.4154358,"\int x^2 (A+B x) \sqrt{a+b x^2} \, dx","IntegrateAlgebraic[x^2*(A + B*x)*Sqrt[a + b*x^2],x]","\int x^2 (A+B x) \sqrt{a+b x^2} \, dx","-\frac{a^2 A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{3/2}}-\frac{\left(a+b x^2\right)^{3/2} (8 a B-15 A b x)}{60 b^2}-\frac{a A x \sqrt{a+b x^2}}{8 b}+\frac{B x^2 \left(a+b x^2\right)^{3/2}}{5 b}",1,"Could not integrate","F",-1
3,0,0,80,3.4494979,"\int x (A+B x) \sqrt{a+b x^2} \, dx","IntegrateAlgebraic[x*(A + B*x)*Sqrt[a + b*x^2],x]","\int x (A+B x) \sqrt{a+b x^2} \, dx","-\frac{a^2 B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{3/2}}+\frac{\left(a+b x^2\right)^{3/2} (4 A+3 B x)}{12 b}-\frac{a B x \sqrt{a+b x^2}}{8 b}",1,"Could not integrate","F",-1
4,0,0,67,3.34665,"\int (A+B x) \sqrt{a+b x^2} \, dx","IntegrateAlgebraic[(A + B*x)*Sqrt[a + b*x^2],x]","\int (A+B x) \sqrt{a+b x^2} \, dx","\frac{1}{2} A x \sqrt{a+b x^2}+\frac{a A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 \sqrt{b}}+\frac{B \left(a+b x^2\right)^{3/2}}{3 b}",1,"Could not integrate","F",-1
5,0,0,79,3.6868059,"\int \frac{(A+B x) \sqrt{a+b x^2}}{x} \, dx","IntegrateAlgebraic[((A + B*x)*Sqrt[a + b*x^2])/x,x]","\int \frac{(A+B x) \sqrt{a+b x^2}}{x} \, dx","\frac{1}{2} \sqrt{a+b x^2} (2 A+B x)-\sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{a B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 \sqrt{b}}",1,"Could not integrate","F",-1
6,0,0,75,4.2756488,"\int \frac{(A+B x) \sqrt{a+b x^2}}{x^2} \, dx","IntegrateAlgebraic[((A + B*x)*Sqrt[a + b*x^2])/x^2,x]","\int \frac{(A+B x) \sqrt{a+b x^2}}{x^2} \, dx","-\frac{\sqrt{a+b x^2} (A-B x)}{x}+A \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\sqrt{a} B \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)",1,"Could not integrate","F",-1
7,0,0,80,4.1331093,"\int \frac{(A+B x) \sqrt{a+b x^2}}{x^3} \, dx","IntegrateAlgebraic[((A + B*x)*Sqrt[a + b*x^2])/x^3,x]","\int \frac{(A+B x) \sqrt{a+b x^2}}{x^3} \, dx","-\frac{\sqrt{a+b x^2} (A+2 B x)}{2 x^2}-\frac{A b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 \sqrt{a}}+\sqrt{b} B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)",1,"Could not integrate","F",-1
8,0,0,150,5.0485711,"\int x^3 (A+B x) \left(a+b x^2\right)^{3/2} \, dx","IntegrateAlgebraic[x^3*(A + B*x)*(a + b*x^2)^(3/2),x]","\int x^3 (A+B x) \left(a+b x^2\right)^{3/2} \, dx","\frac{3 a^4 B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{5/2}}+\frac{3 a^3 B x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 B x \left(a+b x^2\right)^{3/2}}{64 b^2}-\frac{a \left(a+b x^2\right)^{5/2} (32 A+35 B x)}{560 b^2}+\frac{A x^2 \left(a+b x^2\right)^{5/2}}{7 b}+\frac{B x^3 \left(a+b x^2\right)^{5/2}}{8 b}",1,"Could not integrate","F",-1
9,0,0,127,4.5644605,"\int x^2 (A+B x) \left(a+b x^2\right)^{3/2} \, dx","IntegrateAlgebraic[x^2*(A + B*x)*(a + b*x^2)^(3/2),x]","\int x^2 (A+B x) \left(a+b x^2\right)^{3/2} \, dx","-\frac{a^3 A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{3/2}}-\frac{a^2 A x \sqrt{a+b x^2}}{16 b}-\frac{\left(a+b x^2\right)^{5/2} (12 a B-35 A b x)}{210 b^2}-\frac{a A x \left(a+b x^2\right)^{3/2}}{24 b}+\frac{B x^2 \left(a+b x^2\right)^{5/2}}{7 b}",1,"Could not integrate","F",-1
10,0,0,103,4.4731035,"\int x (A+B x) \left(a+b x^2\right)^{3/2} \, dx","IntegrateAlgebraic[x*(A + B*x)*(a + b*x^2)^(3/2),x]","\int x (A+B x) \left(a+b x^2\right)^{3/2} \, dx","-\frac{a^3 B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{3/2}}-\frac{a^2 B x \sqrt{a+b x^2}}{16 b}+\frac{\left(a+b x^2\right)^{5/2} (6 A+5 B x)}{30 b}-\frac{a B x \left(a+b x^2\right)^{3/2}}{24 b}",1,"Could not integrate","F",-1
11,0,0,87,4.4790986,"\int (A+B x) \left(a+b x^2\right)^{3/2} \, dx","IntegrateAlgebraic[(A + B*x)*(a + b*x^2)^(3/2),x]","\int (A+B x) \left(a+b x^2\right)^{3/2} \, dx","\frac{3 a^2 A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 \sqrt{b}}+\frac{1}{4} A x \left(a+b x^2\right)^{3/2}+\frac{3}{8} a A x \sqrt{a+b x^2}+\frac{B \left(a+b x^2\right)^{5/2}}{5 b}",1,"Could not integrate","F",-1
12,0,0,106,4.4348763,"\int \frac{(A+B x) \left(a+b x^2\right)^{3/2}}{x} \, dx","IntegrateAlgebraic[((A + B*x)*(a + b*x^2)^(3/2))/x,x]","\int \frac{(A+B x) \left(a+b x^2\right)^{3/2}}{x} \, dx","-a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{3 a^2 B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 \sqrt{b}}+\frac{1}{8} a \sqrt{a+b x^2} (8 A+3 B x)+\frac{1}{12} \left(a+b x^2\right)^{3/2} (4 A+3 B x)",1,"Could not integrate","F",-1
13,0,0,108,4.4257269,"\int \frac{(A+B x) \left(a+b x^2\right)^{3/2}}{x^2} \, dx","IntegrateAlgebraic[((A + B*x)*(a + b*x^2)^(3/2))/x^2,x]","\int \frac{(A+B x) \left(a+b x^2\right)^{3/2}}{x^2} \, dx","a^{3/2} (-B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)-\frac{\left(a+b x^2\right)^{3/2} (3 A-B x)}{3 x}+\frac{1}{2} \sqrt{a+b x^2} (2 a B+3 A b x)+\frac{3}{2} a A \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)",1,"Could not integrate","F",-1
14,0,0,111,4.4686857,"\int \frac{(A+B x) \left(a+b x^2\right)^{3/2}}{x^3} \, dx","IntegrateAlgebraic[((A + B*x)*(a + b*x^2)^(3/2))/x^3,x]","\int \frac{(A+B x) \left(a+b x^2\right)^{3/2}}{x^3} \, dx","-\frac{\left(a+b x^2\right)^{3/2} (A-B x)}{2 x^2}-\frac{3 \sqrt{a+b x^2} (a B-A b x)}{2 x}-\frac{3}{2} \sqrt{a} A b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{3}{2} a \sqrt{b} B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)",1,"Could not integrate","F",-1
15,0,0,173,4.9931867,"\int x^3 (A+B x) \left(a+b x^2\right)^{5/2} \, dx","IntegrateAlgebraic[x^3*(A + B*x)*(a + b*x^2)^(5/2),x]","\int x^3 (A+B x) \left(a+b x^2\right)^{5/2} \, dx","\frac{3 a^5 B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 b^{5/2}}+\frac{3 a^4 B x \sqrt{a+b x^2}}{256 b^2}+\frac{a^3 B x \left(a+b x^2\right)^{3/2}}{128 b^2}+\frac{a^2 B x \left(a+b x^2\right)^{5/2}}{160 b^2}-\frac{a \left(a+b x^2\right)^{7/2} (160 A+189 B x)}{5040 b^2}+\frac{A x^2 \left(a+b x^2\right)^{7/2}}{9 b}+\frac{B x^3 \left(a+b x^2\right)^{7/2}}{10 b}",1,"Could not integrate","F",-1
16,0,0,150,4.704581,"\int x^2 (A+B x) \left(a+b x^2\right)^{5/2} \, dx","IntegrateAlgebraic[x^2*(A + B*x)*(a + b*x^2)^(5/2),x]","\int x^2 (A+B x) \left(a+b x^2\right)^{5/2} \, dx","-\frac{5 a^4 A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{3/2}}-\frac{5 a^3 A x \sqrt{a+b x^2}}{128 b}-\frac{5 a^2 A x \left(a+b x^2\right)^{3/2}}{192 b}-\frac{\left(a+b x^2\right)^{7/2} (16 a B-63 A b x)}{504 b^2}-\frac{a A x \left(a+b x^2\right)^{5/2}}{48 b}+\frac{B x^2 \left(a+b x^2\right)^{7/2}}{9 b}",1,"Could not integrate","F",-1
17,0,0,126,4.6493007,"\int x (A+B x) \left(a+b x^2\right)^{5/2} \, dx","IntegrateAlgebraic[x*(A + B*x)*(a + b*x^2)^(5/2),x]","\int x (A+B x) \left(a+b x^2\right)^{5/2} \, dx","-\frac{5 a^4 B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{3/2}}-\frac{5 a^3 B x \sqrt{a+b x^2}}{128 b}-\frac{5 a^2 B x \left(a+b x^2\right)^{3/2}}{192 b}+\frac{\left(a+b x^2\right)^{7/2} (8 A+7 B x)}{56 b}-\frac{a B x \left(a+b x^2\right)^{5/2}}{48 b}",1,"Could not integrate","F",-1
18,0,0,107,4.6219881,"\int (A+B x) \left(a+b x^2\right)^{5/2} \, dx","IntegrateAlgebraic[(A + B*x)*(a + b*x^2)^(5/2),x]","\int (A+B x) \left(a+b x^2\right)^{5/2} \, dx","\frac{5 a^3 A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 \sqrt{b}}+\frac{5}{16} a^2 A x \sqrt{a+b x^2}+\frac{1}{6} A x \left(a+b x^2\right)^{5/2}+\frac{5}{24} a A x \left(a+b x^2\right)^{3/2}+\frac{B \left(a+b x^2\right)^{7/2}}{7 b}",1,"Could not integrate","F",-1
19,0,0,132,4.7072587,"\int \frac{(A+B x) \left(a+b x^2\right)^{5/2}}{x} \, dx","IntegrateAlgebraic[((A + B*x)*(a + b*x^2)^(5/2))/x,x]","\int \frac{(A+B x) \left(a+b x^2\right)^{5/2}}{x} \, dx","-a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{5 a^3 B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 \sqrt{b}}+\frac{1}{16} a^2 \sqrt{a+b x^2} (16 A+5 B x)+\frac{1}{24} a \left(a+b x^2\right)^{3/2} (8 A+5 B x)+\frac{1}{30} \left(a+b x^2\right)^{5/2} (6 A+5 B x)",1,"Could not integrate","F",-1
20,0,0,136,4.6936594,"\int \frac{(A+B x) \left(a+b x^2\right)^{5/2}}{x^2} \, dx","IntegrateAlgebraic[((A + B*x)*(a + b*x^2)^(5/2))/x^2,x]","\int \frac{(A+B x) \left(a+b x^2\right)^{5/2}}{x^2} \, dx","a^{5/2} (-B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{15}{8} a^2 A \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\frac{1}{8} a \sqrt{a+b x^2} (8 a B+15 A b x)-\frac{\left(a+b x^2\right)^{5/2} (5 A-B x)}{5 x}+\frac{1}{12} \left(a+b x^2\right)^{3/2} (4 a B+15 A b x)",1,"Could not integrate","F",-1
21,0,0,141,4.7305662,"\int \frac{(A+B x) \left(a+b x^2\right)^{5/2}}{x^3} \, dx","IntegrateAlgebraic[((A + B*x)*(a + b*x^2)^(5/2))/x^3,x]","\int \frac{(A+B x) \left(a+b x^2\right)^{5/2}}{x^3} \, dx","-\frac{5}{2} a^{3/2} A b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{15}{8} a^2 \sqrt{b} B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\frac{\left(a+b x^2\right)^{5/2} (2 A-B x)}{4 x^2}-\frac{5 \left(a+b x^2\right)^{3/2} (3 a B-2 A b x)}{12 x}+\frac{5}{8} a b \sqrt{a+b x^2} (4 A+3 B x)",1,"Could not integrate","F",-1
22,0,0,104,3.1859359,"\int \frac{x^3 (A+B x)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x^3*(A + B*x))/Sqrt[a + b*x^2],x]","\int \frac{x^3 (A+B x)}{\sqrt{a+b x^2}} \, dx","\frac{3 a^2 B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{5/2}}-\frac{a \sqrt{a+b x^2} (16 A+9 B x)}{24 b^2}+\frac{A x^2 \sqrt{a+b x^2}}{3 b}+\frac{B x^3 \sqrt{a+b x^2}}{4 b}",1,"Could not integrate","F",-1
23,0,0,81,3.6105798,"\int \frac{x^2 (A+B x)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x^2*(A + B*x))/Sqrt[a + b*x^2],x]","\int \frac{x^2 (A+B x)}{\sqrt{a+b x^2}} \, dx","-\frac{a A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}-\frac{\sqrt{a+b x^2} (4 a B-3 A b x)}{6 b^2}+\frac{B x^2 \sqrt{a+b x^2}}{3 b}",1,"Could not integrate","F",-1
24,0,0,56,3.4795202,"\int \frac{x (A+B x)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x*(A + B*x))/Sqrt[a + b*x^2],x]","\int \frac{x (A+B x)}{\sqrt{a+b x^2}} \, dx","\frac{\sqrt{a+b x^2} (2 A+B x)}{2 b}-\frac{a B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}",1,"Could not integrate","F",-1
25,0,0,43,4.1379242,"\int \frac{A+B x}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x)/Sqrt[a + b*x^2],x]","\int \frac{A+B x}{\sqrt{a+b x^2}} \, dx","\frac{A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}+\frac{B \sqrt{a+b x^2}}{b}",1,"Could not integrate","F",-1
26,0,0,53,4.6945508,"\int \frac{A+B x}{x \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x)/(x*Sqrt[a + b*x^2]),x]","\int \frac{A+B x}{x \sqrt{a+b x^2}} \, dx","\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"Could not integrate","F",-1
27,0,0,47,6.3372177,"\int \frac{A+B x}{x^2 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x)/(x^2*Sqrt[a + b*x^2]),x]","\int \frac{A+B x}{x^2 \sqrt{a+b x^2}} \, dx","-\frac{A \sqrt{a+b x^2}}{a x}-\frac{B \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"Could not integrate","F",-1
28,0,0,72,5.5491115,"\int \frac{A+B x}{x^3 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(A + B*x)/(x^3*Sqrt[a + b*x^2]),x]","\int \frac{A+B x}{x^3 \sqrt{a+b x^2}} \, dx","\frac{A b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{3/2}}-\frac{A \sqrt{a+b x^2}}{2 a x^2}-\frac{B \sqrt{a+b x^2}}{a x}",1,"Could not integrate","F",-1
29,0,0,81,4.784522,"\int \frac{x^3 (A+B x)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x^3*(A + B*x))/(a + b*x^2)^(3/2),x]","\int \frac{x^3 (A+B x)}{\left(a+b x^2\right)^{3/2}} \, dx","\frac{\sqrt{a+b x^2} (4 A+3 B x)}{2 b^2}-\frac{x^2 (A+B x)}{b \sqrt{a+b x^2}}-\frac{3 a B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{5/2}}",1,"Could not integrate","F",-1
30,0,0,66,4.3829879,"\int \frac{x^2 (A+B x)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x^2*(A + B*x))/(a + b*x^2)^(3/2),x]","\int \frac{x^2 (A+B x)}{\left(a+b x^2\right)^{3/2}} \, dx","\frac{A \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{3/2}}-\frac{x (A+B x)}{b \sqrt{a+b x^2}}+\frac{2 B \sqrt{a+b x^2}}{b^2}",1,"Could not integrate","F",-1
31,0,0,48,4.5212016,"\int \frac{x (A+B x)}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(x*(A + B*x))/(a + b*x^2)^(3/2),x]","\int \frac{x (A+B x)}{\left(a+b x^2\right)^{3/2}} \, dx","\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{3/2}}-\frac{A+B x}{b \sqrt{a+b x^2}}",1,"Could not integrate","F",-1
32,1,27,28,3.5425994,"\int \frac{A+B x}{\left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x)/(a + b*x^2)^(3/2),x]","\frac{A b x-a B}{a b \sqrt{a+b x^2}}","-\frac{a B-A b x}{a b \sqrt{a+b x^2}}",1,"(-(a*B) + A*b*x)/(a*b*Sqrt[a + b*x^2])","A",1
33,0,0,47,4.4915743,"\int \frac{A+B x}{x \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x)/(x*(a + b*x^2)^(3/2)),x]","\int \frac{A+B x}{x \left(a+b x^2\right)^{3/2}} \, dx","\frac{A+B x}{a \sqrt{a+b x^2}}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{3/2}}",1,"Could not integrate","F",-1
34,0,0,70,5.4367294,"\int \frac{A+B x}{x^2 \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x)/(x^2*(a + b*x^2)^(3/2)),x]","\int \frac{A+B x}{x^2 \left(a+b x^2\right)^{3/2}} \, dx","-\frac{B \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{2 A \sqrt{a+b x^2}}{a^2 x}+\frac{A+B x}{a x \sqrt{a+b x^2}}",1,"Could not integrate","F",-1
35,0,0,95,5.8232509,"\int \frac{A+B x}{x^3 \left(a+b x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x)/(x^3*(a + b*x^2)^(3/2)),x]","\int \frac{A+B x}{x^3 \left(a+b x^2\right)^{3/2}} \, dx","\frac{3 A b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{3 A \sqrt{a+b x^2}}{2 a^2 x^2}-\frac{2 B \sqrt{a+b x^2}}{a^2 x}+\frac{A+B x}{a x^2 \sqrt{a+b x^2}}",1,"Could not integrate","F",-1
36,0,0,79,5.0000017,"\int \frac{x^3 (A+B x)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x^3*(A + B*x))/(a + b*x^2)^(5/2),x]","\int \frac{x^3 (A+B x)}{\left(a+b x^2\right)^{5/2}} \, dx","-\frac{2 A+3 B x}{3 b^2 \sqrt{a+b x^2}}-\frac{x^2 (A+B x)}{3 b \left(a+b x^2\right)^{3/2}}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{5/2}}",1,"Could not integrate","F",-1
37,0,0,53,4.5401969,"\int \frac{x^2 (A+B x)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x^2*(A + B*x))/(a + b*x^2)^(5/2),x]","\int \frac{x^2 (A+B x)}{\left(a+b x^2\right)^{5/2}} \, dx","-\frac{x^2 (a B-A b x)}{3 a b \left(a+b x^2\right)^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^2}}",1,"Could not integrate","F",-1
38,0,0,50,4.4808189,"\int \frac{x (A+B x)}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(x*(A + B*x))/(a + b*x^2)^(5/2),x]","\int \frac{x (A+B x)}{\left(a+b x^2\right)^{5/2}} \, dx","\frac{-A-B x}{3 b \left(a+b x^2\right)^{3/2}}+\frac{B x}{3 a b \sqrt{a+b x^2}}",1,"Could not integrate","F",-1
39,0,0,51,4.5288225,"\int \frac{A+B x}{\left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x)/(a + b*x^2)^(5/2),x]","\int \frac{A+B x}{\left(a+b x^2\right)^{5/2}} \, dx","\frac{2 A x}{3 a^2 \sqrt{a+b x^2}}+\frac{A b x-a B}{3 a b \left(a+b x^2\right)^{3/2}}",1,"Could not integrate","F",-1
40,0,0,76,4.713298,"\int \frac{A+B x}{x \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x)/(x*(a + b*x^2)^(5/2)),x]","\int \frac{A+B x}{x \left(a+b x^2\right)^{5/2}} \, dx","-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{3 A+2 B x}{3 a^2 \sqrt{a+b x^2}}+\frac{A+B x}{3 a \left(a+b x^2\right)^{3/2}}",1,"Could not integrate","F",-1
41,0,0,104,5.8698621,"\int \frac{A+B x}{x^2 \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x)/(x^2*(a + b*x^2)^(5/2)),x]","\int \frac{A+B x}{x^2 \left(a+b x^2\right)^{5/2}} \, dx","-\frac{B \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{5/2}}-\frac{8 A \sqrt{a+b x^2}}{3 a^3 x}+\frac{4 A+3 B x}{3 a^2 x \sqrt{a+b x^2}}+\frac{A+B x}{3 a x \left(a+b x^2\right)^{3/2}}",1,"Could not integrate","F",-1
42,0,0,129,5.520484,"\int \frac{A+B x}{x^3 \left(a+b x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x)/(x^3*(a + b*x^2)^(5/2)),x]","\int \frac{A+B x}{x^3 \left(a+b x^2\right)^{5/2}} \, dx","\frac{5 A b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{7/2}}-\frac{5 A \sqrt{a+b x^2}}{2 a^3 x^2}-\frac{8 B \sqrt{a+b x^2}}{3 a^3 x}+\frac{5 A+4 B x}{3 a^2 x^2 \sqrt{a+b x^2}}+\frac{A+B x}{3 a x^2 \left(a+b x^2\right)^{3/2}}",1,"Could not integrate","F",-1
43,1,37,27,0.1184821,"\int \frac{(1-x) x}{\sqrt{1-x^2}} \, dx","IntegrateAlgebraic[((1 - x)*x)/Sqrt[1 - x^2],x]","\frac{1}{2} \sqrt{1-x^2} (x-2)+\tan ^{-1}\left(\frac{\sqrt{1-x^2}}{x+1}\right)","-\frac{1}{2} \sqrt{1-x^2} (2-x)-\frac{1}{2} \sin ^{-1}(x)",1,"((-2 + x)*Sqrt[1 - x^2])/2 + ArcTan[Sqrt[1 - x^2]/(1 + x)]","A",1
44,1,37,27,0.1068144,"\int \frac{x-x^2}{\sqrt{1-x^2}} \, dx","IntegrateAlgebraic[(x - x^2)/Sqrt[1 - x^2],x]","\frac{1}{2} \sqrt{1-x^2} (x-2)+\tan ^{-1}\left(\frac{\sqrt{1-x^2}}{x+1}\right)","-\frac{1}{2} \sqrt{1-x^2} (2-x)-\frac{1}{2} \sin ^{-1}(x)",1,"((-2 + x)*Sqrt[1 - x^2])/2 + ArcTan[Sqrt[1 - x^2]/(1 + x)]","A",1
45,1,17,17,0.0196104,"\int \frac{3+x^2}{-3+x^2} \, dx","IntegrateAlgebraic[(3 + x^2)/(-3 + x^2),x]","x-2 \sqrt{3} \tanh ^{-1}\left(\frac{x}{\sqrt{3}}\right)","x-2 \sqrt{3} \tanh ^{-1}\left(\frac{x}{\sqrt{3}}\right)",1,"x - 2*Sqrt[3]*ArcTanh[x/Sqrt[3]]","A",1
46,1,6,6,0.0068818,"\int \frac{-1+x^2}{1+x^2} \, dx","IntegrateAlgebraic[(-1 + x^2)/(1 + x^2),x]","x-2 \tan ^{-1}(x)","x-2 \tan ^{-1}(x)",1,"x - 2*ArcTan[x]","A",1
47,0,0,213,7.8674831,"\int \frac{x^7 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^7*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\int \frac{x^7 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","-\frac{16 \sqrt{a+b x^2} (A b-8 a C)}{35 a b^5}-\frac{x (35 a B-8 x (A b-8 a C))}{35 a b^4 \sqrt{a+b x^2}}-\frac{x^3 (35 a B-6 x (A b-8 a C))}{105 a b^3 \left(a+b x^2\right)^{3/2}}-\frac{x^5 (7 a B-x (A b-8 a C))}{35 a b^2 \left(a+b x^2\right)^{5/2}}-\frac{x^7 (a B-x (A b-a C))}{7 a b \left(a+b x^2\right)^{7/2}}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{9/2}}",1,"Could not integrate","F",-1
48,0,0,150,7.5868127,"\int \frac{x^6 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^6*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\int \frac{x^6 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","-\frac{x^6 (a B-x (A b-a C))}{7 a b \left(a+b x^2\right)^{7/2}}+\frac{C \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{9/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left(a+b x^2\right)^{3/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left(a+b x^2\right)^{5/2}}",1,"Could not integrate","F",-1
49,0,0,132,7.7254214,"\int \frac{x^5 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^5*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\int \frac{x^5 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","-\frac{4 (6 a C+A b)}{35 a b^4 \sqrt{a+b x^2}}+\frac{4 (6 a C+A b)}{105 b^4 \left(a+b x^2\right)^{3/2}}-\frac{x^4 (6 a C+A b-5 b B x)}{35 a b^2 \left(a+b x^2\right)^{5/2}}-\frac{x^5 (a B-x (A b-a C))}{7 a b \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
50,0,0,149,6.8532011,"\int \frac{x^4 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^4*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\int \frac{x^4 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{x (5 a C+2 A b)}{35 a^2 b^3 \sqrt{a+b x^2}}-\frac{3 x (5 a C+2 A b)+8 a B}{105 a b^3 \left(a+b x^2\right)^{3/2}}-\frac{x^2 (x (5 a C+2 A b)+4 a B)}{35 a b^2 \left(a+b x^2\right)^{5/2}}-\frac{x^4 (a B-x (A b-a C))}{7 a b \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
51,0,0,139,6.8147601,"\int \frac{x^3 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^3*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\int \frac{x^3 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{2 B x}{35 a^2 b^2 \sqrt{a+b x^2}}-\frac{2 (4 a C+3 A b)-3 b B x}{105 a b^3 \left(a+b x^2\right)^{3/2}}-\frac{x (x (4 a C+3 A b)+3 a B)}{35 a b^2 \left(a+b x^2\right)^{5/2}}-\frac{x^3 (a B-x (A b-a C))}{7 a b \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
52,0,0,139,6.2510818,"\int \frac{x^2 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^2*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\int \frac{x^2 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{2 x (3 a C+4 A b)}{105 a^3 b^2 \sqrt{a+b x^2}}+\frac{x (3 a C+4 A b)}{105 a^2 b^2 \left(a+b x^2\right)^{3/2}}-\frac{x (3 a C+4 A b)+2 a B}{35 a b^2 \left(a+b x^2\right)^{5/2}}-\frac{x^2 (a B-x (A b-a C))}{7 a b \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
53,0,0,119,5.7879708,"\int \frac{x \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\int \frac{x \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{8 B x}{105 a^3 b \sqrt{a+b x^2}}+\frac{4 B x}{105 a^2 b \left(a+b x^2\right)^{3/2}}-\frac{2 a C+5 A b-b B x}{35 a b^2 \left(a+b x^2\right)^{5/2}}-\frac{x (a B-x (A b-a C))}{7 a b \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
54,0,0,127,6.3101809,"\int \frac{A+B x+C x^2}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2)/(a + b*x^2)^(9/2),x]","\int \frac{A+B x+C x^2}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{8 x (a C+6 A b)}{105 a^4 b \sqrt{a+b x^2}}+\frac{4 x (a C+6 A b)}{105 a^3 b \left(a+b x^2\right)^{3/2}}+\frac{x (a C+6 A b)}{35 a^2 b \left(a+b x^2\right)^{5/2}}-\frac{a B-x (A b-a C)}{7 a b \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
55,0,0,138,6.105247,"\int \frac{A+B x+C x^2}{x \left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2)/(x*(a + b*x^2)^(9/2)),x]","\int \frac{A+B x+C x^2}{x \left(a+b x^2\right)^{9/2}} \, dx","-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{9/2}}+\frac{35 A+16 B x}{35 a^4 \sqrt{a+b x^2}}+\frac{35 A+24 B x}{105 a^3 \left(a+b x^2\right)^{3/2}}+\frac{7 A+6 B x}{35 a^2 \left(a+b x^2\right)^{5/2}}+\frac{-a C+A b+b B x}{7 a b \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
56,0,0,188,6.6547904,"\int \frac{A+B x+C x^2}{x^2 \left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2)/(x^2*(a + b*x^2)^(9/2)),x]","\int \frac{A+B x+C x^2}{x^2 \left(a+b x^2\right)^{9/2}} \, dx","-\frac{B \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{9/2}}-\frac{A \sqrt{a+b x^2}}{a^5 x}+\frac{35 B-x \left(\frac{93 A b}{a}-16 C\right)}{35 a^4 \sqrt{a+b x^2}}+\frac{35 B-3 x \left(\frac{29 A b}{a}-8 C\right)}{105 a^3 \left(a+b x^2\right)^{3/2}}+\frac{7 B-x \left(\frac{13 A b}{a}-6 C\right)}{35 a^2 \left(a+b x^2\right)^{5/2}}+\frac{B-x \left(\frac{A b}{a}-C\right)}{7 a \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
57,0,0,219,6.5254029,"\int \frac{A+B x+C x^2}{x^3 \left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2)/(x^3*(a + b*x^2)^(9/2)),x]","\int \frac{A+B x+C x^2}{x^3 \left(a+b x^2\right)^{9/2}} \, dx","\frac{(9 A b-2 a C) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{11/2}}-\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{2 a^5 x^2}-\frac{B \sqrt{a+b x^2}}{a^5 x}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left(a+b x^2\right)^{3/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left(a+b x^2\right)^{5/2}}-\frac{a \left(\frac{A b}{a}-C\right)+b B x}{7 a^2 \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
58,0,0,45,0.0150602,"\int \frac{A (c x)^m}{a+b x^2} \, dx","IntegrateAlgebraic[(A*(c*x)^m)/(a + b*x^2),x]","\int \frac{A (c x)^m}{a+b x^2} \, dx","\frac{A (c x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a c (m+1)}",1,"Could not integrate","F",-1
59,0,0,91,0.0325298,"\int \frac{(c x)^m (A+B x)}{a+b x^2} \, dx","IntegrateAlgebraic[((c*x)^m*(A + B*x))/(a + b*x^2),x]","\int \frac{(c x)^m (A+B x)}{a+b x^2} \, dx","\frac{A (c x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a c (m+1)}+\frac{B (c x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right)}{a c^2 (m+2)}",1,"Could not integrate","F",-1
60,0,0,76,0.0264265,"\int \frac{(c x)^m \left(A+C x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[((c*x)^m*(A + C*x^2))/(a + b*x^2),x]","\int \frac{(c x)^m \left(A+C x^2\right)}{a+b x^2} \, dx","\frac{(c x)^{m+1} (A b-a C) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a b c (m+1)}+\frac{C (c x)^{m+1}}{b c (m+1)}",1,"Could not integrate","F",-1
61,0,0,121,0.0353287,"\int \frac{(c x)^m \left(A+B x+C x^2\right)}{a+b x^2} \, dx","IntegrateAlgebraic[((c*x)^m*(A + B*x + C*x^2))/(a + b*x^2),x]","\int \frac{(c x)^m \left(A+B x+C x^2\right)}{a+b x^2} \, dx","\frac{(c x)^{m+1} (A b-a C) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a b c (m+1)}+\frac{B (c x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right)}{a c^2 (m+2)}+\frac{C (c x)^{m+1}}{b c (m+1)}",1,"Could not integrate","F",-1
62,1,61,65,0.0340608,"\int x^3 \left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]","\frac{630 a A x^4+504 a B x^5+420 a C x^6+360 a D x^7+420 A b x^6+360 b B x^7+315 b C x^8+280 b D x^9}{2520}","\frac{1}{6} x^6 (a C+A b)+\frac{1}{4} a A x^4+\frac{1}{7} x^7 (a D+b B)+\frac{1}{5} a B x^5+\frac{1}{8} b C x^8+\frac{1}{9} b D x^9",1,"(630*a*A*x^4 + 504*a*B*x^5 + 420*A*b*x^6 + 420*a*C*x^6 + 360*b*B*x^7 + 360*a*D*x^7 + 315*b*C*x^8 + 280*b*D*x^9)/2520","A",1
63,1,61,65,0.0326344,"\int x^2 \left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]","\frac{1}{840} \left(280 a A x^3+210 a B x^4+168 a C x^5+140 a D x^6+168 A b x^5+140 b B x^6+120 b C x^7+105 b D x^8\right)","\frac{1}{5} x^5 (a C+A b)+\frac{1}{3} a A x^3+\frac{1}{6} x^6 (a D+b B)+\frac{1}{4} a B x^4+\frac{1}{7} b C x^7+\frac{1}{8} b D x^8",1,"(280*a*A*x^3 + 210*a*B*x^4 + 168*A*b*x^5 + 168*a*C*x^5 + 140*b*B*x^6 + 140*a*D*x^6 + 120*b*C*x^7 + 105*b*D*x^8)/840","A",1
64,1,61,65,0.0228274,"\int x \left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[x*(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]","\frac{1}{420} \left(210 a A x^2+140 a B x^3+105 a C x^4+84 a D x^5+105 A b x^4+84 b B x^5+70 b C x^6+60 b D x^7\right)","\frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{5} x^5 (a D+b B)+\frac{1}{3} a B x^3+\frac{1}{6} b C x^6+\frac{1}{7} b D x^7",1,"(210*a*A*x^2 + 140*a*B*x^3 + 105*A*b*x^4 + 105*a*C*x^4 + 84*b*B*x^5 + 84*a*D*x^5 + 70*b*C*x^6 + 60*b*D*x^7)/420","A",1
65,1,59,60,0.828208,"\int \left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]","\frac{1}{60} \left(60 a A x+30 a B x^2+20 a C x^3+15 a D x^4+20 A b x^3+15 b B x^4+12 b C x^5+10 b D x^6\right)","\frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{4} x^4 (a D+b B)+\frac{1}{2} a B x^2+\frac{1}{5} b C x^5+\frac{1}{6} b D x^6",1,"(60*a*A*x + 30*a*B*x^2 + 20*A*b*x^3 + 20*a*C*x^3 + 15*b*B*x^4 + 15*a*D*x^4 + 12*b*C*x^5 + 10*b*D*x^6)/60","A",1
66,1,58,56,0.0271387,"\int \frac{\left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right)}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x + C*x^2 + D*x^3))/x,x]","\frac{1}{60} \left(60 a B x+30 a C x^2+20 a D x^3+30 A b x^2+20 b B x^3+15 b C x^4+12 b D x^5\right)+a A \log (x)","\frac{1}{2} x^2 (a C+A b)+a A \log (x)+\frac{1}{3} x^3 (a D+b B)+a B x+\frac{1}{4} b C x^4+\frac{1}{5} b D x^5",1,"(60*a*B*x + 30*A*b*x^2 + 30*a*C*x^2 + 20*b*B*x^3 + 20*a*D*x^3 + 15*b*C*x^4 + 12*b*D*x^5)/60 + a*A*Log[x]","A",1
67,1,60,54,0.0437991,"\int \frac{\left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right)}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x + C*x^2 + D*x^3))/x^2,x]","\frac{-12 a A+12 a C x^2+6 a D x^3+12 A b x^2+6 b B x^3+4 b C x^4+3 b D x^5}{12 x}+a B \log (x)","x (a C+A b)-\frac{a A}{x}+\frac{1}{2} x^2 (a D+b B)+a B \log (x)+\frac{1}{3} b C x^3+\frac{1}{4} b D x^4",1,"(-12*a*A + 12*A*b*x^2 + 12*a*C*x^2 + 6*b*B*x^3 + 6*a*D*x^3 + 4*b*C*x^4 + 3*b*D*x^5)/(12*x) + a*B*Log[x]","A",1
68,1,56,54,0.0378138,"\int \frac{\left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right)}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x + C*x^2 + D*x^3))/x^3,x]","\frac{-3 a A-6 a B x+6 a D x^3+6 b B x^3+3 b C x^4+2 b D x^5}{6 x^2}+\log (x) (a C+A b)","\log (x) (a C+A b)-\frac{a A}{2 x^2}+x (a D+b B)-\frac{a B}{x}+\frac{1}{2} b C x^2+\frac{1}{3} b D x^3",1,"(-3*a*A - 6*a*B*x + 6*b*B*x^3 + 6*a*D*x^3 + 3*b*C*x^4 + 2*b*D*x^5)/(6*x^2) + (A*b + a*C)*Log[x]","A",1
69,1,56,54,0.0307178,"\int \frac{\left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right)}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)*(A + B*x + C*x^2 + D*x^3))/x^4,x]","\frac{-2 a A-3 a B x-6 a C x^2-6 A b x^2+6 b C x^4+3 b D x^5}{6 x^3}+\log (x) (a D+b B)","-\frac{a C+A b}{x}-\frac{a A}{3 x^3}+\log (x) (a D+b B)-\frac{a B}{2 x^2}+b C x+\frac{1}{2} b D x^2",1,"(-2*a*A - 3*a*B*x - 6*A*b*x^2 - 6*a*C*x^2 + 6*b*C*x^4 + 3*b*D*x^5)/(6*x^3) + (b*B + a*D)*Log[x]","A",1
70,1,109,109,0.0686865,"\int x^3 \left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]","\frac{6930 a^2 A x^4+5544 a^2 B x^5+4620 a^2 C x^6+3960 a^2 D x^7+9240 a A b x^6+7920 a b B x^7+6930 a b C x^8+6160 a b D x^9+3465 A b^2 x^8+3080 b^2 B x^9+2772 b^2 C x^{10}+2520 b^2 D x^{11}}{27720}","\frac{1}{4} a^2 A x^4+\frac{1}{5} a^2 B x^5+\frac{1}{8} b x^8 (2 a C+A b)+\frac{1}{6} a x^6 (a C+2 A b)+\frac{1}{9} b x^9 (2 a D+b B)+\frac{1}{7} a x^7 (a D+2 b B)+\frac{1}{10} b^2 C x^{10}+\frac{1}{11} b^2 D x^{11}",1,"(6930*a^2*A*x^4 + 5544*a^2*B*x^5 + 9240*a*A*b*x^6 + 4620*a^2*C*x^6 + 7920*a*b*B*x^7 + 3960*a^2*D*x^7 + 3465*A*b^2*x^8 + 6930*a*b*C*x^8 + 3080*b^2*B*x^9 + 6160*a*b*D*x^9 + 2772*b^2*C*x^10 + 2520*b^2*D*x^11)/27720","A",1
71,1,109,109,0.0746068,"\int x^2 \left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]","\frac{840 a^2 A x^3+630 a^2 B x^4+504 a^2 C x^5+420 a^2 D x^6+1008 a A b x^5+840 a b B x^6+720 a b C x^7+630 a b D x^8+360 A b^2 x^7+315 b^2 B x^8+280 b^2 C x^9+252 b^2 D x^{10}}{2520}","\frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{1}{7} b x^7 (2 a C+A b)+\frac{1}{5} a x^5 (a C+2 A b)+\frac{1}{8} b x^8 (2 a D+b B)+\frac{1}{6} a x^6 (a D+2 b B)+\frac{1}{9} b^2 C x^9+\frac{1}{10} b^2 D x^{10}",1,"(840*a^2*A*x^3 + 630*a^2*B*x^4 + 1008*a*A*b*x^5 + 504*a^2*C*x^5 + 840*a*b*B*x^6 + 420*a^2*D*x^6 + 360*A*b^2*x^7 + 720*a*b*C*x^7 + 315*b^2*B*x^8 + 630*a*b*D*x^8 + 280*b^2*C*x^9 + 252*b^2*D*x^10)/2520","A",1
72,1,109,104,0.0694266,"\int x \left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[x*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]","\frac{1260 a^2 A x^2+840 a^2 B x^3+630 a^2 C x^4+504 a^2 D x^5+1260 a A b x^4+1008 a b B x^5+840 a b C x^6+720 a b D x^7+420 A b^2 x^6+360 b^2 B x^7+315 b^2 C x^8+280 b^2 D x^9}{2520}","\frac{1}{3} a^2 B x^3+\frac{1}{4} a^2 C x^4+\frac{A \left(a+b x^2\right)^3}{6 b}+\frac{1}{7} b x^7 (2 a D+b B)+\frac{1}{5} a x^5 (a D+2 b B)+\frac{1}{3} a b C x^6+\frac{1}{8} b^2 C x^8+\frac{1}{9} b^2 D x^9",1,"(1260*a^2*A*x^2 + 840*a^2*B*x^3 + 1260*a*A*b*x^4 + 630*a^2*C*x^4 + 1008*a*b*B*x^5 + 504*a^2*D*x^5 + 420*A*b^2*x^6 + 840*a*b*C*x^6 + 360*b^2*B*x^7 + 720*a*b*D*x^7 + 315*b^2*C*x^8 + 280*b^2*D*x^9)/2520","A",1
73,1,107,99,0.0555514,"\int \left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]","\frac{1}{840} \left(840 a^2 A x+420 a^2 B x^2+280 a^2 C x^3+210 a^2 D x^4+560 a A b x^3+420 a b B x^4+336 a b C x^5+280 a b D x^6+168 A b^2 x^5+140 b^2 B x^6+120 b^2 C x^7+105 b^2 D x^8\right)","a^2 A x+\frac{1}{4} a^2 D x^4+\frac{1}{5} b x^5 (2 a C+A b)+\frac{1}{3} a x^3 (a C+2 A b)+\frac{B \left(a+b x^2\right)^3}{6 b}+\frac{1}{3} a b D x^6+\frac{1}{7} b^2 C x^7+\frac{1}{8} b^2 D x^8",1,"(840*a^2*A*x + 420*a^2*B*x^2 + 560*a*A*b*x^3 + 280*a^2*C*x^3 + 420*a*b*B*x^4 + 210*a^2*D*x^4 + 168*A*b^2*x^5 + 336*a*b*C*x^5 + 140*b^2*B*x^6 + 280*a*b*D*x^6 + 120*b^2*C*x^7 + 105*b^2*D*x^8)/840","A",1
74,1,106,92,0.0653218,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right)}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x,x]","\frac{1}{420} \left(420 a^2 B x+210 a^2 C x^2+140 a^2 D x^3+420 a A b x^2+280 a b B x^3+210 a b C x^4+168 a b D x^5+105 A b^2 x^4+84 b^2 B x^5+70 b^2 C x^6+60 b^2 D x^7\right)+a^2 A \log (x)","a^2 A \log (x)+a^2 B x+a A b x^2+\frac{1}{5} b x^5 (2 a D+b B)+\frac{1}{3} a x^3 (a D+2 b B)+\frac{C \left(a+b x^2\right)^3}{6 b}+\frac{1}{4} A b^2 x^4+\frac{1}{7} b^2 D x^7",1,"(420*a^2*B*x + 420*a*A*b*x^2 + 210*a^2*C*x^2 + 280*a*b*B*x^3 + 140*a^2*D*x^3 + 105*A*b^2*x^4 + 210*a*b*C*x^4 + 84*b^2*B*x^5 + 168*a*b*D*x^5 + 70*b^2*C*x^6 + 60*b^2*D*x^7)/420 + a^2*A*Log[x]","A",1
75,1,108,90,0.0681728,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right)}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^2,x]","\frac{-60 a^2 A+60 a^2 C x^2+30 a^2 D x^3+120 a A b x^2+60 a b B x^3+40 a b C x^4+30 a b D x^5+20 A b^2 x^4+15 b^2 B x^5+12 b^2 C x^6+10 b^2 D x^7}{60 x}+a^2 B \log (x)","-\frac{a^2 A}{x}+a^2 B \log (x)+\frac{1}{3} b x^3 (2 a C+A b)+a x (a C+2 A b)+a b B x^2+\frac{D \left(a+b x^2\right)^3}{6 b}+\frac{1}{4} b^2 B x^4+\frac{1}{5} b^2 C x^5",1,"(-60*a^2*A + 120*a*A*b*x^2 + 60*a^2*C*x^2 + 60*a*b*B*x^3 + 30*a^2*D*x^3 + 20*A*b^2*x^4 + 40*a*b*C*x^4 + 15*b^2*B*x^5 + 30*a*b*D*x^5 + 12*b^2*C*x^6 + 10*b^2*D*x^7)/(60*x) + a^2*B*Log[x]","A",1
76,1,105,98,0.0543228,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right)}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^3,x]","\frac{-30 a^2 A-60 a^2 B x+60 a^2 D x^3+120 a b B x^3+60 a b C x^4+40 a b D x^5+30 A b^2 x^4+20 b^2 B x^5+15 b^2 C x^6+12 b^2 D x^7}{60 x^2}+\log (x) \left(a^2 C+2 a A b\right)","-\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{2} b x^2 (2 a C+A b)+a \log (x) (a C+2 A b)+\frac{1}{3} b x^3 (2 a D+b B)+a x (a D+2 b B)+\frac{1}{4} b^2 C x^4+\frac{1}{5} b^2 D x^5",1,"(-30*a^2*A - 60*a^2*B*x + 120*a*b*B*x^3 + 60*a^2*D*x^3 + 30*A*b^2*x^4 + 60*a*b*C*x^4 + 20*b^2*B*x^5 + 40*a*b*D*x^5 + 15*b^2*C*x^6 + 12*b^2*D*x^7)/(60*x^2) + (2*a*A*b + a^2*C)*Log[x]","A",1
77,1,105,98,0.0685632,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right)}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^4,x]","\frac{-4 a^2 A-6 a^2 B x-12 a^2 C x^2-24 a A b x^2+24 a b C x^4+12 a b D x^5+12 A b^2 x^4+6 b^2 B x^5+4 b^2 C x^6+3 b^2 D x^7}{12 x^3}+\log (x) \left(a^2 D+2 a b B\right)","-\frac{a^2 A}{3 x^3}-\frac{a^2 B}{2 x^2}+b x (2 a C+A b)-\frac{a (a C+2 A b)}{x}+\frac{1}{2} b x^2 (2 a D+b B)+a \log (x) (a D+2 b B)+\frac{1}{3} b^2 C x^3+\frac{1}{4} b^2 D x^4",1,"(-4*a^2*A - 6*a^2*B*x - 24*a*A*b*x^2 - 12*a^2*C*x^2 + 12*A*b^2*x^4 + 24*a*b*C*x^4 + 6*b^2*B*x^5 + 12*a*b*D*x^5 + 4*b^2*C*x^6 + 3*b^2*D*x^7)/(12*x^3) + (2*a*b*B + a^2*D)*Log[x]","A",1
78,1,157,149,0.0658507,"\int x^3 \left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[x^3*(a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3),x]","\frac{30030 a^3 A x^4+24024 a^3 B x^5+20020 a^3 C x^6+17160 a^3 D x^7+60060 a^2 A b x^6+51480 a^2 b B x^7+45045 a^2 b C x^8+40040 a^2 b D x^9+45045 a A b^2 x^8+40040 a b^2 B x^9+36036 a b^2 C x^{10}+32760 a b^2 D x^{11}+12012 A b^3 x^{10}+10920 b^3 B x^{11}+10010 b^3 C x^{12}+9240 b^3 D x^{13}}{120120}","\frac{1}{4} a^3 A x^4+\frac{1}{5} a^3 B x^5+\frac{1}{6} a^2 x^6 (a C+3 A b)+\frac{1}{7} a^2 x^7 (a D+3 b B)+\frac{1}{10} b^2 x^{10} (3 a C+A b)+\frac{3}{8} a b x^8 (a C+A b)+\frac{1}{11} b^2 x^{11} (3 a D+b B)+\frac{1}{3} a b x^9 (a D+b B)+\frac{1}{12} b^3 C x^{12}+\frac{1}{13} b^3 D x^{13}",1,"(30030*a^3*A*x^4 + 24024*a^3*B*x^5 + 60060*a^2*A*b*x^6 + 20020*a^3*C*x^6 + 51480*a^2*b*B*x^7 + 17160*a^3*D*x^7 + 45045*a*A*b^2*x^8 + 45045*a^2*b*C*x^8 + 40040*a*b^2*B*x^9 + 40040*a^2*b*D*x^9 + 12012*A*b^3*x^10 + 36036*a*b^2*C*x^10 + 10920*b^3*B*x^11 + 32760*a*b^2*D*x^11 + 10010*b^3*C*x^12 + 9240*b^3*D*x^13)/120120","A",1
79,1,157,149,0.0968535,"\int x^2 \left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[x^2*(a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3),x]","\frac{9240 a^3 A x^3+6930 a^3 B x^4+5544 a^3 C x^5+4620 a^3 D x^6+16632 a^2 A b x^5+13860 a^2 b B x^6+11880 a^2 b C x^7+10395 a^2 b D x^8+11880 a A b^2 x^7+10395 a b^2 B x^8+9240 a b^2 C x^9+8316 a b^2 D x^{10}+3080 A b^3 x^9+2772 b^3 B x^{10}+2520 b^3 C x^{11}+2310 b^3 D x^{12}}{27720}","\frac{1}{3} a^3 A x^3+\frac{1}{4} a^3 B x^4+\frac{1}{5} a^2 x^5 (a C+3 A b)+\frac{1}{6} a^2 x^6 (a D+3 b B)+\frac{1}{9} b^2 x^9 (3 a C+A b)+\frac{3}{7} a b x^7 (a C+A b)+\frac{1}{10} b^2 x^{10} (3 a D+b B)+\frac{3}{8} a b x^8 (a D+b B)+\frac{1}{11} b^3 C x^{11}+\frac{1}{12} b^3 D x^{12}",1,"(9240*a^3*A*x^3 + 6930*a^3*B*x^4 + 16632*a^2*A*b*x^5 + 5544*a^3*C*x^5 + 13860*a^2*b*B*x^6 + 4620*a^3*D*x^6 + 11880*a*A*b^2*x^7 + 11880*a^2*b*C*x^7 + 10395*a*b^2*B*x^8 + 10395*a^2*b*D*x^8 + 3080*A*b^3*x^9 + 9240*a*b^2*C*x^9 + 2772*b^3*B*x^10 + 8316*a*b^2*D*x^10 + 2520*b^3*C*x^11 + 2310*b^3*D*x^12)/27720","A",1
80,1,157,138,0.0966979,"\int x \left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[x*(a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3),x]","\frac{13860 a^3 A x^2+9240 a^3 B x^3+6930 a^3 C x^4+5544 a^3 D x^5+20790 a^2 A b x^4+16632 a^2 b B x^5+13860 a^2 b C x^6+11880 a^2 b D x^7+13860 a A b^2 x^6+11880 a b^2 B x^7+10395 a b^2 C x^8+9240 a b^2 D x^9+3465 A b^3 x^8+3080 b^3 B x^9+2772 b^3 C x^{10}+2520 b^3 D x^{11}}{27720}","\frac{1}{3} a^3 B x^3+\frac{1}{4} a^3 C x^4+\frac{1}{5} a^2 x^5 (a D+3 b B)+\frac{1}{2} a^2 b C x^6+\frac{A \left(a+b x^2\right)^4}{8 b}+\frac{1}{9} b^2 x^9 (3 a D+b B)+\frac{3}{8} a b^2 C x^8+\frac{3}{7} a b x^7 (a D+b B)+\frac{1}{10} b^3 C x^{10}+\frac{1}{11} b^3 D x^{11}",1,"(13860*a^3*A*x^2 + 9240*a^3*B*x^3 + 20790*a^2*A*b*x^4 + 6930*a^3*C*x^4 + 16632*a^2*b*B*x^5 + 5544*a^3*D*x^5 + 13860*a*A*b^2*x^6 + 13860*a^2*b*C*x^6 + 11880*a*b^2*B*x^7 + 11880*a^2*b*D*x^7 + 3465*A*b^3*x^8 + 10395*a*b^2*C*x^8 + 3080*b^3*B*x^9 + 9240*a*b^2*D*x^9 + 2772*b^3*C*x^10 + 2520*b^3*D*x^11)/27720","A",1
81,1,155,133,0.0880851,"\int \left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[(a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3),x]","\frac{2520 a^3 A x+1260 a^3 B x^2+840 a^3 C x^3+630 a^3 D x^4+2520 a^2 A b x^3+1890 a^2 b B x^4+1512 a^2 b C x^5+1260 a^2 b D x^6+1512 a A b^2 x^5+1260 a b^2 B x^6+1080 a b^2 C x^7+945 a b^2 D x^8+360 A b^3 x^7+315 b^3 B x^8+280 b^3 C x^9+252 b^3 D x^{10}}{2520}","a^3 A x+\frac{1}{4} a^3 D x^4+\frac{1}{3} a^2 x^3 (a C+3 A b)+\frac{1}{2} a^2 b D x^6+\frac{1}{7} b^2 x^7 (3 a C+A b)+\frac{3}{5} a b x^5 (a C+A b)+\frac{3}{8} a b^2 D x^8+\frac{B \left(a+b x^2\right)^4}{8 b}+\frac{1}{9} b^3 C x^9+\frac{1}{10} b^3 D x^{10}",1,"(2520*a^3*A*x + 1260*a^3*B*x^2 + 2520*a^2*A*b*x^3 + 840*a^3*C*x^3 + 1890*a^2*b*B*x^4 + 630*a^3*D*x^4 + 1512*a*A*b^2*x^5 + 1512*a^2*b*C*x^5 + 1260*a*b^2*B*x^6 + 1260*a^2*b*D*x^6 + 360*A*b^3*x^7 + 1080*a*b^2*C*x^7 + 315*b^3*B*x^8 + 945*a*b^2*D*x^8 + 280*b^3*C*x^9 + 252*b^3*D*x^10)/2520","A",1
82,1,154,129,0.0963833,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right)}{x} \, dx","IntegrateAlgebraic[((a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3))/x,x]","a^3 A \log (x)+\frac{2520 a^3 B x+1260 a^3 C x^2+840 a^3 D x^3+3780 a^2 A b x^2+2520 a^2 b B x^3+1890 a^2 b C x^4+1512 a^2 b D x^5+1890 a A b^2 x^4+1512 a b^2 B x^5+1260 a b^2 C x^6+1080 a b^2 D x^7+420 A b^3 x^6+360 b^3 B x^7+315 b^3 C x^8+280 b^3 D x^9}{2520}","a^3 A \log (x)+a^3 B x+\frac{3}{2} a^2 A b x^2+\frac{1}{3} a^2 x^3 (a D+3 b B)+\frac{3}{4} a A b^2 x^4+\frac{1}{7} b^2 x^7 (3 a D+b B)+\frac{3}{5} a b x^5 (a D+b B)+\frac{C \left(a+b x^2\right)^4}{8 b}+\frac{1}{6} A b^3 x^6+\frac{1}{9} b^3 D x^9",1,"(2520*a^3*B*x + 3780*a^2*A*b*x^2 + 1260*a^3*C*x^2 + 2520*a^2*b*B*x^3 + 840*a^3*D*x^3 + 1890*a*A*b^2*x^4 + 1890*a^2*b*C*x^4 + 1512*a*b^2*B*x^5 + 1512*a^2*b*D*x^5 + 420*A*b^3*x^6 + 1260*a*b^2*C*x^6 + 360*b^3*B*x^7 + 1080*a*b^2*D*x^7 + 315*b^3*C*x^8 + 280*b^3*D*x^9)/2520 + a^3*A*Log[x]","A",1
83,1,156,124,0.1049175,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right)}{x^2} \, dx","IntegrateAlgebraic[((a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3))/x^2,x]","a^3 B \log (x)+\frac{-840 a^3 A+840 a^3 C x^2+420 a^3 D x^3+2520 a^2 A b x^2+1260 a^2 b B x^3+840 a^2 b C x^4+630 a^2 b D x^5+840 a A b^2 x^4+630 a b^2 B x^5+504 a b^2 C x^6+420 a b^2 D x^7+168 A b^3 x^6+140 b^3 B x^7+120 b^3 C x^8+105 b^3 D x^9}{840 x}","-\frac{a^3 A}{x}+a^3 B \log (x)+a^2 x (a C+3 A b)+\frac{3}{2} a^2 b B x^2+\frac{1}{5} b^2 x^5 (3 a C+A b)+a b x^3 (a C+A b)+\frac{3}{4} a b^2 B x^4+\frac{D \left(a+b x^2\right)^4}{8 b}+\frac{1}{6} b^3 B x^6+\frac{1}{7} b^3 C x^7",1,"(-840*a^3*A + 2520*a^2*A*b*x^2 + 840*a^3*C*x^2 + 1260*a^2*b*B*x^3 + 420*a^3*D*x^3 + 840*a*A*b^2*x^4 + 840*a^2*b*C*x^4 + 630*a*b^2*B*x^5 + 630*a^2*b*D*x^5 + 168*A*b^3*x^6 + 504*a*b^2*C*x^6 + 140*b^3*B*x^7 + 420*a*b^2*D*x^7 + 120*b^3*C*x^8 + 105*b^3*D*x^9)/(840*x) + a^3*B*Log[x]","A",1
84,1,153,135,0.0902365,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right)}{x^3} \, dx","IntegrateAlgebraic[((a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3))/x^3,x]","\frac{-210 a^3 A-420 a^3 B x+420 a^3 D x^3+1260 a^2 b B x^3+630 a^2 b C x^4+420 a^2 b D x^5+630 a A b^2 x^4+420 a b^2 B x^5+315 a b^2 C x^6+252 a b^2 D x^7+105 A b^3 x^6+84 b^3 B x^7+70 b^3 C x^8+60 b^3 D x^9}{420 x^2}+\log (x) \left(a^3 C+3 a^2 A b\right)","-\frac{a^3 A}{2 x^2}-\frac{a^3 B}{x}+a^2 \log (x) (a C+3 A b)+a^2 x (a D+3 b B)+\frac{1}{4} b^2 x^4 (3 a C+A b)+\frac{3}{2} a b x^2 (a C+A b)+\frac{1}{5} b^2 x^5 (3 a D+b B)+a b x^3 (a D+b B)+\frac{1}{6} b^3 C x^6+\frac{1}{7} b^3 D x^7",1,"(-210*a^3*A - 420*a^3*B*x + 1260*a^2*b*B*x^3 + 420*a^3*D*x^3 + 630*a*A*b^2*x^4 + 630*a^2*b*C*x^4 + 420*a*b^2*B*x^5 + 420*a^2*b*D*x^5 + 105*A*b^3*x^6 + 315*a*b^2*C*x^6 + 84*b^3*B*x^7 + 252*a*b^2*D*x^7 + 70*b^3*C*x^8 + 60*b^3*D*x^9)/(420*x^2) + (3*a^2*A*b + a^3*C)*Log[x]","A",1
85,1,153,139,0.0701596,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right)}{x^4} \, dx","IntegrateAlgebraic[((a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3))/x^4,x]","\frac{-20 a^3 A-30 a^3 B x-60 a^3 C x^2-180 a^2 A b x^2+180 a^2 b C x^4+90 a^2 b D x^5+180 a A b^2 x^4+90 a b^2 B x^5+60 a b^2 C x^6+45 a b^2 D x^7+20 A b^3 x^6+15 b^3 B x^7+12 b^3 C x^8+10 b^3 D x^9}{60 x^3}+\log (x) \left(a^3 D+3 a^2 b B\right)","-\frac{a^3 A}{3 x^3}-\frac{a^3 B}{2 x^2}-\frac{a^2 (a C+3 A b)}{x}+a^2 \log (x) (a D+3 b B)+\frac{1}{3} b^2 x^3 (3 a C+A b)+3 a b x (a C+A b)+\frac{1}{4} b^2 x^4 (3 a D+b B)+\frac{3}{2} a b x^2 (a D+b B)+\frac{1}{5} b^3 C x^5+\frac{1}{6} b^3 D x^6",1,"(-20*a^3*A - 30*a^3*B*x - 180*a^2*A*b*x^2 - 60*a^3*C*x^2 + 180*a*A*b^2*x^4 + 180*a^2*b*C*x^4 + 90*a*b^2*B*x^5 + 90*a^2*b*D*x^5 + 20*A*b^3*x^6 + 60*a*b^2*C*x^6 + 15*b^3*B*x^7 + 45*a*b^2*D*x^7 + 12*b^3*C*x^8 + 10*b^3*D*x^9)/(60*x^3) + (3*a^2*b*B + a^3*D)*Log[x]","A",1
86,1,155,151,0.1128009,"\int \frac{x^4 \left(A+B x+C x^2+D x^3\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^4*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]","\frac{\left(a^{3/2} A b-a^{5/2} C\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{60 a^2 C x+30 a^2 D x^2-60 a A b x-30 a b B x^2-20 a b C x^3-15 a b D x^4+20 A b^2 x^3+15 b^2 B x^4+12 b^2 C x^5+10 b^2 D x^6}{60 b^3}+\frac{\left(a^2 b B-a^3 D\right) \log \left(a+b x^2\right)}{2 b^4}","\frac{a^{3/2} (A b-a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{a^2 (b B-a D) \log \left(a+b x^2\right)}{2 b^4}-\frac{a x (A b-a C)}{b^3}+\frac{x^3 (A b-a C)}{3 b^2}-\frac{a x^2 (b B-a D)}{2 b^3}+\frac{x^4 (b B-a D)}{4 b^2}+\frac{C x^5}{5 b}+\frac{D x^6}{6 b}",1,"(-60*a*A*b*x + 60*a^2*C*x - 30*a*b*B*x^2 + 30*a^2*D*x^2 + 20*A*b^2*x^3 - 20*a*b*C*x^3 + 15*b^2*B*x^4 - 15*a*b*D*x^4 + 12*b^2*C*x^5 + 10*b^2*D*x^6)/(60*b^3) + ((a^(3/2)*A*b - a^(5/2)*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2) + ((a^2*b*B - a^3*D)*Log[a + b*x^2])/(2*b^4)","A",1
87,1,136,130,0.123123,"\int \frac{x^3 \left(A+B x+C x^2+D x^3\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^3*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]","\frac{\left(a^{3/2} b B-a^{5/2} D\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{\left(a^2 C-a A b\right) \log \left(a+b x^2\right)}{2 b^3}+\frac{60 a^2 D x-60 a b B x-30 a b C x^2-20 a b D x^3+30 A b^2 x^2+20 b^2 B x^3+15 b^2 C x^4+12 b^2 D x^5}{60 b^3}","\frac{a^{3/2} (b B-a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}-\frac{a (A b-a C) \log \left(a+b x^2\right)}{2 b^3}+\frac{x^2 (A b-a C)}{2 b^2}-\frac{a x (b B-a D)}{b^3}+\frac{x^3 (b B-a D)}{3 b^2}+\frac{C x^4}{4 b}+\frac{D x^5}{5 b}",1,"(-60*a*b*B*x + 60*a^2*D*x + 30*A*b^2*x^2 - 30*a*b*C*x^2 + 20*b^2*B*x^3 - 20*a*b*D*x^3 + 15*b^2*C*x^4 + 12*b^2*D*x^5)/(60*b^3) + ((a^(3/2)*b*B - a^(5/2)*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2) + ((-(a*A*b) + a^2*C)*Log[a + b*x^2])/(2*b^3)","A",1
88,1,109,111,0.0749437,"\int \frac{x^2 \left(A+B x+C x^2+D x^3\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^2*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]","\frac{\left(a^{3/2} C-\sqrt{a} A b\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}+\frac{\left(a^2 D-a b B\right) \log \left(a+b x^2\right)}{2 b^3}+\frac{-12 a C x-6 a D x^2+12 A b x+6 b B x^2+4 b C x^3+3 b D x^4}{12 b^2}","-\frac{\sqrt{a} (A b-a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}+\frac{x (A b-a C)}{b^2}-\frac{a (b B-a D) \log \left(a+b x^2\right)}{2 b^3}+\frac{x^2 (b B-a D)}{2 b^2}+\frac{C x^3}{3 b}+\frac{D x^4}{4 b}",1,"(12*A*b*x - 12*a*C*x + 6*b*B*x^2 - 6*a*D*x^2 + 4*b*C*x^3 + 3*b*D*x^4)/(12*b^2) + ((-(Sqrt[a]*A*b) + a^(3/2)*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2) + ((-(a*b*B) + a^2*D)*Log[a + b*x^2])/(2*b^3)","A",1
89,1,92,92,0.0797097,"\int \frac{x \left(A+B x+C x^2+D x^3\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]","\frac{\left(a^{3/2} D-\sqrt{a} b B\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}+\frac{(A b-a C) \log \left(a+b x^2\right)}{2 b^2}+\frac{-6 a D x+6 b B x+3 b C x^2+2 b D x^3}{6 b^2}","\frac{(A b-a C) \log \left(a+b x^2\right)}{2 b^2}-\frac{\sqrt{a} (b B-a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}+\frac{x (b B-a D)}{b^2}+\frac{C x^2}{2 b}+\frac{D x^3}{3 b}",1,"(6*b*B*x - 6*a*D*x + 3*b*C*x^2 + 2*b*D*x^3)/(6*b^2) + ((-(Sqrt[a]*b*B) + a^(3/2)*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2) + ((A*b - a*C)*Log[a + b*x^2])/(2*b^2)","A",1
90,1,73,73,0.0544656,"\int \frac{A+B x+C x^2+D x^3}{a+b x^2} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2),x]","\frac{(A b-a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+\frac{(b B-a D) \log \left(a+b x^2\right)}{2 b^2}+\frac{2 C x+D x^2}{2 b}","\frac{(A b-a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+\frac{(b B-a D) \log \left(a+b x^2\right)}{2 b^2}+\frac{C x}{b}+\frac{D x^2}{2 b}",1,"(2*C*x + D*x^2)/(2*b) + ((A*b - a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2)) + ((b*B - a*D)*Log[a + b*x^2])/(2*b^2)","A",1
91,1,72,72,0.0675859,"\int \frac{A+B x+C x^2+D x^3}{x \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(x*(a + b*x^2)),x]","\frac{(a C-A b) \log \left(a+b x^2\right)}{2 a b}+\frac{A \log (x)}{a}+\frac{(b B-a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+\frac{D x}{b}","-\frac{(A b-a C) \log \left(a+b x^2\right)}{2 a b}+\frac{A \log (x)}{a}+\frac{(b B-a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}+\frac{D x}{b}",1,"(D*x)/b + ((b*B - a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2)) + (A*Log[x])/a + ((-(A*b) + a*C)*Log[a + b*x^2])/(2*a*b)","A",1
92,1,75,76,0.0639513,"\int \frac{A+B x+C x^2+D x^3}{x^2 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(x^2*(a + b*x^2)),x]","\frac{(a C-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}-\frac{A}{a x}+\frac{(a D-b B) \log \left(a+b x^2\right)}{2 a b}+\frac{B \log (x)}{a}","-\frac{(A b-a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}-\frac{A}{a x}-\frac{(b B-a D) \log \left(a+b x^2\right)}{2 a b}+\frac{B \log (x)}{a}",1,"-(A/(a*x)) + ((-(A*b) + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]) + (B*Log[x])/a + ((-(b*B) + a*D)*Log[a + b*x^2])/(2*a*b)","A",1
93,1,88,92,0.101612,"\int \frac{A+B x+C x^2+D x^3}{x^3 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)),x]","\frac{(a D-b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}+\frac{(A b-a C) \log \left(a+b x^2\right)}{2 a^2}+\frac{\log (x) (a C-A b)}{a^2}+\frac{-A-2 B x}{2 a x^2}","-\frac{(b B-a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2} \sqrt{b}}+\frac{(A b-a C) \log \left(a+b x^2\right)}{2 a^2}-\frac{\log (x) (A b-a C)}{a^2}-\frac{A}{2 a x^2}-\frac{B}{a x}",1,"(-A - 2*B*x)/(2*a*x^2) + ((-(b*B) + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]) + ((-(A*b) + a*C)*Log[x])/a^2 + ((A*b - a*C)*Log[a + b*x^2])/(2*a^2)","A",1
94,1,190,176,0.1565304,"\int \frac{x^4 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^4*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]","\frac{\left(5 a^{3/2} C-3 \sqrt{a} A b\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{\left(3 a^2 D-2 a b B\right) \log \left(a+b x^2\right)}{2 b^4}+\frac{6 a^3 D-6 a^2 b B-30 a^2 b C x-12 a^2 b D x^2+18 a A b^2 x+6 a b^2 B x^2-20 a b^2 C x^3-9 a b^2 D x^4+12 A b^3 x^3+6 b^3 B x^4+4 b^3 C x^5+3 b^3 D x^6}{12 b^4 \left(a+b x^2\right)}","-\frac{\sqrt{a} (3 A b-5 a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{x (3 A b-5 a C)}{2 b^3}-\frac{x^3 (3 A b-5 a C)}{6 a b^2}-\frac{x^4 \left(a \left(B-\frac{a D}{b}\right)-x (A b-a C)\right)}{2 a b \left(a+b x^2\right)}-\frac{a (2 b B-3 a D) \log \left(a+b x^2\right)}{2 b^4}+\frac{x^2 (2 b B-3 a D)}{2 b^3}+\frac{D x^4}{4 b^2}",1,"(-6*a^2*b*B + 6*a^3*D + 18*a*A*b^2*x - 30*a^2*b*C*x + 6*a*b^2*B*x^2 - 12*a^2*b*D*x^2 + 12*A*b^3*x^3 - 20*a*b^2*C*x^3 + 6*b^3*B*x^4 - 9*a*b^2*D*x^4 + 4*b^3*C*x^5 + 3*b^3*D*x^6)/(12*b^4*(a + b*x^2)) + ((-3*Sqrt[a]*A*b + 5*a^(3/2)*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2)) + ((-2*a*b*B + 3*a^2*D)*Log[a + b*x^2])/(2*b^4)","A",1
95,1,148,154,0.103349,"\int \frac{x^3 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^3*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]","\frac{\left(5 a^{3/2} D-3 \sqrt{a} b B\right) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{-3 a^2 C-15 a^2 D x+3 a A b+9 a b B x+3 a b C x^2-10 a b D x^3+6 b^2 B x^3+3 b^2 C x^4+2 b^2 D x^5}{6 b^3 \left(a+b x^2\right)}+\frac{(A b-2 a C) \log \left(a+b x^2\right)}{2 b^3}","\frac{(A b-2 a C) \log \left(a+b x^2\right)}{2 b^3}-\frac{x^2 (A b-2 a C)}{2 a b^2}-\frac{x^3 \left(a \left(B-\frac{a D}{b}\right)-x (A b-a C)\right)}{2 a b \left(a+b x^2\right)}-\frac{\sqrt{a} (3 b B-5 a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{x (3 b B-5 a D)}{2 b^3}+\frac{D x^3}{3 b^2}",1,"(3*a*A*b - 3*a^2*C + 9*a*b*B*x - 15*a^2*D*x + 3*a*b*C*x^2 + 6*b^2*B*x^3 - 10*a*b*D*x^3 + 3*b^2*C*x^4 + 2*b^2*D*x^5)/(6*b^3*(a + b*x^2)) + ((-3*Sqrt[a]*b*B + 5*a^(3/2)*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2)) + ((A*b - 2*a*C)*Log[a + b*x^2])/(2*b^3)","A",1
96,1,123,134,0.0996086,"\int \frac{x^2 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^2*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]","\frac{-a^2 D+a b B+3 a b C x+a b D x^2-A b^2 x+2 b^2 C x^3+b^2 D x^4}{2 b^3 \left(a+b x^2\right)}+\frac{(A b-3 a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{5/2}}+\frac{(b B-2 a D) \log \left(a+b x^2\right)}{2 b^3}","\frac{(A b-3 a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{5/2}}-\frac{x (A b-3 a C)}{2 a b^2}-\frac{x^2 \left(a \left(B-\frac{a D}{b}\right)-x (A b-a C)\right)}{2 a b \left(a+b x^2\right)}+\frac{(b B-2 a D) \log \left(a+b x^2\right)}{2 b^3}+\frac{D x^2}{2 b^2}",1,"(a*b*B - a^2*D - A*b^2*x + 3*a*b*C*x + a*b*D*x^2 + 2*b^2*C*x^3 + b^2*D*x^4)/(2*b^3*(a + b*x^2)) + ((A*b - 3*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(5/2)) + ((b*B - 2*a*D)*Log[a + b*x^2])/(2*b^3)","A",1
97,1,93,101,0.0706018,"\int \frac{x \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]","\frac{a C+3 a D x-A b-b B x+2 b D x^3}{2 b^2 \left(a+b x^2\right)}+\frac{(b B-3 a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{5/2}}+\frac{C \log \left(a+b x^2\right)}{2 b^2}","-\frac{x \left(a \left(B-\frac{a D}{b}\right)-x (A b-a C)\right)}{2 a b \left(a+b x^2\right)}+\frac{(b B-3 a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{5/2}}+\frac{C \log \left(a+b x^2\right)}{2 b^2}+\frac{D x}{b^2}",1,"(-(A*b) + a*C - b*B*x + 3*a*D*x + 2*b*D*x^3)/(2*b^2*(a + b*x^2)) + ((b*B - 3*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(5/2)) + (C*Log[a + b*x^2])/(2*b^2)","A",1
98,1,93,93,0.1032672,"\int \frac{A+B x+C x^2+D x^3}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^2,x]","\frac{(a C+A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}+\frac{a^2 D-a b B-a b C x+A b^2 x}{2 a b^2 \left(a+b x^2\right)}+\frac{D \log \left(a+b x^2\right)}{2 b^2}","\frac{(a C+A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}+\frac{x (A b-a C)-a \left(B-\frac{a D}{b}\right)}{2 a b \left(a+b x^2\right)}+\frac{D \log \left(a+b x^2\right)}{2 b^2}",1,"(-(a*b*B) + a^2*D + A*b^2*x - a*b*C*x)/(2*a*b^2*(a + b*x^2)) + ((A*b + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(3/2)) + (D*Log[a + b*x^2])/(2*b^2)","A",1
99,1,94,95,0.0917658,"\int \frac{A+B x+C x^2+D x^3}{x \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(x*(a + b*x^2)^2),x]","\frac{(a D+b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}-\frac{A \log \left(a+b x^2\right)}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{-a C-a D x+A b+b B x}{2 a b \left(a+b x^2\right)}","\frac{(a D+b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}-\frac{A \log \left(a+b x^2\right)}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{x (b B-a D)-a C+A b}{2 a b \left(a+b x^2\right)}",1,"(A*b - a*C + b*B*x - a*D*x)/(2*a*b*(a + b*x^2)) + ((b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(3/2)) + (A*Log[x])/a^2 - (A*Log[a + b*x^2])/(2*a^2)","A",1
100,1,115,110,0.0900726,"\int \frac{A+B x+C x^2+D x^3}{x^2 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(x^2*(a + b*x^2)^2),x]","\frac{(a C-3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} \sqrt{b}}+\frac{a^2 (-D) x-2 a A b+a b B x+a b C x^2-3 A b^2 x^2}{2 a^2 b x \left(a+b x^2\right)}-\frac{B \log \left(a+b x^2\right)}{2 a^2}+\frac{B \log (x)}{a^2}","-\frac{(3 A b-a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} \sqrt{b}}-\frac{A}{a^2 x}-\frac{B \log \left(a+b x^2\right)}{2 a^2}+\frac{B \log (x)}{a^2}+\frac{-b x \left(\frac{A b}{a}-C\right)-a D+b B}{2 a b \left(a+b x^2\right)}",1,"(-2*a*A*b + a*b*B*x - a^2*D*x - 3*A*b^2*x^2 + a*b*C*x^2)/(2*a^2*b*x*(a + b*x^2)) + ((-3*A*b + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b]) + (B*Log[x])/a^2 - (B*Log[a + b*x^2])/(2*a^2)","A",1
101,1,129,135,0.1277463,"\int \frac{A+B x+C x^2+D x^3}{x^3 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)^2),x]","\frac{(a D-3 b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} \sqrt{b}}+\frac{(2 A b-a C) \log \left(a+b x^2\right)}{2 a^3}+\frac{\log (x) (a C-2 A b)}{a^3}+\frac{-a A-2 a B x+a C x^2+a D x^3-2 A b x^2-3 b B x^3}{2 a^2 x^2 \left(a+b x^2\right)}","-\frac{(3 b B-a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2} \sqrt{b}}+\frac{(2 A b-a C) \log \left(a+b x^2\right)}{2 a^3}-\frac{\log (x) (2 A b-a C)}{a^3}-\frac{A}{2 a^2 x^2}-\frac{B}{a^2 x}-\frac{\frac{A b}{a}+x \left(\frac{b B}{a}-D\right)-C}{2 a \left(a+b x^2\right)}",1,"(-(a*A) - 2*a*B*x - 2*A*b*x^2 + a*C*x^2 - 3*b*B*x^3 + a*D*x^3)/(2*a^2*x^2*(a + b*x^2)) + ((-3*b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b]) + ((-2*A*b + a*C)*Log[x])/a^3 + ((2*A*b - a*C)*Log[a + b*x^2])/(2*a^3)","A",1
102,1,173,185,0.1440869,"\int \frac{x^4 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^4*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^3,x]","\frac{-10 a^3 D+6 a^2 b B+15 a^2 b C x-8 a^2 b D x^2-3 a A b^2 x+8 a b^2 B x^2+25 a b^2 C x^3+8 a b^2 D x^4-5 A b^3 x^3+8 b^3 C x^5+4 b^3 D x^6}{8 b^4 \left(a+b x^2\right)^2}-\frac{3 (5 a C-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 \sqrt{a} b^{7/2}}+\frac{(b B-3 a D) \log \left(a+b x^2\right)}{2 b^4}","\frac{3 (A b-5 a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 \sqrt{a} b^{7/2}}-\frac{3 x (A b-5 a C)}{8 a b^3}+\frac{x^3 (4 x (b B-2 a D)-5 a C+A b)}{8 a b^2 \left(a+b x^2\right)}-\frac{x^4 \left(a \left(B-\frac{a D}{b}\right)-x (A b-a C)\right)}{4 a b \left(a+b x^2\right)^2}+\frac{(b B-3 a D) \log \left(a+b x^2\right)}{2 b^4}-\frac{x^2 (b B-3 a D)}{2 a b^3}",1,"(6*a^2*b*B - 10*a^3*D - 3*a*A*b^2*x + 15*a^2*b*C*x + 8*a*b^2*B*x^2 - 8*a^2*b*D*x^2 - 5*A*b^3*x^3 + 25*a*b^2*C*x^3 + 8*a*b^2*D*x^4 + 8*b^3*C*x^5 + 4*b^3*D*x^6)/(8*b^4*(a + b*x^2)^2) - (3*(-(A*b) + 5*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(7/2)) + ((b*B - 3*a*D)*Log[a + b*x^2])/(2*b^4)","A",1
103,1,137,155,0.1032101,"\int \frac{x^3 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^3*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^3,x]","\frac{6 a^2 C+15 a^2 D x-2 a A b-3 a b B x+8 a b C x^2+25 a b D x^3-4 A b^2 x^2-5 b^2 B x^3+8 b^2 D x^5}{8 b^3 \left(a+b x^2\right)^2}-\frac{3 (5 a D-b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 \sqrt{a} b^{7/2}}+\frac{C \log \left(a+b x^2\right)}{2 b^3}","-\frac{x^3 \left(a \left(B-\frac{a D}{b}\right)-x (A b-a C)\right)}{4 a b \left(a+b x^2\right)^2}+\frac{3 (b B-5 a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 \sqrt{a} b^{7/2}}-\frac{3 x (b B-5 a D)}{8 a b^3}+\frac{C \log \left(a+b x^2\right)}{2 b^3}-\frac{x^2 (4 a C-x (3 b B-7 a D))}{8 a b^2 \left(a+b x^2\right)}",1,"(-2*a*A*b + 6*a^2*C - 3*a*b*B*x + 15*a^2*D*x - 4*A*b^2*x^2 + 8*a*b*C*x^2 - 5*b^2*B*x^3 + 25*a*b*D*x^3 + 8*b^2*D*x^5)/(8*b^3*(a + b*x^2)^2) - (3*(-(b*B) + 5*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(7/2)) + (C*Log[a + b*x^2])/(2*b^3)","A",1
104,1,139,136,0.1204006,"\int \frac{x^2 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^2*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^3,x]","\frac{(3 a C+A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{5/2}}+\frac{6 a^3 D-2 a^2 b B-3 a^2 b C x+8 a^2 b D x^2-a A b^2 x-4 a b^2 B x^2-5 a b^2 C x^3+A b^3 x^3}{8 a b^3 \left(a+b x^2\right)^2}+\frac{D \log \left(a+b x^2\right)}{2 b^3}","\frac{(3 a C+A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{5/2}}-\frac{x (-2 x (b B-3 a D)+3 a C+A b)}{8 a b^2 \left(a+b x^2\right)}-\frac{x^2 \left(a \left(B-\frac{a D}{b}\right)-x (A b-a C)\right)}{4 a b \left(a+b x^2\right)^2}+\frac{D \log \left(a+b x^2\right)}{2 b^3}",1,"(-2*a^2*b*B + 6*a^3*D - a*A*b^2*x - 3*a^2*b*C*x - 4*a*b^2*B*x^2 + 8*a^2*b*D*x^2 + A*b^3*x^3 - 5*a*b^2*C*x^3)/(8*a*b^3*(a + b*x^2)^2) + ((A*b + 3*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(5/2)) + (D*Log[a + b*x^2])/(2*b^3)","A",1
105,1,104,119,0.1292715,"\int \frac{x \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^3,x]","\frac{(3 a D+b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{5/2}}+\frac{-2 a^2 C-3 a^2 D x-2 a A b-a b B x-4 a b C x^2-5 a b D x^3+b^2 B x^3}{8 a b^2 \left(a+b x^2\right)^2}","\frac{(3 a D+b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{5/2}}-\frac{2 (a C+A b)-x (b B-5 a D)}{8 a b^2 \left(a+b x^2\right)}-\frac{x \left(a \left(B-\frac{a D}{b}\right)-x (A b-a C)\right)}{4 a b \left(a+b x^2\right)^2}",1,"(-2*a*A*b - 2*a^2*C - a*b*B*x - 3*a^2*D*x - 4*a*b*C*x^2 + b^2*B*x^3 - 5*a*b*D*x^3)/(8*a*b^2*(a + b*x^2)^2) + ((b*B + 3*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(5/2))","A",1
106,1,113,116,0.125048,"\int \frac{A+B x+C x^2+D x^3}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^3,x]","\frac{(a C+3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2}}+\frac{-2 a^3 D-2 a^2 b B-a^2 b C x-4 a^2 b D x^2+5 a A b^2 x+a b^2 C x^3+3 A b^3 x^3}{8 a^2 b^2 \left(a+b x^2\right)^2}","\frac{(a C+3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2}}-\frac{4 a^2 D-b x (a C+3 A b)}{8 a^2 b^2 \left(a+b x^2\right)}+\frac{x (A b-a C)-a \left(B-\frac{a D}{b}\right)}{4 a b \left(a+b x^2\right)^2}",1,"(-2*a^2*b*B - 2*a^3*D + 5*a*A*b^2*x - a^2*b*C*x - 4*a^2*b*D*x^2 + 3*A*b^3*x^3 + a*b^2*C*x^3)/(8*a^2*b^2*(a + b*x^2)^2) + ((3*A*b + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2))","A",1
107,1,128,130,0.1312308,"\int \frac{A+B x+C x^2+D x^3}{x \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(x*(a + b*x^2)^3),x]","\frac{(a D+3 b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2}}-\frac{A \log \left(a+b x^2\right)}{2 a^3}+\frac{A \log (x)}{a^3}+\frac{-2 a^2 C-a^2 D x+6 a A b+5 a b B x+a b D x^3+4 A b^2 x^2+3 b^2 B x^3}{8 a^2 b \left(a+b x^2\right)^2}","\frac{(a D+3 b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^{3/2}}-\frac{A \log \left(a+b x^2\right)}{2 a^3}+\frac{A \log (x)}{a^3}+\frac{x (a D+3 b B)+4 A b}{8 a^2 b \left(a+b x^2\right)}+\frac{x (b B-a D)-a C+A b}{4 a b \left(a+b x^2\right)^2}",1,"(6*a*A*b - 2*a^2*C + 5*a*b*B*x - a^2*D*x + 4*A*b^2*x^2 + 3*b^2*B*x^3 + a*b*D*x^3)/(8*a^2*b*(a + b*x^2)^2) + ((3*b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2)) + (A*Log[x])/a^3 - (A*Log[a + b*x^2])/(2*a^3)","A",1
108,1,153,144,0.1276947,"\int \frac{A+B x+C x^2+D x^3}{x^2 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(x^2*(a + b*x^2)^3),x]","\frac{3 (a C-5 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2} \sqrt{b}}-\frac{B \log \left(a+b x^2\right)}{2 a^3}+\frac{B \log (x)}{a^3}+\frac{-2 a^3 D x-8 a^2 A b+6 a^2 b B x+5 a^2 b C x^2-25 a A b^2 x^2+4 a b^2 B x^3+3 a b^2 C x^4-15 A b^3 x^4}{8 a^3 b x \left(a+b x^2\right)^2}","-\frac{3 (5 A b-a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2} \sqrt{b}}-\frac{A}{a^3 x}-\frac{B \log \left(a+b x^2\right)}{2 a^3}+\frac{B \log (x)}{a^3}+\frac{4 B-x \left(\frac{7 A b}{a}-3 C\right)}{8 a^2 \left(a+b x^2\right)}+\frac{-b x \left(\frac{A b}{a}-C\right)-a D+b B}{4 a b \left(a+b x^2\right)^2}",1,"(-8*a^2*A*b + 6*a^2*b*B*x - 2*a^3*D*x - 25*a*A*b^2*x^2 + 5*a^2*b*C*x^2 + 4*a*b^2*B*x^3 - 15*A*b^3*x^4 + 3*a*b^2*C*x^4)/(8*a^3*b*x*(a + b*x^2)^2) + (3*(-5*A*b + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[b]) + (B*Log[x])/a^3 - (B*Log[a + b*x^2])/(2*a^3)","A",1
109,1,175,174,0.1870109,"\int \frac{A+B x+C x^2+D x^3}{x^3 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)^3),x]","\frac{3 (a D-5 b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2} \sqrt{b}}+\frac{(3 A b-a C) \log \left(a+b x^2\right)}{2 a^4}+\frac{\log (x) (a C-3 A b)}{a^4}+\frac{-4 a^2 A-8 a^2 B x+6 a^2 C x^2+5 a^2 D x^3-18 a A b x^2-25 a b B x^3+4 a b C x^4+3 a b D x^5-12 A b^2 x^4-15 b^2 B x^5}{8 a^3 x^2 \left(a+b x^2\right)^2}","-\frac{3 (5 b B-a D) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2} \sqrt{b}}+\frac{(3 A b-a C) \log \left(a+b x^2\right)}{2 a^4}-\frac{\log (x) (3 A b-a C)}{a^4}-\frac{4 (2 A b-a C)+x (7 b B-3 a D)}{8 a^3 \left(a+b x^2\right)}-\frac{A}{2 a^3 x^2}-\frac{B}{a^3 x}-\frac{\frac{A b}{a}+x \left(\frac{b B}{a}-D\right)-C}{4 a \left(a+b x^2\right)^2}",1,"(-4*a^2*A - 8*a^2*B*x - 18*a*A*b*x^2 + 6*a^2*C*x^2 - 25*a*b*B*x^3 + 5*a^2*D*x^3 - 12*A*b^2*x^4 + 4*a*b*C*x^4 - 15*b^2*B*x^5 + 3*a*b*D*x^5)/(8*a^3*x^2*(a + b*x^2)^2) + (3*(-5*b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[b]) + ((-3*A*b + a*C)*Log[x])/a^4 + ((3*A*b - a*C)*Log[a + b*x^2])/(2*a^4)","A",1
110,1,20,20,0.0122681,"\int \frac{-x+4 x^3}{\left(5+x^2\right)^2} \, dx","IntegrateAlgebraic[(-x + 4*x^3)/(5 + x^2)^2,x]","\frac{21}{2 \left(x^2+5\right)}+2 \log \left(x^2+5\right)","\frac{21}{2 \left(x^2+5\right)}+2 \log \left(x^2+5\right)",1,"21/(2*(5 + x^2)) + 2*Log[5 + x^2]","A",1
111,1,18,23,0.0325827,"\int \frac{-x+x^3}{\sqrt{-2+x^2}} \, dx","IntegrateAlgebraic[(-x + x^3)/Sqrt[-2 + x^2],x]","\frac{1}{3} \sqrt{x^2-2} \left(x^2+1\right)","\frac{1}{3} \left(x^2-2\right)^{3/2}+\sqrt{x^2-2}",1,"(Sqrt[-2 + x^2]*(1 + x^2))/3","A",1
112,1,25,25,0.0206047,"\int \frac{-x^2+2 x^4}{1+2 x^2} \, dx","IntegrateAlgebraic[(-x^2 + 2*x^4)/(1 + 2*x^2),x]","\frac{1}{3} x \left(x^2-3\right)+\frac{\tan ^{-1}\left(\sqrt{2} x\right)}{\sqrt{2}}","\frac{x^3}{3}-x+\frac{\tan ^{-1}\left(\sqrt{2} x\right)}{\sqrt{2}}",1,"(x*(-3 + x^2))/3 + ArcTan[Sqrt[2]*x]/Sqrt[2]","A",1
113,1,28,30,0.011077,"\int \frac{x^3+x^4}{1+x^2} \, dx","IntegrateAlgebraic[(x^3 + x^4)/(1 + x^2),x]","\frac{1}{6} x \left(2 x^2+3 x-6\right)-\frac{1}{2} \log \left(x^2+1\right)+\tan ^{-1}(x)","\frac{x^3}{3}+\frac{x^2}{2}-\frac{1}{2} \log \left(x^2+1\right)-x+\tan ^{-1}(x)",1,"(x*(-6 + 3*x + 2*x^2))/6 + ArcTan[x] - Log[1 + x^2]/2","A",1
114,1,244,210,0.2118927,"\int \frac{x^6 \left(c+d x^2+e x^4+f x^6\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-a^{5/2} b^3 c+a^{7/2} b^2 d-a^{9/2} b e+a^{11/2} f\right)}{b^{13/2}}+\frac{x \left(-3465 a^5 f+3465 a^4 b e+1155 a^4 b f x^2-3465 a^3 b^2 d-1155 a^3 b^2 e x^2-693 a^3 b^2 f x^4+3465 a^2 b^3 c+1155 a^2 b^3 d x^2+693 a^2 b^3 e x^4+495 a^2 b^3 f x^6-1155 a b^4 c x^2-693 a b^4 d x^4-495 a b^4 e x^6-385 a b^4 f x^8+693 b^5 c x^4+495 b^5 d x^6+385 b^5 e x^8+315 b^5 f x^{10}\right)}{3465 b^6}","\frac{x^7 \left(a^2 f-a b e+b^2 d\right)}{7 b^3}+\frac{a^2 x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^6}-\frac{a x^3 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^5}+\frac{x^5 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{5 b^4}-\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^{13/2}}+\frac{x^9 (b e-a f)}{9 b^2}+\frac{f x^{11}}{11 b}",1,"(x*(3465*a^2*b^3*c - 3465*a^3*b^2*d + 3465*a^4*b*e - 3465*a^5*f - 1155*a*b^4*c*x^2 + 1155*a^2*b^3*d*x^2 - 1155*a^3*b^2*e*x^2 + 1155*a^4*b*f*x^2 + 693*b^5*c*x^4 - 693*a*b^4*d*x^4 + 693*a^2*b^3*e*x^4 - 693*a^3*b^2*f*x^4 + 495*b^5*d*x^6 - 495*a*b^4*e*x^6 + 495*a^2*b^3*f*x^6 + 385*b^5*e*x^8 - 385*a*b^4*f*x^8 + 315*b^5*f*x^10))/(3465*b^6) + ((-(a^(5/2)*b^3*c) + a^(7/2)*b^2*d - a^(9/2)*b*e + a^(11/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(13/2)","A",1
115,1,196,172,0.1623091,"\int \frac{x^4 \left(c+d x^2+e x^4+f x^6\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^{3/2} b^3 c-a^{5/2} b^2 d+a^{7/2} b e+a^{9/2} (-f)\right)}{b^{11/2}}+\frac{x \left(315 a^4 f-315 a^3 b e-105 a^3 b f x^2+315 a^2 b^2 d+105 a^2 b^2 e x^2+63 a^2 b^2 f x^4-315 a b^3 c-105 a b^3 d x^2-63 a b^3 e x^4-45 a b^3 f x^6+105 b^4 c x^2+63 b^4 d x^4+45 b^4 e x^6+35 b^4 f x^8\right)}{315 b^5}","\frac{x^5 \left(a^2 f-a b e+b^2 d\right)}{5 b^3}-\frac{a x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^5}+\frac{x^3 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^4}+\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^{11/2}}+\frac{x^7 (b e-a f)}{7 b^2}+\frac{f x^9}{9 b}",1,"(x*(-315*a*b^3*c + 315*a^2*b^2*d - 315*a^3*b*e + 315*a^4*f + 105*b^4*c*x^2 - 105*a*b^3*d*x^2 + 105*a^2*b^2*e*x^2 - 105*a^3*b*f*x^2 + 63*b^4*d*x^4 - 63*a*b^3*e*x^4 + 63*a^2*b^2*f*x^4 + 45*b^4*e*x^6 - 45*a*b^3*f*x^6 + 35*b^4*f*x^8))/(315*b^5) + ((a^(3/2)*b^3*c - a^(5/2)*b^2*d + a^(7/2)*b*e - a^(9/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2)","A",1
116,1,150,136,0.1269767,"\int \frac{x^2 \left(c+d x^2+e x^4+f x^6\right)}{a+b x^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^{3/2} b^2 d-a^{5/2} b e+a^{7/2} f-\sqrt{a} b^3 c\right)}{b^{9/2}}+\frac{x \left(-105 a^3 f+105 a^2 b e+35 a^2 b f x^2-105 a b^2 d-35 a b^2 e x^2-21 a b^2 f x^4+105 b^3 c+35 b^3 d x^2+21 b^3 e x^4+15 b^3 f x^6\right)}{105 b^4}","\frac{x^3 \left(a^2 f-a b e+b^2 d\right)}{3 b^3}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^{9/2}}+\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^4}+\frac{x^5 (b e-a f)}{5 b^2}+\frac{f x^7}{7 b}",1,"(x*(105*b^3*c - 105*a*b^2*d + 105*a^2*b*e - 105*a^3*f + 35*b^3*d*x^2 - 35*a*b^2*e*x^2 + 35*a^2*b*f*x^2 + 21*b^3*e*x^4 - 21*a*b^2*f*x^4 + 15*b^3*f*x^6))/(105*b^4) + ((-(Sqrt[a]*b^3*c) + a^(3/2)*b^2*d - a^(5/2)*b*e + a^(7/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)","A",1
117,1,102,100,0.1052162,"\int \frac{c+d x^2+e x^4+f x^6}{a+b x^2} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(a + b*x^2),x]","\frac{x \left(15 a^2 f-15 a b e-5 a b f x^2+15 b^2 d+5 b^2 e x^2+3 b^2 f x^4\right)}{15 b^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{a} b^{7/2}}","\frac{x \left(a^2 f-a b e+b^2 d\right)}{b^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{a} b^{7/2}}+\frac{x^3 (b e-a f)}{3 b^2}+\frac{f x^5}{5 b}",1,"(x*(15*b^2*d - 15*a*b*e + 15*a^2*f + 5*b^2*e*x^2 - 5*a*b*f*x^2 + 3*b^2*f*x^4))/(15*b^3) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(7/2))","A",1
118,1,94,84,0.0932504,"\int \frac{c+d x^2+e x^4+f x^6}{x^2 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)),x]","\frac{-3 a^2 f x^2+3 a b e x^2+a b f x^4-3 b^2 c}{3 a b^2 x}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^{3/2} b^{5/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{3/2} b^{5/2}}+\frac{x (b e-a f)}{b^2}-\frac{c}{a x}+\frac{f x^3}{3 b}",1,"(-3*b^2*c + 3*a*b*e*x^2 - 3*a^2*f*x^2 + a*b*f*x^4)/(3*a*b^2*x) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*b^(5/2))","A",1
119,1,95,82,0.1122244,"\int \frac{c+d x^2+e x^4+f x^6}{x^4 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^4*(a + b*x^2)),x]","\frac{3 a^2 f x^4-a b c-3 a b d x^2+3 b^2 c x^2}{3 a^2 b x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{5/2} b^{3/2}}","\frac{b c-a d}{a^2 x}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{5/2} b^{3/2}}-\frac{c}{3 a x^3}+\frac{f x}{b}",1,"(-(a*b*c) + 3*b^2*c*x^2 - 3*a*b*d*x^2 + 3*a^2*f*x^4)/(3*a^2*b*x^3) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*b^(3/2))","A",1
120,1,110,104,0.1170062,"\int \frac{c+d x^2+e x^4+f x^6}{x^6 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^6*(a + b*x^2)),x]","\frac{-3 a^2 c-5 a^2 d x^2-15 a^2 e x^4+5 a b c x^2+15 a b d x^4-15 b^2 c x^4}{15 a^3 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^{7/2} \sqrt{b}}","\frac{b c-a d}{3 a^2 x^3}-\frac{a^2 e-a b d+b^2 c}{a^3 x}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{7/2} \sqrt{b}}-\frac{c}{5 a x^5}",1,"(-3*a^2*c + 5*a*b*c*x^2 - 5*a^2*d*x^2 - 15*b^2*c*x^4 + 15*a*b*d*x^4 - 15*a^2*e*x^4)/(15*a^3*x^5) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*Sqrt[b])","A",1
121,1,161,137,0.1603512,"\int \frac{c+d x^2+e x^4+f x^6}{x^8 \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-a^3 \sqrt{b} f+a^2 b^{3/2} e-a b^{5/2} d+b^{7/2} c\right)}{a^{9/2}}+\frac{-15 a^3 c-21 a^3 d x^2-35 a^3 e x^4-105 a^3 f x^6+21 a^2 b c x^2+35 a^2 b d x^4+105 a^2 b e x^6-35 a b^2 c x^4-105 a b^2 d x^6+105 b^3 c x^6}{105 a^4 x^7}","\frac{b c-a d}{5 a^2 x^5}-\frac{a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{9/2}}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x}-\frac{c}{7 a x^7}",1,"(-15*a^3*c + 21*a^2*b*c*x^2 - 21*a^3*d*x^2 - 35*a*b^2*c*x^4 + 35*a^2*b*d*x^4 - 35*a^3*e*x^4 + 105*b^3*c*x^6 - 105*a*b^2*d*x^6 + 105*a^2*b*e*x^6 - 105*a^3*f*x^6)/(105*a^4*x^7) + ((b^(7/2)*c - a*b^(5/2)*d + a^2*b^(3/2)*e - a^3*Sqrt[b]*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)","A",1
122,1,207,175,0.1903522,"\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 b^{3/2} f-a^2 b^{5/2} e+a b^{7/2} d+b^{9/2} (-c)\right)}{a^{11/2}}+\frac{-35 a^4 c-45 a^4 d x^2-63 a^4 e x^4-105 a^4 f x^6+45 a^3 b c x^2+63 a^3 b d x^4+105 a^3 b e x^6+315 a^3 b f x^8-63 a^2 b^2 c x^4-105 a^2 b^2 d x^6-315 a^2 b^2 e x^8+105 a b^3 c x^6+315 a b^3 d x^8-315 b^4 c x^8}{315 a^5 x^9}","\frac{b c-a d}{7 a^2 x^7}-\frac{a^2 e-a b d+b^2 c}{5 a^3 x^5}-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{11/2}}-\frac{b \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^5 x}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 x^3}-\frac{c}{9 a x^9}",1,"(-35*a^4*c + 45*a^3*b*c*x^2 - 45*a^4*d*x^2 - 63*a^2*b^2*c*x^4 + 63*a^3*b*d*x^4 - 63*a^4*e*x^4 + 105*a*b^3*c*x^6 - 105*a^2*b^2*d*x^6 + 105*a^3*b*e*x^6 - 105*a^4*f*x^6 - 315*b^4*c*x^8 + 315*a*b^3*d*x^8 - 315*a^2*b^2*e*x^8 + 315*a^3*b*f*x^8)/(315*a^5*x^9) + ((-(b^(9/2)*c) + a*b^(7/2)*d - a^2*b^(5/2)*e + a^3*b^(3/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(11/2)","A",1
123,1,255,211,0.2278185,"\int \frac{c+d x^2+e x^4+f x^6}{x^{12} \left(a+b x^2\right)} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^12*(a + b*x^2)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-a^3 b^{5/2} f+a^2 b^{7/2} e-a b^{9/2} d+b^{11/2} c\right)}{a^{13/2}}+\frac{-315 a^5 c-385 a^5 d x^2-495 a^5 e x^4-693 a^5 f x^6+385 a^4 b c x^2+495 a^4 b d x^4+693 a^4 b e x^6+1155 a^4 b f x^8-495 a^3 b^2 c x^4-693 a^3 b^2 d x^6-1155 a^3 b^2 e x^8-3465 a^3 b^2 f x^{10}+693 a^2 b^3 c x^6+1155 a^2 b^3 d x^8+3465 a^2 b^3 e x^{10}-1155 a b^4 c x^8-3465 a b^4 d x^{10}+3465 b^5 c x^{10}}{3465 a^6 x^{11}}","\frac{b c-a d}{9 a^2 x^9}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{13/2}}+\frac{b^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^6 x}-\frac{b \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^5 x^3}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{5 a^4 x^5}-\frac{c}{11 a x^{11}}",1,"(-315*a^5*c + 385*a^4*b*c*x^2 - 385*a^5*d*x^2 - 495*a^3*b^2*c*x^4 + 495*a^4*b*d*x^4 - 495*a^5*e*x^4 + 693*a^2*b^3*c*x^6 - 693*a^3*b^2*d*x^6 + 693*a^4*b*e*x^6 - 693*a^5*f*x^6 - 1155*a*b^4*c*x^8 + 1155*a^2*b^3*d*x^8 - 1155*a^3*b^2*e*x^8 + 1155*a^4*b*f*x^8 + 3465*b^5*c*x^10 - 3465*a*b^4*d*x^10 + 3465*a^2*b^3*e*x^10 - 3465*a^3*b^2*f*x^10)/(3465*a^6*x^11) + ((b^(11/2)*c - a*b^(9/2)*d + a^2*b^(7/2)*e - a^3*b^(5/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(13/2)","A",1
124,1,258,240,0.1918142,"\int \frac{x^6 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(5 a^{3/2} b^3 c-7 a^{5/2} b^2 d+9 a^{7/2} b e-11 a^{9/2} f\right)}{2 b^{13/2}}+\frac{x \left(3465 a^5 f-2835 a^4 b e+2310 a^4 b f x^2+2205 a^3 b^2 d-1890 a^3 b^2 e x^2-462 a^3 b^2 f x^4-1575 a^2 b^3 c+1470 a^2 b^3 d x^2+378 a^2 b^3 e x^4+198 a^2 b^3 f x^6-1050 a b^4 c x^2-294 a b^4 d x^4-162 a b^4 e x^6-110 a b^4 f x^8+210 b^5 c x^4+126 b^5 d x^6+90 b^5 e x^8+70 b^5 f x^{10}\right)}{630 b^6 \left(a+b x^2\right)}","\frac{x^7 \left(c-\frac{a \left(a^2 f-a b e+b^2 d\right)}{b^3}\right)}{2 a \left(a+b x^2\right)}-\frac{a x \left(-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right)}{2 b^6}+\frac{x^3 \left(-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right)}{6 b^5}-\frac{x^5 \left(-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right)}{10 a b^4}+\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right)}{2 b^{13/2}}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^9}{9 b^2}",1,"(x*(-1575*a^2*b^3*c + 2205*a^3*b^2*d - 2835*a^4*b*e + 3465*a^5*f - 1050*a*b^4*c*x^2 + 1470*a^2*b^3*d*x^2 - 1890*a^3*b^2*e*x^2 + 2310*a^4*b*f*x^2 + 210*b^5*c*x^4 - 294*a*b^4*d*x^4 + 378*a^2*b^3*e*x^4 - 462*a^3*b^2*f*x^4 + 126*b^5*d*x^6 - 162*a*b^4*e*x^6 + 198*a^2*b^3*f*x^6 + 90*b^5*e*x^8 - 110*a*b^4*f*x^8 + 70*b^5*f*x^10))/(630*b^6*(a + b*x^2)) + ((5*a^(3/2)*b^3*c - 7*a^(5/2)*b^2*d + 9*a^(7/2)*b*e - 11*a^(9/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))","A",1
125,1,210,202,0.1598675,"\int \frac{x^4 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(5 a^{3/2} b^2 d-7 a^{5/2} b e+9 a^{7/2} f-3 \sqrt{a} b^3 c\right)}{2 b^{11/2}}+\frac{x \left(-945 a^4 f+735 a^3 b e-630 a^3 b f x^2-525 a^2 b^2 d+490 a^2 b^2 e x^2+126 a^2 b^2 f x^4+315 a b^3 c-350 a b^3 d x^2-98 a b^3 e x^4-54 a b^3 f x^6+210 b^4 c x^2+70 b^4 d x^4+42 b^4 e x^6+30 b^4 f x^8\right)}{210 b^5 \left(a+b x^2\right)}","\frac{x^5 \left(c-\frac{a \left(a^2 f-a b e+b^2 d\right)}{b^3}\right)}{2 a \left(a+b x^2\right)}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right)}{2 b^{11/2}}+\frac{x \left(-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right)}{2 b^5}-\frac{x^3 \left(-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right)}{6 a b^4}+\frac{x^5 (b e-2 a f)}{5 b^3}+\frac{f x^7}{7 b^2}",1,"(x*(315*a*b^3*c - 525*a^2*b^2*d + 735*a^3*b*e - 945*a^4*f + 210*b^4*c*x^2 - 350*a*b^3*d*x^2 + 490*a^2*b^2*e*x^2 - 630*a^3*b*f*x^2 + 70*b^4*d*x^4 - 98*a*b^3*e*x^4 + 126*a^2*b^2*f*x^4 + 42*b^4*e*x^6 - 54*a*b^3*f*x^6 + 30*b^4*f*x^8))/(210*b^5*(a + b*x^2)) + ((-3*Sqrt[a]*b^3*c + 5*a^(3/2)*b^2*d - 7*a^(5/2)*b*e + 9*a^(7/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))","A",1
126,1,155,163,0.127334,"\int \frac{x^2 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-7 a^3 f+5 a^2 b e-3 a b^2 d+b^3 c\right)}{2 \sqrt{a} b^{9/2}}+\frac{x \left(105 a^3 f-75 a^2 b e+70 a^2 b f x^2+45 a b^2 d-50 a b^2 e x^2-14 a b^2 f x^4-15 b^3 c+30 b^3 d x^2+10 b^3 e x^4+6 b^3 f x^6\right)}{30 b^4 \left(a+b x^2\right)}","\frac{x^3 \left(c-\frac{a \left(a^2 f-a b e+b^2 d\right)}{b^3}\right)}{2 a \left(a+b x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-7 a^3 f+5 a^2 b e-3 a b^2 d+b^3 c\right)}{2 \sqrt{a} b^{9/2}}-\frac{x \left(-7 a^3 f+5 a^2 b e-3 a b^2 d+b^3 c\right)}{2 a b^4}+\frac{x^3 (b e-2 a f)}{3 b^3}+\frac{f x^5}{5 b^2}",1,"(x*(-15*b^3*c + 45*a*b^2*d - 75*a^2*b*e + 105*a^3*f + 30*b^3*d*x^2 - 50*a*b^2*e*x^2 + 70*a^2*b*f*x^2 + 10*b^3*e*x^4 - 14*a*b^2*f*x^4 + 6*b^3*f*x^6))/(30*b^4*(a + b*x^2)) + ((b^3*c - 3*a*b^2*d + 5*a^2*b*e - 7*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(9/2))","A",1
127,1,130,118,0.1244434,"\int \frac{c+d x^2+e x^4+f x^6}{\left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(a + b*x^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(5 a^3 f-3 a^2 b e+a b^2 d+b^3 c\right)}{2 a^{3/2} b^{7/2}}-\frac{x \left(15 a^3 f-9 a^2 b e+10 a^2 b f x^2+3 a b^2 d-6 a b^2 e x^2-2 a b^2 f x^4-3 b^3 c\right)}{6 a b^3 \left(a+b x^2\right)}","\frac{x \left(c-\frac{a \left(a^2 f-a b e+b^2 d\right)}{b^3}\right)}{2 a \left(a+b x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(5 a^3 f-3 a^2 b e+a b^2 d+b^3 c\right)}{2 a^{3/2} b^{7/2}}+\frac{x (b e-2 a f)}{b^3}+\frac{f x^3}{3 b^2}",1,"-1/6*(x*(-3*b^3*c + 3*a*b^2*d - 9*a^2*b*e + 15*a^3*f - 6*a*b^2*e*x^2 + 10*a^2*b*f*x^2 - 2*a*b^2*f*x^4))/(a*b^3*(a + b*x^2)) + ((b^3*c + a*b^2*d - 3*a^2*b*e + 5*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(7/2))","A",1
128,1,130,112,0.1012791,"\int \frac{c+d x^2+e x^4+f x^6}{x^2 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^2),x]","\frac{3 a^3 f x^2-a^2 b e x^2+2 a^2 b f x^4-2 a b^2 c+a b^2 d x^2-3 b^3 c x^2}{2 a^2 b^2 x \left(a+b x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-3 a^3 f+a^2 b e+a b^2 d-3 b^3 c\right)}{2 a^{5/2} b^{5/2}}","-\frac{x \left(-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right)}{2 a \left(a+b x^2\right)}-\frac{c}{a^2 x}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(3 a^3 f-a^2 b e-a b^2 d+3 b^3 c\right)}{2 a^{5/2} b^{5/2}}+\frac{f x}{b^2}",1,"(-2*a*b^2*c - 3*b^3*c*x^2 + a*b^2*d*x^2 - a^2*b*e*x^2 + 3*a^3*f*x^2 + 2*a^2*b*f*x^4)/(2*a^2*b^2*x*(a + b*x^2)) + ((-3*b^3*c + a*b^2*d + a^2*b*e - 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*b^(5/2))","A",1
129,1,141,121,0.1054221,"\int \frac{c+d x^2+e x^4+f x^6}{x^4 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^4*(a + b*x^2)^2),x]","\frac{-3 a^3 f x^4-2 a^2 b c-6 a^2 b d x^2+3 a^2 b e x^4+10 a b^2 c x^2-9 a b^2 d x^4+15 b^3 c x^4}{6 a^3 b x^3 \left(a+b x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 f+a^2 b e-3 a b^2 d+5 b^3 c\right)}{2 a^{7/2} b^{3/2}}","\frac{2 b c-a d}{a^3 x}+\frac{x \left(\frac{b^2 c}{a^2}-\frac{b d}{a}-\frac{a f}{b}+e\right)}{2 a \left(a+b x^2\right)}-\frac{c}{3 a^2 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 f+a^2 b e-3 a b^2 d+5 b^3 c\right)}{2 a^{7/2} b^{3/2}}",1,"(-2*a^2*b*c + 10*a*b^2*c*x^2 - 6*a^2*b*d*x^2 + 15*b^3*c*x^4 - 9*a*b^2*d*x^4 + 3*a^2*b*e*x^4 - 3*a^3*f*x^4)/(6*a^3*b*x^3*(a + b*x^2)) + ((5*b^3*c - 3*a*b^2*d + a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*b^(3/2))","A",1
130,1,166,152,0.1263004,"\int \frac{c+d x^2+e x^4+f x^6}{x^6 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^6*(a + b*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 f-3 a^2 b e+5 a b^2 d-7 b^3 c\right)}{2 a^{9/2} \sqrt{b}}+\frac{-6 a^3 c-10 a^3 d x^2-30 a^3 e x^4+15 a^3 f x^6+14 a^2 b c x^2+50 a^2 b d x^4-45 a^2 b e x^6-70 a b^2 c x^4+75 a b^2 d x^6-105 b^3 c x^6}{30 a^4 x^5 \left(a+b x^2\right)}","\frac{2 b c-a d}{3 a^3 x^3}-\frac{c}{5 a^2 x^5}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 (-f)+3 a^2 b e-5 a b^2 d+7 b^3 c\right)}{2 a^{9/2} \sqrt{b}}-\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{2 a^4 \left(a+b x^2\right)}",1,"(-6*a^3*c + 14*a^2*b*c*x^2 - 10*a^3*d*x^2 - 70*a*b^2*c*x^4 + 50*a^2*b*d*x^4 - 30*a^3*e*x^4 - 105*b^3*c*x^6 + 75*a*b^2*d*x^6 - 45*a^2*b*e*x^6 + 15*a^3*f*x^6)/(30*a^4*x^5*(a + b*x^2)) + ((-7*b^3*c + 5*a*b^2*d - 3*a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2)*Sqrt[b])","A",1
131,1,221,189,0.1697666,"\int \frac{c+d x^2+e x^4+f x^6}{x^8 \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-3 a^3 \sqrt{b} f+5 a^2 b^{3/2} e-7 a b^{5/2} d+9 b^{7/2} c\right)}{2 a^{11/2}}+\frac{-30 a^4 c-42 a^4 d x^2-70 a^4 e x^4-210 a^4 f x^6+54 a^3 b c x^2+98 a^3 b d x^4+350 a^3 b e x^6-315 a^3 b f x^8-126 a^2 b^2 c x^4-490 a^2 b^2 d x^6+525 a^2 b^2 e x^8+630 a b^3 c x^6-735 a b^3 d x^8+945 b^4 c x^8}{210 a^5 x^7 \left(a+b x^2\right)}","\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{7 a^2 x^7}-\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-3 a^3 f+5 a^2 b e-7 a b^2 d+9 b^3 c\right)}{2 a^{11/2}}+\frac{b x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{2 a^5 \left(a+b x^2\right)}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x}",1,"(-30*a^4*c + 54*a^3*b*c*x^2 - 42*a^4*d*x^2 - 126*a^2*b^2*c*x^4 + 98*a^3*b*d*x^4 - 70*a^4*e*x^4 + 630*a*b^3*c*x^6 - 490*a^2*b^2*d*x^6 + 350*a^3*b*e*x^6 - 210*a^4*f*x^6 + 945*b^4*c*x^8 - 735*a*b^3*d*x^8 + 525*a^2*b^2*e*x^8 - 315*a^3*b*f*x^8)/(210*a^5*x^7*(a + b*x^2)) + ((9*b^(7/2)*c - 7*a*b^(5/2)*d + 5*a^2*b^(3/2)*e - 3*a^3*Sqrt[b]*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(11/2))","A",1
132,1,269,230,0.1893769,"\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \left(a+b x^2\right)^2} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(5 a^3 b^{3/2} f-7 a^2 b^{5/2} e+9 a b^{7/2} d-11 b^{9/2} c\right)}{2 a^{13/2}}+\frac{-70 a^5 c-90 a^5 d x^2-126 a^5 e x^4-210 a^5 f x^6+110 a^4 b c x^2+162 a^4 b d x^4+294 a^4 b e x^6+1050 a^4 b f x^8-198 a^3 b^2 c x^4-378 a^3 b^2 d x^6-1470 a^3 b^2 e x^8+1575 a^3 b^2 f x^{10}+462 a^2 b^3 c x^6+1890 a^2 b^3 d x^8-2205 a^2 b^3 e x^{10}-2310 a b^4 c x^8+2835 a b^4 d x^{10}-3465 b^5 c x^{10}}{630 a^6 x^9 \left(a+b x^2\right)}","\frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{9 a^2 x^9}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-5 a^3 f+7 a^2 b e-9 a b^2 d+11 b^3 c\right)}{2 a^{13/2}}-\frac{b^2 x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{2 a^6 \left(a+b x^2\right)}-\frac{b \left(-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right)}{a^6 x}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{3 a^5 x^3}",1,"(-70*a^5*c + 110*a^4*b*c*x^2 - 90*a^5*d*x^2 - 198*a^3*b^2*c*x^4 + 162*a^4*b*d*x^4 - 126*a^5*e*x^4 + 462*a^2*b^3*c*x^6 - 378*a^3*b^2*d*x^6 + 294*a^4*b*e*x^6 - 210*a^5*f*x^6 - 2310*a*b^4*c*x^8 + 1890*a^2*b^3*d*x^8 - 1470*a^3*b^2*e*x^8 + 1050*a^4*b*f*x^8 - 3465*b^5*c*x^10 + 2835*a*b^4*d*x^10 - 2205*a^2*b^3*e*x^10 + 1575*a^3*b^2*f*x^10)/(630*a^6*x^9*(a + b*x^2)) + ((-11*b^(9/2)*c + 9*a*b^(7/2)*d - 7*a^2*b^(5/2)*e + 5*a^3*b^(3/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(13/2))","A",1
133,1,306,287,0.2468934,"\int \frac{x^8 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^8*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(35 a^{3/2} b^3 c-63 a^{5/2} b^2 d+99 a^{7/2} b e-143 a^{9/2} f\right)}{8 b^{15/2}}+\frac{x \left(45045 a^6 f-31185 a^5 b e+75075 a^5 b f x^2+19845 a^4 b^2 d-51975 a^4 b^2 e x^2+24024 a^4 b^2 f x^4-11025 a^3 b^3 c+33075 a^3 b^3 d x^2-16632 a^3 b^3 e x^4-3432 a^3 b^3 f x^6-18375 a^2 b^4 c x^2+10584 a^2 b^4 d x^4+2376 a^2 b^4 e x^6+1144 a^2 b^4 f x^8-5880 a b^5 c x^4-1512 a b^5 d x^6-792 a b^5 e x^8-520 a b^5 f x^{10}+840 b^6 c x^6+504 b^6 d x^8+360 b^6 e x^{10}+280 b^6 f x^{12}\right)}{2520 b^7 \left(a+b x^2\right)^2}","\frac{x^9 \left(c-\frac{a \left(a^2 f-a b e+b^2 d\right)}{b^3}\right)}{4 a \left(a+b x^2\right)^2}-\frac{a^2 x \left(-17 a^3 f+13 a^2 b e-9 a b^2 d+5 b^3 c\right)}{8 b^7 \left(a+b x^2\right)}-\frac{a x \left(-63 a^3 f+43 a^2 b e-27 a b^2 d+15 b^3 c\right)}{4 b^7}+\frac{x^3 \left(-23 a^3 f+15 a^2 b e-9 a b^2 d+5 b^3 c\right)}{6 b^6}-\frac{x^5 \left(-29 a^3 f+17 a^2 b e-9 a b^2 d+5 b^3 c\right)}{20 a b^5}+\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-143 a^3 f+99 a^2 b e-63 a b^2 d+35 b^3 c\right)}{8 b^{15/2}}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^9}{9 b^3}",1,"(x*(-11025*a^3*b^3*c + 19845*a^4*b^2*d - 31185*a^5*b*e + 45045*a^6*f - 18375*a^2*b^4*c*x^2 + 33075*a^3*b^3*d*x^2 - 51975*a^4*b^2*e*x^2 + 75075*a^5*b*f*x^2 - 5880*a*b^5*c*x^4 + 10584*a^2*b^4*d*x^4 - 16632*a^3*b^3*e*x^4 + 24024*a^4*b^2*f*x^4 + 840*b^6*c*x^6 - 1512*a*b^5*d*x^6 + 2376*a^2*b^4*e*x^6 - 3432*a^3*b^3*f*x^6 + 504*b^6*d*x^8 - 792*a*b^5*e*x^8 + 1144*a^2*b^4*f*x^8 + 360*b^6*e*x^10 - 520*a*b^5*f*x^10 + 280*b^6*f*x^12))/(2520*b^7*(a + b*x^2)^2) + ((35*a^(3/2)*b^3*c - 63*a^(5/2)*b^2*d + 99*a^(7/2)*b*e - 143*a^(9/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(15/2))","A",1
134,1,258,247,0.220479,"\int \frac{x^6 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(35 a^{3/2} b^2 d-63 a^{5/2} b e+99 a^{7/2} f-15 \sqrt{a} b^3 c\right)}{8 b^{13/2}}+\frac{x \left(-10395 a^5 f+6615 a^4 b e-17325 a^4 b f x^2-3675 a^3 b^2 d+11025 a^3 b^2 e x^2-5544 a^3 b^2 f x^4+1575 a^2 b^3 c-6125 a^2 b^3 d x^2+3528 a^2 b^3 e x^4+792 a^2 b^3 f x^6+2625 a b^4 c x^2-1960 a b^4 d x^4-504 a b^4 e x^6-264 a b^4 f x^8+840 b^5 c x^4+280 b^5 d x^6+168 b^5 e x^8+120 b^5 f x^{10}\right)}{840 b^6 \left(a+b x^2\right)^2}","\frac{x^7 \left(c-\frac{a \left(a^2 f-a b e+b^2 d\right)}{b^3}\right)}{4 a \left(a+b x^2\right)^2}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right)}{8 b^{13/2}}+\frac{a x \left(-15 a^3 f+11 a^2 b e-7 a b^2 d+3 b^3 c\right)}{8 b^6 \left(a+b x^2\right)}+\frac{x \left(-21 a^3 f+13 a^2 b e-7 a b^2 d+3 b^3 c\right)}{2 b^6}-\frac{x^3 \left(-27 a^3 f+15 a^2 b e-7 a b^2 d+3 b^3 c\right)}{12 a b^5}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^7}{7 b^3}",1,"(x*(1575*a^2*b^3*c - 3675*a^3*b^2*d + 6615*a^4*b*e - 10395*a^5*f + 2625*a*b^4*c*x^2 - 6125*a^2*b^3*d*x^2 + 11025*a^3*b^2*e*x^2 - 17325*a^4*b*f*x^2 + 840*b^5*c*x^4 - 1960*a*b^4*d*x^4 + 3528*a^2*b^3*e*x^4 - 5544*a^3*b^2*f*x^4 + 280*b^5*d*x^6 - 504*a*b^4*e*x^6 + 792*a^2*b^3*f*x^6 + 168*b^5*e*x^8 - 264*a*b^4*f*x^8 + 120*b^5*f*x^10))/(840*b^6*(a + b*x^2)^2) + ((-15*Sqrt[a]*b^3*c + 35*a^(3/2)*b^2*d - 63*a^(5/2)*b*e + 99*a^(7/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))","A",1
135,1,202,207,0.2230125,"\int \frac{x^4 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right)}{8 \sqrt{a} b^{11/2}}+\frac{x \left(945 a^4 f-525 a^3 b e+1575 a^3 b f x^2+225 a^2 b^2 d-875 a^2 b^2 e x^2+504 a^2 b^2 f x^4-45 a b^3 c+375 a b^3 d x^2-280 a b^3 e x^4-72 a b^3 f x^6-75 b^4 c x^2+120 b^4 d x^4+40 b^4 e x^6+24 b^4 f x^8\right)}{120 b^5 \left(a+b x^2\right)^2}","\frac{x^5 \left(c-\frac{a \left(a^2 f-a b e+b^2 d\right)}{b^3}\right)}{4 a \left(a+b x^2\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right)}{8 \sqrt{a} b^{11/2}}-\frac{x \left(-13 a^3 f+9 a^2 b e-5 a b^2 d+b^3 c\right)}{8 b^5 \left(a+b x^2\right)}-\frac{x \left(-25 a^3 f+13 a^2 b e-5 a b^2 d+b^3 c\right)}{4 a b^5}+\frac{x^3 (b e-3 a f)}{3 b^4}+\frac{f x^5}{5 b^3}",1,"(x*(-45*a*b^3*c + 225*a^2*b^2*d - 525*a^3*b*e + 945*a^4*f - 75*b^4*c*x^2 + 375*a*b^3*d*x^2 - 875*a^2*b^2*e*x^2 + 1575*a^3*b*f*x^2 + 120*b^4*d*x^4 - 280*a*b^3*e*x^4 + 504*a^2*b^2*f*x^4 + 40*b^4*e*x^6 - 72*a*b^3*f*x^6 + 24*b^4*f*x^8))/(120*b^5*(a + b*x^2)^2) + ((3*b^3*c - 15*a*b^2*d + 35*a^2*b*e - 63*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(11/2))","A",1
136,1,177,167,0.1737964,"\int \frac{x^2 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(35 a^3 f-15 a^2 b e+3 a b^2 d+b^3 c\right)}{8 a^{3/2} b^{9/2}}-\frac{x \left(105 a^4 f-45 a^3 b e+175 a^3 b f x^2+9 a^2 b^2 d-75 a^2 b^2 e x^2+56 a^2 b^2 f x^4+3 a b^3 c+15 a b^3 d x^2-24 a b^3 e x^4-8 a b^3 f x^6-3 b^4 c x^2\right)}{24 a b^4 \left(a+b x^2\right)^2}","\frac{x^3 \left(c-\frac{a \left(a^2 f-a b e+b^2 d\right)}{b^3}\right)}{4 a \left(a+b x^2\right)^2}-\frac{x \left(11 a^3 f-7 a^2 b e+3 a b^2 d+b^3 c\right)}{8 a b^4 \left(a+b x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(35 a^3 f-15 a^2 b e+3 a b^2 d+b^3 c\right)}{8 a^{3/2} b^{9/2}}+\frac{x (b e-3 a f)}{b^4}+\frac{f x^3}{3 b^3}",1,"-1/24*(x*(3*a*b^3*c + 9*a^2*b^2*d - 45*a^3*b*e + 105*a^4*f - 3*b^4*c*x^2 + 15*a*b^3*d*x^2 - 75*a^2*b^2*e*x^2 + 175*a^3*b*f*x^2 - 24*a*b^3*e*x^4 + 56*a^2*b^2*f*x^4 - 8*a*b^3*f*x^6))/(a*b^4*(a + b*x^2)^2) + ((b^3*c + 3*a*b^2*d - 15*a^2*b*e + 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(9/2))","A",1
137,1,156,147,0.1607526,"\int \frac{c+d x^2+e x^4+f x^6}{\left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(a + b*x^2)^3,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-15 a^3 f+3 a^2 b e+a b^2 d+3 b^3 c\right)}{8 a^{5/2} b^{7/2}}+\frac{x \left(15 a^4 f-3 a^3 b e+25 a^3 b f x^2-a^2 b^2 d-5 a^2 b^2 e x^2+8 a^2 b^2 f x^4+5 a b^3 c+a b^3 d x^2+3 b^4 c x^2\right)}{8 a^2 b^3 \left(a+b x^2\right)^2}","\frac{x \left(c-\frac{a \left(a^2 f-a b e+b^2 d\right)}{b^3}\right)}{4 a \left(a+b x^2\right)^2}+\frac{x \left(9 a^3 f-5 a^2 b e+a b^2 d+3 b^3 c\right)}{8 a^2 b^3 \left(a+b x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-15 a^3 f+3 a^2 b e+a b^2 d+3 b^3 c\right)}{8 a^{5/2} b^{7/2}}+\frac{f x}{b^3}",1,"(x*(5*a*b^3*c - a^2*b^2*d - 3*a^3*b*e + 15*a^4*f + 3*b^4*c*x^2 + a*b^3*d*x^2 - 5*a^2*b^2*e*x^2 + 25*a^3*b*f*x^2 + 8*a^2*b^2*f*x^4))/(8*a^2*b^3*(a + b*x^2)^2) + ((3*b^3*c + a*b^2*d + 3*a^2*b*e - 15*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(7/2))","A",1
138,1,167,153,0.1692594,"\int \frac{c+d x^2+e x^4+f x^6}{x^2 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^3),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(3 a^3 f+a^2 b e+3 a b^2 d-15 b^3 c\right)}{8 a^{7/2} b^{5/2}}+\frac{-3 a^4 f x^2-a^3 b e x^2-5 a^3 b f x^4-8 a^2 b^2 c+5 a^2 b^2 d x^2+a^2 b^2 e x^4-25 a b^3 c x^2+3 a b^3 d x^4-15 b^4 c x^4}{8 a^3 b^2 x \left(a+b x^2\right)^2}","-\frac{c}{a^3 x}-\frac{x \left(-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right)}{4 a \left(a+b x^2\right)^2}-\frac{x \left(5 a^3 f-a^2 b e-3 a b^2 d+7 b^3 c\right)}{8 a^3 b^2 \left(a+b x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-3 a^3 f-a^2 b e-3 a b^2 d+15 b^3 c\right)}{8 a^{7/2} b^{5/2}}",1,"(-8*a^2*b^2*c - 25*a*b^3*c*x^2 + 5*a^2*b^2*d*x^2 - a^3*b*e*x^2 - 3*a^4*f*x^2 - 15*b^4*c*x^4 + 3*a*b^3*d*x^4 + a^2*b^2*e*x^4 - 5*a^3*b*f*x^4)/(8*a^3*b^2*x*(a + b*x^2)^2) + ((-15*b^3*c + 3*a*b^2*d + a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*b^(5/2))","A",1
139,1,188,168,0.1968144,"\int \frac{c+d x^2+e x^4+f x^6}{x^4 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^4*(a + b*x^2)^3),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 f+3 a^2 b e-15 a b^2 d+35 b^3 c\right)}{8 a^{9/2} b^{3/2}}+\frac{-3 a^4 f x^4-8 a^3 b c-24 a^3 b d x^2+15 a^3 b e x^4+3 a^3 b f x^6+56 a^2 b^2 c x^2-75 a^2 b^2 d x^4+9 a^2 b^2 e x^6+175 a b^3 c x^4-45 a b^3 d x^6+105 b^4 c x^6}{24 a^4 b x^3 \left(a+b x^2\right)^2}","\frac{3 b c-a d}{a^4 x}-\frac{c}{3 a^3 x^3}+\frac{x \left(\frac{b^2 c}{a^2}-\frac{b d}{a}-\frac{a f}{b}+e\right)}{4 a \left(a+b x^2\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(a^3 f+3 a^2 b e-15 a b^2 d+35 b^3 c\right)}{8 a^{9/2} b^{3/2}}+\frac{x \left(a^3 f+3 a^2 b e-7 a b^2 d+11 b^3 c\right)}{8 a^4 b \left(a+b x^2\right)}",1,"(-8*a^3*b*c + 56*a^2*b^2*c*x^2 - 24*a^3*b*d*x^2 + 175*a*b^3*c*x^4 - 75*a^2*b^2*d*x^4 + 15*a^3*b*e*x^4 - 3*a^4*f*x^4 + 105*b^4*c*x^6 - 45*a*b^3*d*x^6 + 9*a^2*b^2*e*x^6 + 3*a^3*b*f*x^6)/(24*a^4*b*x^3*(a + b*x^2)^2) + ((35*b^3*c - 15*a*b^2*d + 3*a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2)*b^(3/2))","A",1
140,1,213,196,0.1764574,"\int \frac{c+d x^2+e x^4+f x^6}{x^6 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^6*(a + b*x^2)^3),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(3 a^3 f-15 a^2 b e+35 a b^2 d-63 b^3 c\right)}{8 a^{11/2} \sqrt{b}}+\frac{-24 a^4 c-40 a^4 d x^2-120 a^4 e x^4+75 a^4 f x^6+72 a^3 b c x^2+280 a^3 b d x^4-375 a^3 b e x^6+45 a^3 b f x^8-504 a^2 b^2 c x^4+875 a^2 b^2 d x^6-225 a^2 b^2 e x^8-1575 a b^3 c x^6+525 a b^3 d x^8-945 b^4 c x^8}{120 a^5 x^5 \left(a+b x^2\right)^2}","\frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{5 a^3 x^5}-\frac{a^2 e-3 a b d+6 b^2 c}{a^5 x}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-3 a^3 f+15 a^2 b e-35 a b^2 d+63 b^3 c\right)}{8 a^{11/2} \sqrt{b}}-\frac{x \left(-3 a^3 f+7 a^2 b e-11 a b^2 d+15 b^3 c\right)}{8 a^5 \left(a+b x^2\right)}-\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{4 a^4 \left(a+b x^2\right)^2}",1,"(-24*a^4*c + 72*a^3*b*c*x^2 - 40*a^4*d*x^2 - 504*a^2*b^2*c*x^4 + 280*a^3*b*d*x^4 - 120*a^4*e*x^4 - 1575*a*b^3*c*x^6 + 875*a^2*b^2*d*x^6 - 375*a^3*b*e*x^6 + 75*a^4*f*x^6 - 945*b^4*c*x^8 + 525*a*b^3*d*x^8 - 225*a^2*b^2*e*x^8 + 45*a^3*b*f*x^8)/(120*a^5*x^5*(a + b*x^2)^2) + ((-63*b^3*c + 35*a*b^2*d - 15*a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(11/2)*Sqrt[b])","A",1
141,1,269,234,0.2212751,"\int \frac{c+d x^2+e x^4+f x^6}{x^8 \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^3),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-15 a^3 \sqrt{b} f+35 a^2 b^{3/2} e-63 a b^{5/2} d+99 b^{7/2} c\right)}{8 a^{13/2}}+\frac{-120 a^5 c-168 a^5 d x^2-280 a^5 e x^4-840 a^5 f x^6+264 a^4 b c x^2+504 a^4 b d x^4+1960 a^4 b e x^6-2625 a^4 b f x^8-792 a^3 b^2 c x^4-3528 a^3 b^2 d x^6+6125 a^3 b^2 e x^8-1575 a^3 b^2 f x^{10}+5544 a^2 b^3 c x^6-11025 a^2 b^3 d x^8+3675 a^2 b^3 e x^{10}+17325 a b^4 c x^8-6615 a b^4 d x^{10}+10395 b^5 c x^{10}}{840 a^6 x^7 \left(a+b x^2\right)^2}","\frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{7 a^3 x^7}-\frac{a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-15 a^3 f+35 a^2 b e-63 a b^2 d+99 b^3 c\right)}{8 a^{13/2}}+\frac{b x \left(-7 a^3 f+11 a^2 b e-15 a b^2 d+19 b^3 c\right)}{8 a^6 \left(a+b x^2\right)}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{b x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{4 a^5 \left(a+b x^2\right)^2}",1,"(-120*a^5*c + 264*a^4*b*c*x^2 - 168*a^5*d*x^2 - 792*a^3*b^2*c*x^4 + 504*a^4*b*d*x^4 - 280*a^5*e*x^4 + 5544*a^2*b^3*c*x^6 - 3528*a^3*b^2*d*x^6 + 1960*a^4*b*e*x^6 - 840*a^5*f*x^6 + 17325*a*b^4*c*x^8 - 11025*a^2*b^3*d*x^8 + 6125*a^3*b^2*e*x^8 - 2625*a^4*b*f*x^8 + 10395*b^5*c*x^10 - 6615*a*b^4*d*x^10 + 3675*a^2*b^3*e*x^10 - 1575*a^3*b^2*f*x^10)/(840*a^6*x^7*(a + b*x^2)^2) + ((99*b^(7/2)*c - 63*a*b^(5/2)*d + 35*a^2*b^(3/2)*e - 15*a^3*Sqrt[b]*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(13/2))","A",1
142,1,317,277,0.2448929,"\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \left(a+b x^2\right)^3} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^3),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(35 a^3 b^{3/2} f-63 a^2 b^{5/2} e+99 a b^{7/2} d-143 b^{9/2} c\right)}{8 a^{15/2}}+\frac{-280 a^6 c-360 a^6 d x^2-504 a^6 e x^4-840 a^6 f x^6+520 a^5 b c x^2+792 a^5 b d x^4+1512 a^5 b e x^6+5880 a^5 b f x^8-1144 a^4 b^2 c x^4-2376 a^4 b^2 d x^6-10584 a^4 b^2 e x^8+18375 a^4 b^2 f x^{10}+3432 a^3 b^3 c x^6+16632 a^3 b^3 d x^8-33075 a^3 b^3 e x^{10}+11025 a^3 b^3 f x^{12}-24024 a^2 b^4 c x^8+51975 a^2 b^4 d x^{10}-19845 a^2 b^4 e x^{12}-75075 a b^5 c x^{10}+31185 a b^5 d x^{12}-45045 b^6 c x^{12}}{2520 a^7 x^9 \left(a+b x^2\right)^2}","\frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{9 a^3 x^9}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(-35 a^3 f+63 a^2 b e-99 a b^2 d+143 b^3 c\right)}{8 a^{15/2}}-\frac{b^2 x \left(-11 a^3 f+15 a^2 b e-19 a b^2 d+23 b^3 c\right)}{8 a^7 \left(a+b x^2\right)}-\frac{b \left(-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right)}{a^7 x}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{3 a^6 x^3}-\frac{b^2 x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{4 a^6 \left(a+b x^2\right)^2}",1,"(-280*a^6*c + 520*a^5*b*c*x^2 - 360*a^6*d*x^2 - 1144*a^4*b^2*c*x^4 + 792*a^5*b*d*x^4 - 504*a^6*e*x^4 + 3432*a^3*b^3*c*x^6 - 2376*a^4*b^2*d*x^6 + 1512*a^5*b*e*x^6 - 840*a^6*f*x^6 - 24024*a^2*b^4*c*x^8 + 16632*a^3*b^3*d*x^8 - 10584*a^4*b^2*e*x^8 + 5880*a^5*b*f*x^8 - 75075*a*b^5*c*x^10 + 51975*a^2*b^4*d*x^10 - 33075*a^3*b^3*e*x^10 + 18375*a^4*b^2*f*x^10 - 45045*b^6*c*x^12 + 31185*a*b^5*d*x^12 - 19845*a^2*b^4*e*x^12 + 11025*a^3*b^3*f*x^12)/(2520*a^7*x^9*(a + b*x^2)^2) + ((-143*b^(9/2)*c + 99*a*b^(7/2)*d - 63*a^2*b^(5/2)*e + 35*a^3*b^(3/2)*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(15/2))","A",1
143,1,196,214,0.133203,"\int \frac{x^5 \left(c+d x^2+e x^4+f x^6\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x^5*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(-1280 a^5 f+1408 a^4 b e+640 a^4 b f x^2-1584 a^3 b^2 d-704 a^3 b^2 e x^2-480 a^3 b^2 f x^4+1848 a^2 b^3 c+792 a^2 b^3 d x^2+528 a^2 b^3 e x^4+400 a^2 b^3 f x^6-924 a b^4 c x^2-594 a b^4 d x^4-440 a b^4 e x^6-350 a b^4 f x^8+693 b^5 c x^4+495 b^5 d x^6+385 b^5 e x^8+315 b^5 f x^{10}\right)}{3465 b^6}","\frac{\left(a+b x^2\right)^{7/2} \left(10 a^2 f-4 a b e+b^2 d\right)}{7 b^6}+\frac{\left(a+b x^2\right)^{5/2} \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{5 b^6}-\frac{a \left(a+b x^2\right)^{3/2} \left(-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right)}{3 b^6}+\frac{a^2 \sqrt{a+b x^2} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^6}+\frac{\left(a+b x^2\right)^{9/2} (b e-5 a f)}{9 b^6}+\frac{f \left(a+b x^2\right)^{11/2}}{11 b^6}",1,"(Sqrt[a + b*x^2]*(1848*a^2*b^3*c - 1584*a^3*b^2*d + 1408*a^4*b*e - 1280*a^5*f - 924*a*b^4*c*x^2 + 792*a^2*b^3*d*x^2 - 704*a^3*b^2*e*x^2 + 640*a^4*b*f*x^2 + 693*b^5*c*x^4 - 594*a*b^4*d*x^4 + 528*a^2*b^3*e*x^4 - 480*a^3*b^2*f*x^4 + 495*b^5*d*x^6 - 440*a*b^4*e*x^6 + 400*a^2*b^3*f*x^6 + 385*b^5*e*x^8 - 350*a*b^4*f*x^8 + 315*b^5*f*x^10))/(3465*b^6)","A",1
144,1,148,167,0.1057608,"\int \frac{x^3 \left(c+d x^2+e x^4+f x^6\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x^3*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(128 a^4 f-144 a^3 b e-64 a^3 b f x^2+168 a^2 b^2 d+72 a^2 b^2 e x^2+48 a^2 b^2 f x^4-210 a b^3 c-84 a b^3 d x^2-54 a b^3 e x^4-40 a b^3 f x^6+105 b^4 c x^2+63 b^4 d x^4+45 b^4 e x^6+35 b^4 f x^8\right)}{315 b^5}","\frac{\left(a+b x^2\right)^{5/2} \left(6 a^2 f-3 a b e+b^2 d\right)}{5 b^5}+\frac{\left(a+b x^2\right)^{3/2} \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{3 b^5}-\frac{a \sqrt{a+b x^2} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^5}+\frac{\left(a+b x^2\right)^{7/2} (b e-4 a f)}{7 b^5}+\frac{f \left(a+b x^2\right)^{9/2}}{9 b^5}",1,"(Sqrt[a + b*x^2]*(-210*a*b^3*c + 168*a^2*b^2*d - 144*a^3*b*e + 128*a^4*f + 105*b^4*c*x^2 - 84*a*b^3*d*x^2 + 72*a^2*b^2*e*x^2 - 64*a^3*b*f*x^2 + 63*b^4*d*x^4 - 54*a*b^3*e*x^4 + 48*a^2*b^2*f*x^4 + 45*b^4*e*x^6 - 40*a*b^3*f*x^6 + 35*b^4*f*x^8))/(315*b^5)","A",1
145,1,102,121,0.0673795,"\int \frac{x \left(c+d x^2+e x^4+f x^6\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(-48 a^3 f+56 a^2 b e+24 a^2 b f x^2-70 a b^2 d-28 a b^2 e x^2-18 a b^2 f x^4+105 b^3 c+35 b^3 d x^2+21 b^3 e x^4+15 b^3 f x^6\right)}{105 b^4}","\frac{\left(a+b x^2\right)^{3/2} \left(3 a^2 f-2 a b e+b^2 d\right)}{3 b^4}+\frac{\sqrt{a+b x^2} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^4}+\frac{\left(a+b x^2\right)^{5/2} (b e-3 a f)}{5 b^4}+\frac{f \left(a+b x^2\right)^{7/2}}{7 b^4}",1,"(Sqrt[a + b*x^2]*(105*b^3*c - 70*a*b^2*d + 56*a^2*b*e - 48*a^3*f + 35*b^3*d*x^2 - 28*a*b^2*e*x^2 + 24*a^2*b*f*x^2 + 21*b^3*e*x^4 - 18*a*b^2*f*x^4 + 15*b^3*f*x^6))/(105*b^4)","A",1
146,1,89,103,0.1534928,"\int \frac{c+d x^2+e x^4+f x^6}{x \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(8 a^2 f-10 a b e-4 a b f x^2+15 b^2 d+5 b^2 e x^2+3 b^2 f x^4\right)}{15 b^3}-\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}","\frac{\sqrt{a+b x^2} \left(a^2 f-a b e+b^2 d\right)}{b^3}+\frac{\left(a+b x^2\right)^{3/2} (b e-2 a f)}{3 b^3}+\frac{f \left(a+b x^2\right)^{5/2}}{5 b^3}-\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(Sqrt[a + b*x^2]*(15*b^2*d - 10*a*b*e + 8*a^2*f + 5*b^2*e*x^2 - 4*a*b*f*x^2 + 3*b^2*f*x^4))/(15*b^3) - (c*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]","A",1
147,1,92,100,0.2194158,"\int \frac{c+d x^2+e x^4+f x^6}{x^3 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^3*Sqrt[a + b*x^2]),x]","\frac{(b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{3/2}}+\frac{\sqrt{a+b x^2} \left(-4 a^2 f x^2+6 a b e x^2+2 a b f x^4-3 b^2 c\right)}{6 a b^2 x^2}","\frac{(b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{3/2}}+\frac{\sqrt{a+b x^2} (b e-a f)}{b^2}+\frac{f \left(a+b x^2\right)^{3/2}}{3 b^2}-\frac{c \sqrt{a+b x^2}}{2 a x^2}",1,"(Sqrt[a + b*x^2]*(-3*b^2*c + 6*a*b*e*x^2 - 4*a^2*f*x^2 + 2*a*b*f*x^4))/(6*a*b^2*x^2) + ((b*c - 2*a*d)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(3/2))","A",1
148,1,102,114,0.2021353,"\int \frac{c+d x^2+e x^4+f x^6}{x^5 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^5*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(8 a^2 f x^4-2 a b c-4 a b d x^2+3 b^2 c x^2\right)}{8 a^2 b x^4}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right) \left(-8 a^2 e+4 a b d-3 b^2 c\right)}{8 a^{5/2}}","\frac{\sqrt{a+b x^2} (3 b c-4 a d)}{8 a^2 x^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right) \left(8 a^2 e-4 a b d+3 b^2 c\right)}{8 a^{5/2}}-\frac{c \sqrt{a+b x^2}}{4 a x^4}+\frac{f \sqrt{a+b x^2}}{b}",1,"(Sqrt[a + b*x^2]*(-2*a*b*c + 3*b^2*c*x^2 - 4*a*b*d*x^2 + 8*a^2*f*x^4))/(8*a^2*b*x^4) + ((-3*b^2*c + 4*a*b*d - 8*a^2*e)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(5/2))","A",1
149,1,126,146,0.2957817,"\int \frac{c+d x^2+e x^4+f x^6}{x^7 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^7*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(-8 a^2 c-12 a^2 d x^2-24 a^2 e x^4+10 a b c x^2+18 a b d x^4-15 b^2 c x^4\right)}{48 a^3 x^6}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right) \left(-16 a^3 f+8 a^2 b e-6 a b^2 d+5 b^3 c\right)}{16 a^{7/2}}","\frac{\sqrt{a+b x^2} (5 b c-6 a d)}{24 a^2 x^4}-\frac{\sqrt{a+b x^2} \left(8 a^2 e-6 a b d+5 b^2 c\right)}{16 a^3 x^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right) \left(-16 a^3 f+8 a^2 b e-6 a b^2 d+5 b^3 c\right)}{16 a^{7/2}}-\frac{c \sqrt{a+b x^2}}{6 a x^6}",1,"(Sqrt[a + b*x^2]*(-8*a^2*c + 10*a*b*c*x^2 - 12*a^2*d*x^2 - 15*b^2*c*x^4 + 18*a*b*d*x^4 - 24*a^2*e*x^4))/(48*a^3*x^6) + ((5*b^3*c - 6*a*b^2*d + 8*a^2*b*e - 16*a^3*f)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(7/2))","A",1
150,1,172,195,0.3904033,"\int \frac{c+d x^2+e x^4+f x^6}{x^9 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^9*Sqrt[a + b*x^2]),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right) \left(64 a^3 b f-48 a^2 b^2 e+40 a b^3 d-35 b^4 c\right)}{128 a^{9/2}}+\frac{\sqrt{a+b x^2} \left(-48 a^3 c-64 a^3 d x^2-96 a^3 e x^4-192 a^3 f x^6+56 a^2 b c x^2+80 a^2 b d x^4+144 a^2 b e x^6-70 a b^2 c x^4-120 a b^2 d x^6+105 b^3 c x^6\right)}{384 a^4 x^8}","\frac{\sqrt{a+b x^2} (7 b c-8 a d)}{48 a^2 x^6}-\frac{\sqrt{a+b x^2} \left(48 a^2 e-40 a b d+35 b^2 c\right)}{192 a^3 x^4}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right) \left(-64 a^3 f+48 a^2 b e-40 a b^2 d+35 b^3 c\right)}{128 a^{9/2}}+\frac{\sqrt{a+b x^2} \left(-64 a^3 f+48 a^2 b e-40 a b^2 d+35 b^3 c\right)}{128 a^4 x^2}-\frac{c \sqrt{a+b x^2}}{8 a x^8}",1,"(Sqrt[a + b*x^2]*(-48*a^3*c + 56*a^2*b*c*x^2 - 64*a^3*d*x^2 - 70*a*b^2*c*x^4 + 80*a^2*b*d*x^4 - 96*a^3*e*x^4 + 105*b^3*c*x^6 - 120*a*b^2*d*x^6 + 144*a^2*b*e*x^6 - 192*a^3*f*x^6))/(384*a^4*x^8) + ((-35*b^4*c + 40*a*b^3*d - 48*a^2*b^2*e + 64*a^3*b*f)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(128*a^(9/2))","A",1
151,1,213,245,1.0374357,"\int \frac{x^4 \left(c+d x^2+e x^4+f x^6\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(945 a^4 f x-1050 a^3 b e x-630 a^3 b f x^3+1200 a^2 b^2 d x+700 a^2 b^2 e x^3+504 a^2 b^2 f x^5-1440 a b^3 c x-800 a b^3 d x^3-560 a b^3 e x^5-432 a b^3 f x^7+960 b^4 c x^3+640 b^4 d x^5+480 b^4 e x^7+384 b^4 f x^9\right)}{3840 b^5}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right) \left(-63 a^5 f+70 a^4 b e-80 a^3 b^2 d+96 a^2 b^3 c\right)}{256 b^{11/2}}","\frac{x^5 \sqrt{a+b x^2} \left(63 a^2 f-70 a b e+80 b^2 d\right)}{480 b^3}+\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right) \left(-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right)}{256 b^{11/2}}-\frac{a x \sqrt{a+b x^2} \left(-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right)}{256 b^5}+\frac{x^3 \sqrt{a+b x^2} \left(-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right)}{384 b^4}+\frac{x^7 \sqrt{a+b x^2} (10 b e-9 a f)}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b}",1,"(Sqrt[a + b*x^2]*(-1440*a*b^3*c*x + 1200*a^2*b^2*d*x - 1050*a^3*b*e*x + 945*a^4*f*x + 960*b^4*c*x^3 - 800*a*b^3*d*x^3 + 700*a^2*b^2*e*x^3 - 630*a^3*b*f*x^3 + 640*b^4*d*x^5 - 560*a*b^3*e*x^5 + 504*a^2*b^2*f*x^5 + 480*b^4*e*x^7 - 432*a*b^3*f*x^7 + 384*b^4*f*x^9))/(3840*b^5) + ((96*a^2*b^3*c - 80*a^3*b^2*d + 70*a^4*b*e - 63*a^5*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(11/2))","A",1
152,1,165,194,0.6602957,"\int \frac{x^2 \left(c+d x^2+e x^4+f x^6\right)}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(-105 a^3 f x+120 a^2 b e x+70 a^2 b f x^3-144 a b^2 d x-80 a b^2 e x^3-56 a b^2 f x^5+192 b^3 c x+96 b^3 d x^3+64 b^3 e x^5+48 b^3 f x^7\right)}{384 b^4}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right) \left(35 a^4 f-40 a^3 b e+48 a^2 b^2 d-64 a b^3 c\right)}{128 b^{9/2}}","\frac{x^3 \sqrt{a+b x^2} \left(35 a^2 f-40 a b e+48 b^2 d\right)}{192 b^3}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right) \left(-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right)}{128 b^{9/2}}+\frac{x \sqrt{a+b x^2} \left(-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right)}{128 b^4}+\frac{x^5 \sqrt{a+b x^2} (8 b e-7 a f)}{48 b^2}+\frac{f x^7 \sqrt{a+b x^2}}{8 b}",1,"(Sqrt[a + b*x^2]*(192*b^3*c*x - 144*a*b^2*d*x + 120*a^2*b*e*x - 105*a^3*f*x + 96*b^3*d*x^3 - 80*a*b^2*e*x^3 + 70*a^2*b*f*x^3 + 64*b^3*e*x^5 - 56*a*b^2*f*x^5 + 48*b^3*f*x^7))/(384*b^4) + ((-64*a*b^3*c + 48*a^2*b^2*d - 40*a^3*b*e + 35*a^4*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(9/2))","A",1
153,1,121,145,0.4362898,"\int \frac{c+d x^2+e x^4+f x^6}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(15 a^2 f x-18 a b e x-10 a b f x^3+24 b^2 d x+12 b^2 e x^3+8 b^2 f x^5\right)}{48 b^3}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right) \left(-5 a^3 f+6 a^2 b e-8 a b^2 d+16 b^3 c\right)}{16 b^{7/2}}","\frac{x \sqrt{a+b x^2} \left(5 a^2 f-6 a b e+8 b^2 d\right)}{16 b^3}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right) \left(-5 a^3 f+6 a^2 b e-8 a b^2 d+16 b^3 c\right)}{16 b^{7/2}}+\frac{x^3 \sqrt{a+b x^2} (6 b e-5 a f)}{24 b^2}+\frac{f x^5 \sqrt{a+b x^2}}{6 b}",1,"(Sqrt[a + b*x^2]*(24*b^2*d*x - 18*a*b*e*x + 15*a^2*f*x + 12*b^2*e*x^3 - 10*a*b*f*x^3 + 8*b^2*f*x^5))/(48*b^3) + ((16*b^3*c - 8*a*b^2*d + 6*a^2*b*e - 5*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2))","A",1
154,1,103,117,0.3808707,"\int \frac{c+d x^2+e x^4+f x^6}{x^2 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(-3 a^2 f x^2+4 a b e x^2+2 a b f x^4-8 b^2 c\right)}{8 a b^2 x}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right) \left(3 a^2 f-4 a b e+8 b^2 d\right)}{8 b^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right) \left(3 a^2 f-4 a b e+8 b^2 d\right)}{8 b^{5/2}}+\frac{x \sqrt{a+b x^2} (4 b e-3 a f)}{8 b^2}-\frac{c \sqrt{a+b x^2}}{a x}+\frac{f x^3 \sqrt{a+b x^2}}{4 b}",1,"(Sqrt[a + b*x^2]*(-8*b^2*c + 4*a*b*e*x^2 - 3*a^2*f*x^2 + 2*a*b*f*x^4))/(8*a*b^2*x) + ((8*b^2*d - 4*a*b*e + 3*a^2*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(5/2))","A",1
155,1,94,110,0.3184182,"\int \frac{c+d x^2+e x^4+f x^6}{x^4 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^4*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(3 a^2 f x^4-2 a b c-6 a b d x^2+4 b^2 c x^2\right)}{6 a^2 b x^3}+\frac{(2 b e-a f) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}","\frac{\sqrt{a+b x^2} (2 b c-3 a d)}{3 a^2 x}+\frac{(2 b e-a f) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}-\frac{c \sqrt{a+b x^2}}{3 a x^3}+\frac{f x \sqrt{a+b x^2}}{2 b}",1,"(Sqrt[a + b*x^2]*(-2*a*b*c + 4*b^2*c*x^2 - 6*a*b*d*x^2 + 3*a^2*f*x^4))/(6*a^2*b*x^3) + ((2*b*e - a*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))","A",1
156,1,98,118,0.2749919,"\int \frac{c+d x^2+e x^4+f x^6}{x^6 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^6*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(-3 a^2 c-5 a^2 d x^2-15 a^2 e x^4+4 a b c x^2+10 a b d x^4-8 b^2 c x^4\right)}{15 a^3 x^5}+\frac{f \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}","\frac{\sqrt{a+b x^2} (4 b c-5 a d)}{15 a^2 x^3}-\frac{\sqrt{a+b x^2} \left(15 a^2 e-10 a b d+8 b^2 c\right)}{15 a^3 x}-\frac{c \sqrt{a+b x^2}}{5 a x^5}+\frac{f \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}",1,"(Sqrt[a + b*x^2]*(-3*a^2*c + 4*a*b*c*x^2 - 5*a^2*d*x^2 - 8*b^2*c*x^4 + 10*a*b*d*x^4 - 15*a^2*e*x^4))/(15*a^3*x^5) + (f*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b]","A",1
157,1,114,140,0.1783651,"\int \frac{c+d x^2+e x^4+f x^6}{x^8 \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(-15 a^3 c-21 a^3 d x^2-35 a^3 e x^4-105 a^3 f x^6+18 a^2 b c x^2+28 a^2 b d x^4+70 a^2 b e x^6-24 a b^2 c x^4-56 a b^2 d x^6+48 b^3 c x^6\right)}{105 a^4 x^7}","\frac{\sqrt{a+b x^2} (6 b c-7 a d)}{35 a^2 x^5}-\frac{\sqrt{a+b x^2} \left(35 a^2 e-28 a b d+24 b^2 c\right)}{105 a^3 x^3}+\frac{\sqrt{a+b x^2} \left(-105 a^3 f+70 a^2 b e-56 a b^2 d+48 b^3 c\right)}{105 a^4 x}-\frac{c \sqrt{a+b x^2}}{7 a x^7}",1,"(Sqrt[a + b*x^2]*(-15*a^3*c + 18*a^2*b*c*x^2 - 21*a^3*d*x^2 - 24*a*b^2*c*x^4 + 28*a^2*b*d*x^4 - 35*a^3*e*x^4 + 48*b^3*c*x^6 - 56*a*b^2*d*x^6 + 70*a^2*b*e*x^6 - 105*a^3*f*x^6))/(105*a^4*x^7)","A",1
158,1,160,189,0.2514206,"\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(-35 a^4 c-45 a^4 d x^2-63 a^4 e x^4-105 a^4 f x^6+40 a^3 b c x^2+54 a^3 b d x^4+84 a^3 b e x^6+210 a^3 b f x^8-48 a^2 b^2 c x^4-72 a^2 b^2 d x^6-168 a^2 b^2 e x^8+64 a b^3 c x^6+144 a b^3 d x^8-128 b^4 c x^8\right)}{315 a^5 x^9}","\frac{\sqrt{a+b x^2} (8 b c-9 a d)}{63 a^2 x^7}-\frac{\sqrt{a+b x^2} \left(21 a^2 e-18 a b d+16 b^2 c\right)}{105 a^3 x^5}-\frac{2 b \sqrt{a+b x^2} \left(-105 a^3 f+84 a^2 b e-72 a b^2 d+64 b^3 c\right)}{315 a^5 x}+\frac{\sqrt{a+b x^2} \left(-105 a^3 f+84 a^2 b e-72 a b^2 d+64 b^3 c\right)}{315 a^4 x^3}-\frac{c \sqrt{a+b x^2}}{9 a x^9}",1,"(Sqrt[a + b*x^2]*(-35*a^4*c + 40*a^3*b*c*x^2 - 45*a^4*d*x^2 - 48*a^2*b^2*c*x^4 + 54*a^3*b*d*x^4 - 63*a^4*e*x^4 + 64*a*b^3*c*x^6 - 72*a^2*b^2*d*x^6 + 84*a^3*b*e*x^6 - 105*a^4*f*x^6 - 128*b^4*c*x^8 + 144*a*b^3*d*x^8 - 168*a^2*b^2*e*x^8 + 210*a^3*b*f*x^8))/(315*a^5*x^9)","A",1
159,0,0,381,2.3580637,"\int \frac{x^8 \left(A+B x^2+C x^4+D x^6\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^8*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]","\int \frac{x^8 \left(A+B x^2+C x^4+D x^6\right)}{\left(a+b x^2\right)^{9/2}} \, dx","-\frac{x^9 \left(2 A b^3-a \left(23 a^2 D-16 a b C+9 b^2 B\right)\right)}{35 a^2 b^3 \left(a+b x^2\right)^{5/2}}+\frac{x^9 \left(A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}\right)}{7 a \left(a+b x^2\right)^{7/2}}-\frac{x \sqrt{a+b x^2} \left(16 A b^3-3 a \left(143 a^2 D-66 a b C+24 b^2 B\right)\right)}{16 a b^7}+\frac{x^3 \sqrt{a+b x^2} \left(16 A b^3-3 a \left(143 a^2 D-66 a b C+24 b^2 B\right)\right)}{24 a^2 b^6}-\frac{x^5 \left(16 A b^3-3 a \left(143 a^2 D-66 a b C+24 b^2 B\right)\right)}{30 a^2 b^5 \sqrt{a+b x^2}}-\frac{x^7 \left(16 A b^3-3 a \left(143 a^2 D-66 a b C+24 b^2 B\right)\right)}{210 a^2 b^4 \left(a+b x^2\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right) \left(-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right)}{16 b^{15/2}}+\frac{D x^9}{6 b^3 \left(a+b x^2\right)^{3/2}}",1,"Could not integrate","F",-1
160,0,0,279,2.0869783,"\int \frac{x^6 \left(A+B x^2+C x^4+D x^6\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^6*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]","\int \frac{x^6 \left(A+B x^2+C x^4+D x^6\right)}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{x^7 \left(A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}\right)}{7 a \left(a+b x^2\right)^{7/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right) \left(99 a^2 D-36 a b C+8 b^2 B\right)}{8 b^{13/2}}-\frac{x \sqrt{a+b x^2} \left(99 a^2 D-36 a b C+8 b^2 B\right)}{8 a b^6}+\frac{x^3 \left(99 a^2 D-36 a b C+8 b^2 B\right)}{12 a b^5 \sqrt{a+b x^2}}+\frac{x^5 \left(99 a^2 D-36 a b C+8 b^2 B\right)}{60 a b^4 \left(a+b x^2\right)^{3/2}}+\frac{x^7 \left(3 a^2 D-2 a b C+b^2 B\right)}{5 a b^3 \left(a+b x^2\right)^{5/2}}+\frac{D x^7}{4 b^3 \left(a+b x^2\right)^{3/2}}",1,"Could not integrate","F",-1
161,0,0,210,2.0902415,"\int \frac{x^4 \left(A+B x^2+C x^4+D x^6\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^4*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]","\int \frac{x^4 \left(A+B x^2+C x^4+D x^6\right)}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{x^5 \left(a \left(19 a^2 D-12 a b C+5 b^2 B\right)+2 A b^3\right)}{35 a^2 b^3 \left(a+b x^2\right)^{5/2}}+\frac{x^5 \left(A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}\right)}{7 a \left(a+b x^2\right)^{7/2}}+\frac{(2 b C-9 a D) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{11/2}}-\frac{x (4 b C-15 a D)}{3 b^5 \sqrt{a+b x^2}}+\frac{a x (b C-3 a D)}{3 b^5 \left(a+b x^2\right)^{3/2}}+\frac{D x \sqrt{a+b x^2}}{2 b^5}",1,"Could not integrate","F",-1
162,0,0,179,1.8951658,"\int \frac{x^2 \left(A+B x^2+C x^4+D x^6\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^2*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]","\int \frac{x^2 \left(A+B x^2+C x^4+D x^6\right)}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{x^3 \left(A b^3-10 a^3 D\right)}{3 a b^3 \left(a+b x^2\right)^{7/2}}-\frac{a^3 D x}{b^4 \left(a+b x^2\right)^{7/2}}+\frac{x^7 \left(-176 a^3 D+15 a^2 b C+6 a b^2 B+8 A b^3\right)}{105 a^3 b \left(a+b x^2\right)^{7/2}}+\frac{x^5 \left(-58 a^3 D+3 a b^2 B+4 A b^3\right)}{15 a^2 b^2 \left(a+b x^2\right)^{7/2}}+\frac{D \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{9/2}}",1,"Could not integrate","F",-1
163,0,0,134,1.8666484,"\int \frac{A+B x^2+C x^4+D x^6}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2)^(9/2),x]","\int \frac{A+B x^2+C x^4+D x^6}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{x^5 \left(a (3 a C+4 b B)+24 A b^2\right)}{15 a^3 \left(a+b x^2\right)^{7/2}}+\frac{x^3 (a B+6 A b)}{3 a^2 \left(a+b x^2\right)^{7/2}}+\frac{x^7 \left(a \left(15 a^2 D+6 a b C+8 b^2 B\right)+48 A b^3\right)}{105 a^4 \left(a+b x^2\right)^{7/2}}+\frac{A x}{a \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
164,0,0,185,2.0094443,"\int \frac{A+B x^2+C x^4+D x^6}{x^2 \left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x^2 + C*x^4 + D*x^6)/(x^2*(a + b*x^2)^(9/2)),x]","\int \frac{A+B x^2+C x^4+D x^6}{x^2 \left(a+b x^2\right)^{9/2}} \, dx","-\frac{x^3 \left(48 A b^2-a (a C+6 b B)\right)}{3 a^3 \left(a+b x^2\right)^{7/2}}-\frac{x (8 A b-a B)}{a^2 \left(a+b x^2\right)^{7/2}}-\frac{2 b x^7 \left(4 b \left(48 A b^2-a (a C+6 b B)\right)-3 a^3 D\right)}{105 a^5 \left(a+b x^2\right)^{7/2}}-\frac{x^5 \left(4 b \left(48 A b^2-a (a C+6 b B)\right)-3 a^3 D\right)}{15 a^4 \left(a+b x^2\right)^{7/2}}-\frac{A}{a x \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
165,0,0,242,2.0978066,"\int \frac{A+B x^2+C x^4+D x^6}{x^4 \left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x^2 + C*x^4 + D*x^6)/(x^4*(a + b*x^2)^(9/2)),x]","\int \frac{A+B x^2+C x^4+D x^6}{x^4 \left(a+b x^2\right)^{9/2}} \, dx","\frac{x \left(80 A b^2-3 a (8 b B-a C)\right)}{3 a^3 \left(a+b x^2\right)^{7/2}}+\frac{10 A b-3 a B}{3 a^2 x \left(a+b x^2\right)^{7/2}}+\frac{8 b^2 x^7 \left(160 A b^3-a \left(a^2 (-D)-6 a b C+48 b^2 B\right)\right)}{105 a^6 \left(a+b x^2\right)^{7/2}}+\frac{4 b x^5 \left(160 A b^3-a \left(a^2 (-D)-6 a b C+48 b^2 B\right)\right)}{15 a^5 \left(a+b x^2\right)^{7/2}}+\frac{x^3 \left(160 A b^3-a \left(a^2 (-D)-6 a b C+48 b^2 B\right)\right)}{3 a^4 \left(a+b x^2\right)^{7/2}}-\frac{A}{3 a x^3 \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
166,0,0,281,2.0879078,"\int \frac{A+B x^2+C x^4+D x^6}{x^6 \left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x^2 + C*x^4 + D*x^6)/(x^6*(a + b*x^2)^(9/2)),x]","\int \frac{A+B x^2+C x^4+D x^6}{x^6 \left(a+b x^2\right)^{9/2}} \, dx","-\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left(a+b x^2\right)^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left(a+b x^2\right)^{7/2}}-\frac{16 x \left(192 A b^3-a \left(3 a^2 D-24 a b C+80 b^2 B\right)\right)}{105 a^7 \sqrt{a+b x^2}}-\frac{8 x \left(192 A b^3-a \left(3 a^2 D-24 a b C+80 b^2 B\right)\right)}{105 a^6 \left(a+b x^2\right)^{3/2}}-\frac{2 x \left(192 A b^3-a \left(3 a^2 D-24 a b C+80 b^2 B\right)\right)}{35 a^5 \left(a+b x^2\right)^{5/2}}-\frac{x \left(192 A b^3-a \left(3 a^2 D-24 a b C+80 b^2 B\right)\right)}{21 a^4 \left(a+b x^2\right)^{7/2}}-\frac{A}{5 a x^5 \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
167,0,0,334,2.0906762,"\int \frac{A+B x^2+C x^4+D x^6}{x^8 \left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x^2 + C*x^4 + D*x^6)/(x^8*(a + b*x^2)^(9/2)),x]","\int \frac{A+B x^2+C x^4+D x^6}{x^8 \left(a+b x^2\right)^{9/2}} \, dx","-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left(a+b x^2\right)^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left(a+b x^2\right)^{7/2}}+\frac{128 b x \left(48 A b^3-a \left(3 a^2 D-10 a b C+24 b^2 B\right)\right)}{105 a^8 \sqrt{a+b x^2}}+\frac{64 b x \left(48 A b^3-a \left(3 a^2 D-10 a b C+24 b^2 B\right)\right)}{105 a^7 \left(a+b x^2\right)^{3/2}}+\frac{16 b x \left(48 A b^3-a \left(3 a^2 D-10 a b C+24 b^2 B\right)\right)}{35 a^6 \left(a+b x^2\right)^{5/2}}+\frac{8 b x \left(48 A b^3-a \left(3 a^2 D-10 a b C+24 b^2 B\right)\right)}{21 a^5 \left(a+b x^2\right)^{7/2}}+\frac{48 A b^3-a \left(3 a^2 D-10 a b C+24 b^2 B\right)}{3 a^4 x \left(a+b x^2\right)^{7/2}}-\frac{A}{7 a x^7 \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
168,0,0,392,2.0899485,"\int \frac{A+B x^2+C x^4+D x^6}{x^{10} \left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*(a + b*x^2)^(9/2)),x]","\int \frac{A+B x^2+C x^4+D x^6}{x^{10} \left(a+b x^2\right)^{9/2}} \, dx","-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left(a+b x^2\right)^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left(a+b x^2\right)^{7/2}}-\frac{256 b^2 x \left(128 A b^3-3 a \left(5 a^2 D-12 a b C+24 b^2 B\right)\right)}{315 a^9 \sqrt{a+b x^2}}-\frac{128 b^2 x \left(128 A b^3-3 a \left(5 a^2 D-12 a b C+24 b^2 B\right)\right)}{315 a^8 \left(a+b x^2\right)^{3/2}}-\frac{32 b^2 x \left(128 A b^3-3 a \left(5 a^2 D-12 a b C+24 b^2 B\right)\right)}{105 a^7 \left(a+b x^2\right)^{5/2}}-\frac{16 b^2 x \left(128 A b^3-3 a \left(5 a^2 D-12 a b C+24 b^2 B\right)\right)}{63 a^6 \left(a+b x^2\right)^{7/2}}-\frac{2 b \left(128 A b^3-3 a \left(5 a^2 D-12 a b C+24 b^2 B\right)\right)}{9 a^5 x \left(a+b x^2\right)^{7/2}}+\frac{128 A b^3-3 a \left(5 a^2 D-12 a b C+24 b^2 B\right)}{45 a^4 x^3 \left(a+b x^2\right)^{7/2}}-\frac{A}{9 a x^9 \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1
169,1,196,214,0.1377221,"\int \frac{c x^5+d x^7+e x^9+f x^{11}}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c*x^5 + d*x^7 + e*x^9 + f*x^11)/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(-1280 a^5 f+1408 a^4 b e+640 a^4 b f x^2-1584 a^3 b^2 d-704 a^3 b^2 e x^2-480 a^3 b^2 f x^4+1848 a^2 b^3 c+792 a^2 b^3 d x^2+528 a^2 b^3 e x^4+400 a^2 b^3 f x^6-924 a b^4 c x^2-594 a b^4 d x^4-440 a b^4 e x^6-350 a b^4 f x^8+693 b^5 c x^4+495 b^5 d x^6+385 b^5 e x^8+315 b^5 f x^{10}\right)}{3465 b^6}","\frac{\left(a+b x^2\right)^{7/2} \left(10 a^2 f-4 a b e+b^2 d\right)}{7 b^6}+\frac{\left(a+b x^2\right)^{5/2} \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{5 b^6}-\frac{a \left(a+b x^2\right)^{3/2} \left(-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right)}{3 b^6}+\frac{a^2 \sqrt{a+b x^2} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^6}+\frac{\left(a+b x^2\right)^{9/2} (b e-5 a f)}{9 b^6}+\frac{f \left(a+b x^2\right)^{11/2}}{11 b^6}",1,"(Sqrt[a + b*x^2]*(1848*a^2*b^3*c - 1584*a^3*b^2*d + 1408*a^4*b*e - 1280*a^5*f - 924*a*b^4*c*x^2 + 792*a^2*b^3*d*x^2 - 704*a^3*b^2*e*x^2 + 640*a^4*b*f*x^2 + 693*b^5*c*x^4 - 594*a*b^4*d*x^4 + 528*a^2*b^3*e*x^4 - 480*a^3*b^2*f*x^4 + 495*b^5*d*x^6 - 440*a*b^4*e*x^6 + 400*a^2*b^3*f*x^6 + 385*b^5*e*x^8 - 350*a*b^4*f*x^8 + 315*b^5*f*x^10))/(3465*b^6)","A",1
170,1,148,167,0.1024292,"\int \frac{c x^3+d x^5+e x^7+f x^9}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c*x^3 + d*x^5 + e*x^7 + f*x^9)/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(128 a^4 f-144 a^3 b e-64 a^3 b f x^2+168 a^2 b^2 d+72 a^2 b^2 e x^2+48 a^2 b^2 f x^4-210 a b^3 c-84 a b^3 d x^2-54 a b^3 e x^4-40 a b^3 f x^6+105 b^4 c x^2+63 b^4 d x^4+45 b^4 e x^6+35 b^4 f x^8\right)}{315 b^5}","\frac{\left(a+b x^2\right)^{5/2} \left(6 a^2 f-3 a b e+b^2 d\right)}{5 b^5}+\frac{\left(a+b x^2\right)^{3/2} \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{3 b^5}-\frac{a \sqrt{a+b x^2} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^5}+\frac{\left(a+b x^2\right)^{7/2} (b e-4 a f)}{7 b^5}+\frac{f \left(a+b x^2\right)^{9/2}}{9 b^5}",1,"(Sqrt[a + b*x^2]*(-210*a*b^3*c + 168*a^2*b^2*d - 144*a^3*b*e + 128*a^4*f + 105*b^4*c*x^2 - 84*a*b^3*d*x^2 + 72*a^2*b^2*e*x^2 - 64*a^3*b*f*x^2 + 63*b^4*d*x^4 - 54*a*b^3*e*x^4 + 48*a^2*b^2*f*x^4 + 45*b^4*e*x^6 - 40*a*b^3*f*x^6 + 35*b^4*f*x^8))/(315*b^5)","A",1
171,1,102,121,0.0675649,"\int \frac{c x+d x^3+e x^5+f x^7}{\sqrt{a+b x^2}} \, dx","IntegrateAlgebraic[(c*x + d*x^3 + e*x^5 + f*x^7)/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(-48 a^3 f+56 a^2 b e+24 a^2 b f x^2-70 a b^2 d-28 a b^2 e x^2-18 a b^2 f x^4+105 b^3 c+35 b^3 d x^2+21 b^3 e x^4+15 b^3 f x^6\right)}{105 b^4}","\frac{\left(a+b x^2\right)^{3/2} \left(3 a^2 f-2 a b e+b^2 d\right)}{3 b^4}+\frac{\sqrt{a+b x^2} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^4}+\frac{\left(a+b x^2\right)^{5/2} (b e-3 a f)}{5 b^4}+\frac{f \left(a+b x^2\right)^{7/2}}{7 b^4}",1,"(Sqrt[a + b*x^2]*(105*b^3*c - 70*a*b^2*d + 56*a^2*b*e - 48*a^3*f + 35*b^3*d*x^2 - 28*a*b^2*e*x^2 + 24*a^2*b*f*x^2 + 21*b^3*e*x^4 - 18*a*b^2*f*x^4 + 15*b^3*f*x^6))/(105*b^4)","A",1
172,0,0,261,2.4034614,"\int \frac{x^2 \left(A+B x^2+C x^4+D x^6+F x^8\right)}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(x^2*(A + B*x^2 + C*x^4 + D*x^6 + F*x^8))/(a + b*x^2)^(9/2),x]","\int \frac{x^2 \left(A+B x^2+C x^4+D x^6+F x^8\right)}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{x^3 \left(a \left(162 a^3 F-71 a^2 b D+15 a b^2 C+6 b^3 B\right)+8 A b^4\right)}{105 a^3 b^4 \left(a+b x^2\right)^{3/2}}+\frac{x^3 \left(a \left(-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right)+4 A b^4\right)}{35 a^2 b^4 \left(a+b x^2\right)^{5/2}}+\frac{x^3 \left(A b^4-a \left(a^3 (-F)+a^2 b D-a b^2 C+b^3 B\right)\right)}{7 a b^4 \left(a+b x^2\right)^{7/2}}+\frac{(2 b D-9 a F) \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{11/2}}-\frac{x (b D-4 a F)}{b^5 \sqrt{a+b x^2}}+\frac{F x \sqrt{a+b x^2}}{2 b^5}",1,"Could not integrate","F",-1
173,0,0,214,2.3502177,"\int \frac{A+B x^2+C x^4+D x^6+F x^8}{\left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x^2 + C*x^4 + D*x^6 + F*x^8)/(a + b*x^2)^(9/2),x]","\int \frac{A+B x^2+C x^4+D x^6+F x^8}{\left(a+b x^2\right)^{9/2}} \, dx","\frac{x \left(A b^4-a^4 F\right)}{a b^4 \left(a+b x^2\right)^{7/2}}+\frac{x^5 \left(a \left(-58 a^3 F+3 a b^2 C+4 b^3 B\right)+24 A b^4\right)}{15 a^3 b^2 \left(a+b x^2\right)^{7/2}}+\frac{x^3 \left(-10 a^4 F+a b^3 B+6 A b^4\right)}{3 a^2 b^3 \left(a+b x^2\right)^{7/2}}+\frac{x^7 \left(a \left(-176 a^3 F+15 a^2 b D+6 a b^2 C+8 b^3 B\right)+48 A b^4\right)}{105 a^4 b \left(a+b x^2\right)^{7/2}}+\frac{F \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{9/2}}",1,"Could not integrate","F",-1
174,0,0,193,2.3521187,"\int \frac{A+B x^2+C x^4+D x^6+F x^8}{x^2 \left(a+b x^2\right)^{9/2}} \, dx","IntegrateAlgebraic[(A + B*x^2 + C*x^4 + D*x^6 + F*x^8)/(x^2*(a + b*x^2)^(9/2)),x]","\int \frac{A+B x^2+C x^4+D x^6+F x^8}{x^2 \left(a+b x^2\right)^{9/2}} \, dx","-\frac{x^3 \left(48 A b^2-a (a C+6 b B)\right)}{3 a^3 \left(a+b x^2\right)^{7/2}}-\frac{x (8 A b-a B)}{a^2 \left(a+b x^2\right)^{7/2}}-\frac{x^5 \left(192 A b^3-a \left(3 a^2 D+4 a b C+24 b^2 B\right)\right)}{15 a^4 \left(a+b x^2\right)^{7/2}}-\frac{x^7 \left(384 A b^4-a \left(15 a^3 F+6 a^2 b D+8 a b^2 C+48 b^3 B\right)\right)}{105 a^5 \left(a+b x^2\right)^{7/2}}-\frac{A}{a x \left(a+b x^2\right)^{7/2}}",1,"Could not integrate","F",-1