1,1,107,127,0.2314453,"\int x^3 (A+B x) \sqrt{a+b x^2} \, dx","Integrate[x^3*(A + B*x)*Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(\frac{15 a^{5/2} B \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}+\sqrt{b} \left(-a^2 (32 A+15 B x)+2 a b x^2 (8 A+5 B x)+8 b^2 x^4 (6 A+5 B x)\right)\right)}{240 b^{5/2}}","\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,"(Sqrt[a + b*x^2]*(Sqrt[b]*(8*b^2*x^4*(6*A + 5*B*x) + 2*a*b*x^2*(8*A + 5*B*x) - a^2*(32*A + 15*B*x)) + (15*a^(5/2)*B*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(240*b^(5/2))","A",1
2,1,93,104,0.1918514,"\int x^2 (A+B x) \sqrt{a+b x^2} \, dx","Integrate[x^2*(A + B*x)*Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(-\frac{15 a^{3/2} A \sqrt{b} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}-16 a^2 B+a b x (15 A+8 B x)+6 b^2 x^3 (5 A+4 B x)\right)}{120 b^2}","-\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,"(Sqrt[a + b*x^2]*(-16*a^2*B + 6*b^2*x^3*(5*A + 4*B*x) + a*b*x*(15*A + 8*B*x) - (15*a^(3/2)*A*Sqrt[b]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(120*b^2)","A",1
3,1,86,80,0.151791,"\int x (A+B x) \sqrt{a+b x^2} \, dx","Integrate[x*(A + B*x)*Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} \left(8 a A+3 a B x+8 A b x^2+6 b B x^3\right)-\frac{3 a^{3/2} B \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{24 b^{3/2}}","-\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,"(Sqrt[a + b*x^2]*(Sqrt[b]*(8*a*A + 3*a*B*x + 8*A*b*x^2 + 6*b*B*x^3) - (3*a^(3/2)*B*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(24*b^(3/2))","A",1
4,1,67,67,0.0572216,"\int (A+B x) \sqrt{a+b x^2} \, dx","Integrate[(A + B*x)*Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} (2 a B+b x (3 A+2 B x))+3 a A \sqrt{b} \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{6 b}","\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,"(Sqrt[a + b*x^2]*(2*a*B + b*x*(3*A + 2*B*x)) + 3*a*A*Sqrt[b]*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(6*b)","A",1
5,1,100,79,0.2533485,"\int \frac{(A+B x) \sqrt{a+b x^2}}{x} \, dx","Integrate[((A + B*x)*Sqrt[a + b*x^2])/x,x]","\frac{1}{2} \left(\frac{a^{3/2} B \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b x^2}}+\sqrt{a+b x^2} (2 A+B x)-2 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)\right)","\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,"((2*A + B*x)*Sqrt[a + b*x^2] + (a^(3/2)*B*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*Sqrt[a + b*x^2]) - 2*Sqrt[a]*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/2","A",1
6,1,99,75,0.1778449,"\int \frac{(A+B x) \sqrt{a+b x^2}}{x^2} \, dx","Integrate[((A + B*x)*Sqrt[a + b*x^2])/x^2,x]","\frac{\sqrt{a+b x^2} (B x-A)}{x}+\frac{\sqrt{a} A \sqrt{b} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a+b x^2}}-\sqrt{a} B \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","-\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,"((-A + B*x)*Sqrt[a + b*x^2])/x + (Sqrt[a]*A*Sqrt[b]*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a + b*x^2] - Sqrt[a]*B*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
7,1,108,80,0.0928925,"\int \frac{(A+B x) \sqrt{a+b x^2}}{x^3} \, dx","Integrate[((A + B*x)*Sqrt[a + b*x^2])/x^3,x]","-\frac{\sqrt{a+b x^2} \left(a \sqrt{\frac{b x^2}{a}+1} (A+2 B x)+A b x^2 \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)-2 \sqrt{a} \sqrt{b} B x^2 \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right)}{2 a x^2 \sqrt{\frac{b x^2}{a}+1}}","-\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,"-1/2*(Sqrt[a + b*x^2]*(a*(A + 2*B*x)*Sqrt[1 + (b*x^2)/a] - 2*Sqrt[a]*Sqrt[b]*B*x^2*ArcSinh[(Sqrt[b]*x)/Sqrt[a]] + A*b*x^2*ArcTanh[Sqrt[1 + (b*x^2)/a]]))/(a*x^2*Sqrt[1 + (b*x^2)/a])","A",1
8,1,126,150,0.258417,"\int x^3 (A+B x) \left(a+b x^2\right)^{3/2} \, dx","Integrate[x^3*(A + B*x)*(a + b*x^2)^(3/2),x]","\frac{\sqrt{a+b x^2} \left(\frac{105 a^{7/2} B \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}+\sqrt{b} \left(-a^3 (256 A+105 B x)+2 a^2 b x^2 (64 A+35 B x)+8 a b^2 x^4 (128 A+105 B x)+80 b^3 x^6 (8 A+7 B x)\right)\right)}{4480 b^{5/2}}","\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,"(Sqrt[a + b*x^2]*(Sqrt[b]*(80*b^3*x^6*(8*A + 7*B*x) + 2*a^2*b*x^2*(64*A + 35*B*x) + 8*a*b^2*x^4*(128*A + 105*B*x) - a^3*(256*A + 105*B*x)) + (105*a^(7/2)*B*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(4480*b^(5/2))","A",1
9,1,113,127,0.2399177,"\int x^2 (A+B x) \left(a+b x^2\right)^{3/2} \, dx","Integrate[x^2*(A + B*x)*(a + b*x^2)^(3/2),x]","\frac{\sqrt{a+b x^2} \left(-\frac{105 a^{5/2} A \sqrt{b} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}-96 a^3 B+3 a^2 b x (35 A+16 B x)+2 a b^2 x^3 (245 A+192 B x)+40 b^3 x^5 (7 A+6 B x)\right)}{1680 b^2}","-\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,"(Sqrt[a + b*x^2]*(-96*a^3*B + 40*b^3*x^5*(7*A + 6*B*x) + 3*a^2*b*x*(35*A + 16*B*x) + 2*a*b^2*x^3*(245*A + 192*B*x) - (105*a^(5/2)*A*Sqrt[b]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(1680*b^2)","A",1
10,1,107,103,0.193855,"\int x (A+B x) \left(a+b x^2\right)^{3/2} \, dx","Integrate[x*(A + B*x)*(a + b*x^2)^(3/2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} \left(3 a^2 (16 A+5 B x)+2 a b x^2 (48 A+35 B x)+8 b^2 x^4 (6 A+5 B x)\right)-\frac{15 a^{5/2} B \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{240 b^{3/2}}","-\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,"(Sqrt[a + b*x^2]*(Sqrt[b]*(8*b^2*x^4*(6*A + 5*B*x) + 3*a^2*(16*A + 5*B*x) + 2*a*b*x^2*(48*A + 35*B*x)) - (15*a^(5/2)*B*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(240*b^(3/2))","A",1
11,1,88,87,0.0777821,"\int (A+B x) \left(a+b x^2\right)^{3/2} \, dx","Integrate[(A + B*x)*(a + b*x^2)^(3/2),x]","\frac{\sqrt{a+b x^2} \left(8 a^2 B+a b x (25 A+16 B x)+2 b^2 x^3 (5 A+4 B x)\right)+15 a^2 A \sqrt{b} \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{40 b}","\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,"(Sqrt[a + b*x^2]*(8*a^2*B + 2*b^2*x^3*(5*A + 4*B*x) + a*b*x*(25*A + 16*B*x)) + 15*a^2*A*Sqrt[b]*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(40*b)","A",1
12,1,118,106,0.3057913,"\int \frac{(A+B x) \left(a+b x^2\right)^{3/2}}{x} \, dx","Integrate[((A + B*x)*(a + b*x^2)^(3/2))/x,x]","\frac{1}{24} \left(-24 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{9 a^{5/2} B \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b x^2}}+\sqrt{a+b x^2} \left(32 a A+15 a B x+8 A b x^2+6 b B x^3\right)\right)","-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,"(Sqrt[a + b*x^2]*(32*a*A + 15*a*B*x + 8*A*b*x^2 + 6*b*B*x^3) + (9*a^(5/2)*B*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*Sqrt[a + b*x^2]) - 24*a^(3/2)*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/24","A",1
13,1,105,108,0.1835825,"\int \frac{(A+B x) \left(a+b x^2\right)^{3/2}}{x^2} \, dx","Integrate[((A + B*x)*(a + b*x^2)^(3/2))/x^2,x]","-a^{3/2} B \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)-\frac{a^2 A \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt{a+b x^2}}+\frac{1}{3} B \sqrt{a+b x^2} \left(4 a+b x^2\right)","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,"(B*Sqrt[a + b*x^2]*(4*a + b*x^2))/3 - a^(3/2)*B*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]] - (a^2*A*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((b*x^2)/a)])/(x*Sqrt[a + b*x^2])","C",1
14,1,90,111,0.0634583,"\int \frac{(A+B x) \left(a+b x^2\right)^{3/2}}{x^3} \, dx","Integrate[((A + B*x)*(a + b*x^2)^(3/2))/x^3,x]","\frac{A b \left(a+b x^2\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^2}{a}+1\right)}{5 a^2}-\frac{a B \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt{\frac{b x^2}{a}+1}}","-\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,"-((a*B*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((b*x^2)/a)])/(x*Sqrt[1 + (b*x^2)/a])) + (A*b*(a + b*x^2)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^2)/a])/(5*a^2)","C",1
15,1,145,173,0.4018807,"\int x^3 (A+B x) \left(a+b x^2\right)^{5/2} \, dx","Integrate[x^3*(A + B*x)*(a + b*x^2)^(5/2),x]","\frac{\sqrt{a+b x^2} \left(\frac{945 a^{9/2} B \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}+\sqrt{b} \left(-5 a^4 (512 A+189 B x)+10 a^3 b x^2 (128 A+63 B x)+24 a^2 b^2 x^4 (800 A+651 B x)+16 a b^3 x^6 (1520 A+1323 B x)+896 b^4 x^8 (10 A+9 B x)\right)\right)}{80640 b^{5/2}}","\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,"(Sqrt[a + b*x^2]*(Sqrt[b]*(896*b^4*x^8*(10*A + 9*B*x) + 10*a^3*b*x^2*(128*A + 63*B*x) - 5*a^4*(512*A + 189*B*x) + 24*a^2*b^2*x^4*(800*A + 651*B*x) + 16*a*b^3*x^6*(1520*A + 1323*B*x)) + (945*a^(9/2)*B*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(80640*b^(5/2))","A",1
16,1,131,150,0.4064972,"\int x^2 (A+B x) \left(a+b x^2\right)^{5/2} \, dx","Integrate[x^2*(A + B*x)*(a + b*x^2)^(5/2),x]","\frac{\sqrt{a+b x^2} \left(-\frac{315 a^{7/2} A \sqrt{b} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}-256 a^4 B+a^3 b x (315 A+128 B x)+6 a^2 b^2 x^3 (413 A+320 B x)+8 a b^3 x^5 (357 A+304 B x)+112 b^4 x^7 (9 A+8 B x)\right)}{8064 b^2}","-\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,"(Sqrt[a + b*x^2]*(-256*a^4*B + 112*b^4*x^7*(9*A + 8*B*x) + a^3*b*x*(315*A + 128*B*x) + 8*a*b^3*x^5*(357*A + 304*B*x) + 6*a^2*b^2*x^3*(413*A + 320*B*x) - (315*a^(7/2)*A*Sqrt[b]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(8064*b^2)","A",1
17,1,112,126,0.5504948,"\int x (A+B x) \left(a+b x^2\right)^{5/2} \, dx","Integrate[x*(A + B*x)*(a + b*x^2)^(5/2),x]","\frac{\left(a+b x^2\right)^{7/2} \left(-\frac{7 a B x \left(\frac{15 a^{7/2} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} x}+\left(a+b x^2\right) \left(33 a^2+26 a b x^2+8 b^2 x^4\right)\right)}{\left(a+b x^2\right)^4}+384 A+336 B x\right)}{2688 b}","-\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,"((a + b*x^2)^(7/2)*(384*A + 336*B*x - (7*a*B*x*((a + b*x^2)*(33*a^2 + 26*a*b*x^2 + 8*b^2*x^4) + (15*a^(7/2)*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*x)))/(a + b*x^2)^4))/(2688*b)","A",1
18,1,108,107,0.0876285,"\int (A+B x) \left(a+b x^2\right)^{5/2} \, dx","Integrate[(A + B*x)*(a + b*x^2)^(5/2),x]","\frac{105 a^3 A \sqrt{b} \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)+\sqrt{a+b x^2} \left(48 a^3 B+3 a^2 b x (77 A+48 B x)+2 a b^2 x^3 (91 A+72 B x)+8 b^3 x^5 (7 A+6 B x)\right)}{336 b}","\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,"(Sqrt[a + b*x^2]*(48*a^3*B + 8*b^3*x^5*(7*A + 6*B*x) + 3*a^2*b*x*(77*A + 48*B*x) + 2*a*b^2*x^3*(91*A + 72*B*x)) + 105*a^3*A*Sqrt[b]*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(336*b)","A",1
19,1,139,132,0.3609366,"\int \frac{(A+B x) \left(a+b x^2\right)^{5/2}}{x} \, dx","Integrate[((A + B*x)*(a + b*x^2)^(5/2))/x,x]","-a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{5 a^{7/2} B \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{16 \sqrt{b} \sqrt{a+b x^2}}+\frac{1}{240} \sqrt{a+b x^2} \left(a^2 (368 A+165 B x)+2 a b x^2 (88 A+65 B x)+8 b^2 x^4 (6 A+5 B x)\right)","-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,"(Sqrt[a + b*x^2]*(8*b^2*x^4*(6*A + 5*B*x) + 2*a*b*x^2*(88*A + 65*B*x) + a^2*(368*A + 165*B*x)))/240 + (5*a^(7/2)*B*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(16*Sqrt[b]*Sqrt[a + b*x^2]) - a^(5/2)*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
20,1,117,136,0.2317628,"\int \frac{(A+B x) \left(a+b x^2\right)^{5/2}}{x^2} \, dx","Integrate[((A + B*x)*(a + b*x^2)^(5/2))/x^2,x]","-a^{5/2} B \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)-\frac{a^3 A \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt{a+b x^2}}+\frac{1}{15} B \sqrt{a+b x^2} \left(23 a^2+11 a b x^2+3 b^2 x^4\right)","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,"(B*Sqrt[a + b*x^2]*(23*a^2 + 11*a*b*x^2 + 3*b^2*x^4))/15 - a^(5/2)*B*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]] - (a^3*A*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-5/2, -1/2, 1/2, -((b*x^2)/a)])/(x*Sqrt[a + b*x^2])","C",1
21,1,92,141,0.0352842,"\int \frac{(A+B x) \left(a+b x^2\right)^{5/2}}{x^3} \, dx","Integrate[((A + B*x)*(a + b*x^2)^(5/2))/x^3,x]","\frac{A b \left(a+b x^2\right)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{b x^2}{a}+1\right)}{7 a^2}-\frac{a^2 B \sqrt{a+b x^2} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt{\frac{b x^2}{a}+1}}","-\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,"-((a^2*B*Sqrt[a + b*x^2]*Hypergeometric2F1[-5/2, -1/2, 1/2, -((b*x^2)/a)])/(x*Sqrt[1 + (b*x^2)/a])) + (A*b*(a + b*x^2)^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, 1 + (b*x^2)/a])/(7*a^2)","C",1
22,1,76,104,0.0506261,"\int \frac{x^3 (A+B x)}{\sqrt{a+b x^2}} \, dx","Integrate[(x^3*(A + B*x))/Sqrt[a + b*x^2],x]","\frac{9 a^2 B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\sqrt{b} \sqrt{a+b x^2} \left(-16 a A-9 a B x+8 A b x^2+6 b B x^3\right)}{24 b^{5/2}}","\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,"(Sqrt[b]*Sqrt[a + b*x^2]*(-16*a*A - 9*a*B*x + 8*A*b*x^2 + 6*b*B*x^3) + 9*a^2*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(24*b^(5/2))","A",1
23,1,64,81,0.0596638,"\int \frac{x^2 (A+B x)}{\sqrt{a+b x^2}} \, dx","Integrate[(x^2*(A + B*x))/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} (b x (3 A+2 B x)-4 a B)-3 a A \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{6 b^2}","-\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,"(Sqrt[a + b*x^2]*(-4*a*B + b*x*(3*A + 2*B*x)) - 3*a*A*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(6*b^2)","A",1
24,1,57,56,0.0424739,"\int \frac{x (A+B x)}{\sqrt{a+b x^2}} \, dx","Integrate[(x*(A + B*x))/Sqrt[a + b*x^2],x]","\frac{\sqrt{b} \sqrt{a+b x^2} (2 A+B x)-a B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}","\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,"(Sqrt[b]*(2*A + B*x)*Sqrt[a + b*x^2] - a*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))","A",1
25,1,46,43,0.0592928,"\int \frac{A+B x}{\sqrt{a+b x^2}} \, dx","Integrate[(A + B*x)/Sqrt[a + b*x^2],x]","\frac{A \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{\sqrt{b}}+\frac{B \sqrt{a+b x^2}}{b}","\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,"(B*Sqrt[a + b*x^2])/b + (A*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/Sqrt[b]","A",1
26,1,53,53,0.0198075,"\int \frac{A+B x}{x \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x)/(x*Sqrt[a + b*x^2]),x]","\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}}","\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,"(B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - (A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]","A",1
27,1,47,47,0.0394194,"\int \frac{A+B x}{x^2 \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x)/(x^2*Sqrt[a + b*x^2]),x]","-\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}}","-\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,"-((A*Sqrt[a + b*x^2])/(a*x)) - (B*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]","A",1
28,1,63,72,0.1362354,"\int \frac{A+B x}{x^3 \sqrt{a+b x^2}} \, dx","Integrate[(A + B*x)/(x^3*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a+b x^2} \left(\frac{A b \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{\sqrt{\frac{b x^2}{a}+1}}-\frac{a (A+2 B x)}{x^2}\right)}{2 a^2}","\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,"(Sqrt[a + b*x^2]*(-((a*(A + 2*B*x))/x^2) + (A*b*ArcTanh[Sqrt[1 + (b*x^2)/a]])/Sqrt[1 + (b*x^2)/a]))/(2*a^2)","A",1
29,1,72,81,0.0628685,"\int \frac{x^3 (A+B x)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x^3*(A + B*x))/(a + b*x^2)^(3/2),x]","\frac{a (4 A+3 B x)+b x^2 (2 A+B x)}{2 b^2 \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}}","\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,"(b*x^2*(2*A + B*x) + a*(4*A + 3*B*x))/(2*b^2*Sqrt[a + b*x^2]) - (3*a*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(5/2))","A",1
30,1,67,66,0.0591019,"\int \frac{x^2 (A+B x)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x^2*(A + B*x))/(a + b*x^2)^(3/2),x]","\frac{A \sqrt{b} \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+2 a B+b x (B x-A)}{b^2 \sqrt{a+b x^2}}","\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,"(2*a*B + b*x*(-A + B*x) + A*Sqrt[b]*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(b^2*Sqrt[a + b*x^2])","A",1
31,1,64,48,0.0808498,"\int \frac{x (A+B x)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x*(A + B*x))/(a + b*x^2)^(3/2),x]","\frac{\sqrt{a} B \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)-\sqrt{b} (A+B x)}{b^{3/2} \sqrt{a+b x^2}}","\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,"(-(Sqrt[b]*(A + B*x)) + Sqrt[a]*B*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(b^(3/2)*Sqrt[a + b*x^2])","A",1
32,1,27,28,0.0318371,"\int \frac{A+B x}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(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
33,1,47,47,0.04772,"\int \frac{A+B x}{x \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x)/(x*(a + b*x^2)^(3/2)),x]","\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}}","\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,"(A + B*x)/(a*Sqrt[a + b*x^2]) - (A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/2)","A",1
34,1,72,70,0.0441682,"\int \frac{A+B x}{x^2 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x)/(x^2*(a + b*x^2)^(3/2)),x]","-\frac{a (A-B x)+\sqrt{a} B x \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+2 A b x^2}{a^2 x \sqrt{a+b x^2}}","-\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,"-((2*A*b*x^2 + a*(A - B*x) + Sqrt[a]*B*x*Sqrt[a + b*x^2]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(a^2*x*Sqrt[a + b*x^2]))","A",1
35,1,75,95,0.2233741,"\int \frac{A+B x}{x^3 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(A + B*x)/(x^3*(a + b*x^2)^(3/2)),x]","\frac{3 A b \sqrt{\frac{b x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)-\frac{a (A+2 B x)}{x^2}-b (3 A+4 B x)}{2 a^2 \sqrt{a+b x^2}}","\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,"(-((a*(A + 2*B*x))/x^2) - b*(3*A + 4*B*x) + 3*A*b*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]])/(2*a^2*Sqrt[a + b*x^2])","A",1
36,1,69,79,0.123873,"\int \frac{x^3 (A+B x)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x^3*(A + B*x))/(a + b*x^2)^(5/2),x]","\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{5/2}}-\frac{a (2 A+3 B x)+b x^2 (3 A+4 B x)}{3 b^2 \left(a+b x^2\right)^{3/2}}","-\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,"-1/3*(a*(2*A + 3*B*x) + b*x^2*(3*A + 4*B*x))/(b^2*(a + b*x^2)^(3/2)) + (B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(5/2)","A",1
37,1,44,53,0.0545413,"\int \frac{x^2 (A+B x)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x^2*(A + B*x))/(a + b*x^2)^(5/2),x]","\frac{-2 a^2 B-3 a b B x^2+A b^2 x^3}{3 a b^2 \left(a+b x^2\right)^{3/2}}","-\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,"(-2*a^2*B - 3*a*b*B*x^2 + A*b^2*x^3)/(3*a*b^2*(a + b*x^2)^(3/2))","A",1
38,1,32,50,0.0227764,"\int \frac{x (A+B x)}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x*(A + B*x))/(a + b*x^2)^(5/2),x]","\frac{b B x^3-a A}{3 a b \left(a+b x^2\right)^{3/2}}","\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,"(-(a*A) + b*B*x^3)/(3*a*b*(a + b*x^2)^(3/2))","A",1
39,1,43,51,0.046942,"\int \frac{A+B x}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x)/(a + b*x^2)^(5/2),x]","\frac{-a^2 B+3 a A b x+2 A b^2 x^3}{3 a^2 b \left(a+b x^2\right)^{3/2}}","\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,"(-(a^2*B) + 3*a*A*b*x + 2*A*b^2*x^3)/(3*a^2*b*(a + b*x^2)^(3/2))","A",1
40,1,69,76,0.0981565,"\int \frac{A+B x}{x \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x)/(x*(a + b*x^2)^(5/2)),x]","\frac{a (4 A+3 B x)+b x^2 (3 A+2 B x)}{3 a^2 \left(a+b x^2\right)^{3/2}}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{5/2}}","-\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,"(b*x^2*(3*A + 2*B*x) + a*(4*A + 3*B*x))/(3*a^2*(a + b*x^2)^(3/2)) - (A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(5/2)","A",1
41,1,95,104,0.0726853,"\int \frac{A+B x}{x^2 \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x)/(x^2*(a + b*x^2)^(5/2)),x]","\frac{a^2 (4 B x-3 A)+3 a b x^2 (B x-4 A)-3 \sqrt{a} B x \left(a+b x^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)-8 A b^2 x^4}{3 a^3 x \left(a+b x^2\right)^{3/2}}","-\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,"(-8*A*b^2*x^4 + 3*a*b*x^2*(-4*A + B*x) + a^2*(-3*A + 4*B*x) - 3*Sqrt[a]*B*x*(a + b*x^2)^(3/2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(3*a^3*x*(a + b*x^2)^(3/2))","A",1
42,1,106,129,0.2154959,"\int \frac{A+B x}{x^3 \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(A + B*x)/(x^3*(a + b*x^2)^(5/2)),x]","\frac{-\frac{3 a^3 (A+2 B x)}{x^2}-4 a^2 b (5 A+6 B x)-a b^2 x^2 (15 A+16 B x)+\frac{15 A b \left(a+b x^2\right)^2 \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{\sqrt{\frac{b x^2}{a}+1}}}{6 a^4 \left(a+b x^2\right)^{3/2}}","\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,"((-3*a^3*(A + 2*B*x))/x^2 - 4*a^2*b*(5*A + 6*B*x) - a*b^2*x^2*(15*A + 16*B*x) + (15*A*b*(a + b*x^2)^2*ArcTanh[Sqrt[1 + (b*x^2)/a]])/Sqrt[1 + (b*x^2)/a])/(6*a^4*(a + b*x^2)^(3/2))","A",1
43,1,24,27,0.0297192,"\int \frac{(1-x) x}{\sqrt{1-x^2}} \, dx","Integrate[((1 - x)*x)/Sqrt[1 - x^2],x]","\frac{1}{2} \left((x-2) \sqrt{1-x^2}-\sin ^{-1}(x)\right)","-\frac{1}{2} \sqrt{1-x^2} (2-x)-\frac{1}{2} \sin ^{-1}(x)",1,"((-2 + x)*Sqrt[1 - x^2] - ArcSin[x])/2","A",1
44,1,24,27,0.020012,"\int \frac{x-x^2}{\sqrt{1-x^2}} \, dx","Integrate[(x - x^2)/Sqrt[1 - x^2],x]","\frac{1}{2} \left((x-2) \sqrt{1-x^2}-\sin ^{-1}(x)\right)","-\frac{1}{2} \sqrt{1-x^2} (2-x)-\frac{1}{2} \sin ^{-1}(x)",1,"((-2 + x)*Sqrt[1 - x^2] - ArcSin[x])/2","A",1
45,1,33,17,0.0091993,"\int \frac{3+x^2}{-3+x^2} \, dx","Integrate[(3 + x^2)/(-3 + x^2),x]","x+\sqrt{3} \log \left(\sqrt{3}-x\right)-\sqrt{3} \log \left(x+\sqrt{3}\right)","x-2 \sqrt{3} \tanh ^{-1}\left(\frac{x}{\sqrt{3}}\right)",1,"x + Sqrt[3]*Log[Sqrt[3] - x] - Sqrt[3]*Log[Sqrt[3] + x]","A",1
46,1,6,6,0.0052977,"\int \frac{-1+x^2}{1+x^2} \, dx","Integrate[(-1 + x^2)/(1 + x^2),x]","x-2 \tan ^{-1}(x)","x-2 \tan ^{-1}(x)",1,"x - 2*ArcTan[x]","A",1
47,1,165,213,0.6491765,"\int \frac{x^7 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(x^7*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\frac{384 a^4 C-3 a^3 b (16 A+7 x (5 B-64 C x))+14 a^2 b^2 x^2 (5 x (24 C x-5 B)-12 A)+14 a b^3 x^4 (x (60 C x-29 B)-15 A)+105 \sqrt{a} \sqrt{b} B \left(a+b x^2\right)^3 \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+b^4 x^6 (x (105 C x-176 B)-105 A)}{105 b^5 \left(a+b x^2\right)^{7/2}}","-\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,"(384*a^4*C - 3*a^3*b*(16*A + 7*x*(5*B - 64*C*x)) + 14*a^2*b^2*x^2*(-12*A + 5*x*(-5*B + 24*C*x)) + 14*a*b^3*x^4*(-15*A + x*(-29*B + 60*C*x)) + b^4*x^6*(-105*A + x*(-176*B + 105*C*x)) + 105*Sqrt[a]*Sqrt[b]*B*(a + b*x^2)^3*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(105*b^5*(a + b*x^2)^(7/2))","A",1
48,1,147,150,0.4078767,"\int \frac{x^6 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(x^6*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\frac{\sqrt{a} C \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2} \sqrt{a+b x^2}}-\frac{3 a^4 (16 B+35 C x)+14 a^3 b x^2 (12 B+25 C x)+14 a^2 b^2 x^4 (15 B+29 C x)+a b^3 x^6 (105 B+176 C x)-15 A b^4 x^7}{105 a b^4 \left(a+b x^2\right)^{7/2}}","-\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,"-1/105*(-15*A*b^4*x^7 + 14*a^3*b*x^2*(12*B + 25*C*x) + 14*a^2*b^2*x^4*(15*B + 29*C*x) + 3*a^4*(16*B + 35*C*x) + a*b^3*x^6*(105*B + 176*C*x))/(a*b^4*(a + b*x^2)^(7/2)) + (Sqrt[a]*C*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(b^(9/2)*Sqrt[a + b*x^2])","A",1
49,1,89,132,0.1137221,"\int \frac{x^5 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(x^5*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\frac{-48 a^4 C-8 a^3 b \left(A+21 C x^2\right)-14 a^2 b^2 x^2 \left(2 A+15 C x^2\right)-35 a b^3 x^4 \left(A+3 C x^2\right)+15 b^4 B x^7}{105 a b^4 \left(a+b x^2\right)^{7/2}}","-\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,"(-48*a^4*C + 15*b^4*B*x^7 - 35*a*b^3*x^4*(A + 3*C*x^2) - 14*a^2*b^2*x^2*(2*A + 15*C*x^2) - 8*a^3*b*(A + 21*C*x^2))/(105*a*b^4*(a + b*x^2)^(7/2))","A",1
50,1,78,149,0.1038036,"\int \frac{x^4 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(x^4*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\frac{-8 a^4 B-28 a^3 b B x^2-35 a^2 b^2 B x^4+3 a b^3 x^5 \left(7 A+5 C x^2\right)+6 A b^4 x^7}{105 a^2 b^3 \left(a+b x^2\right)^{7/2}}","\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,"(-8*a^4*B - 28*a^3*b*B*x^2 - 35*a^2*b^2*B*x^4 + 6*A*b^4*x^7 + 3*a*b^3*x^5*(7*A + 5*C*x^2))/(105*a^2*b^3*(a + b*x^2)^(7/2))","A",1
51,1,84,139,0.1256967,"\int \frac{x^3 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(x^3*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\frac{-8 a^4 C-2 a^3 b \left(3 A+14 C x^2\right)-7 a^2 b^2 x^2 \left(3 A+5 C x^2\right)+21 a b^3 B x^5+6 b^4 B x^7}{105 a^2 b^3 \left(a+b x^2\right)^{7/2}}","\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,"(-8*a^4*C + 21*a*b^3*B*x^5 + 6*b^4*B*x^7 - 7*a^2*b^2*x^2*(3*A + 5*C*x^2) - 2*a^3*b*(3*A + 14*C*x^2))/(105*a^2*b^3*(a + b*x^2)^(7/2))","A",1
52,1,87,139,0.1424501,"\int \frac{x^2 \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(x^2*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\frac{-6 a^4 B-21 a^3 b B x^2+7 a^2 b^2 x^3 \left(5 A+3 C x^2\right)+2 a b^3 x^5 \left(14 A+3 C x^2\right)+8 A b^4 x^7}{105 a^3 b^2 \left(a+b x^2\right)^{7/2}}","\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,"(-6*a^4*B - 21*a^3*b*B*x^2 + 8*A*b^4*x^7 + 7*a^2*b^2*x^3*(5*A + 3*C*x^2) + 2*a*b^3*x^5*(14*A + 3*C*x^2))/(105*a^3*b^2*(a + b*x^2)^(7/2))","A",1
53,1,75,119,0.0883982,"\int \frac{x \left(A+B x+C x^2\right)}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(x*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]","\frac{-6 a^4 C-3 a^3 b \left(5 A+7 C x^2\right)+35 a^2 b^2 B x^3+28 a b^3 B x^5+8 b^4 B x^7}{105 a^3 b^2 \left(a+b x^2\right)^{7/2}}","\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,"(-6*a^4*C + 35*a^2*b^2*B*x^3 + 28*a*b^3*B*x^5 + 8*b^4*B*x^7 - 3*a^3*b*(5*A + 7*C*x^2))/(105*a^3*b^2*(a + b*x^2)^(7/2))","A",1
54,1,92,127,0.086178,"\int \frac{A+B x+C x^2}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x + C*x^2)/(a + b*x^2)^(9/2),x]","\frac{-15 a^4 B+35 a^3 b x \left(3 A+C x^2\right)+14 a^2 b^2 x^3 \left(15 A+2 C x^2\right)+8 a b^3 x^5 \left(21 A+C x^2\right)+48 A b^4 x^7}{105 a^4 b \left(a+b x^2\right)^{7/2}}","\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,"(-15*a^4*B + 48*A*b^4*x^7 + 35*a^3*b*x*(3*A + C*x^2) + 8*a*b^3*x^5*(21*A + C*x^2) + 14*a^2*b^2*x^3*(15*A + 2*C*x^2))/(105*a^4*b*(a + b*x^2)^(7/2))","A",1
55,1,120,138,0.2287214,"\int \frac{A+B x+C x^2}{x \left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x + C*x^2)/(x*(a + b*x^2)^(9/2)),x]","\frac{-15 a^4 C+a^3 b (176 A+105 B x)+14 a^2 b^2 x^2 (29 A+15 B x)+14 a b^3 x^4 (25 A+12 B x)+3 b^4 x^6 (35 A+16 B x)}{105 a^4 b \left(a+b x^2\right)^{7/2}}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{9/2}}","-\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,"(-15*a^4*C + 14*a*b^3*x^4*(25*A + 12*B*x) + 14*a^2*b^2*x^2*(29*A + 15*B*x) + 3*b^4*x^6*(35*A + 16*B*x) + a^3*b*(176*A + 105*B*x))/(105*a^4*b*(a + b*x^2)^(7/2)) - (A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(9/2)","A",1
56,1,158,188,0.2174068,"\int \frac{A+B x+C x^2}{x^2 \left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x + C*x^2)/(x^2*(a + b*x^2)^(9/2)),x]","\frac{a^4 (x (176 B+105 C x)-105 A)+14 a^3 b x^2 (x (29 B+15 C x)-60 A)+14 a^2 b^2 x^4 (x (25 B+12 C x)-120 A)+3 a b^3 x^6 (x (35 B+16 C x)-448 A)-105 \sqrt{a} B x \left(a+b x^2\right)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)-384 A b^4 x^8}{105 a^5 x \left(a+b x^2\right)^{7/2}}","-\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,"(-384*A*b^4*x^8 + 14*a^2*b^2*x^4*(-120*A + x*(25*B + 12*C*x)) + 14*a^3*b*x^2*(-60*A + x*(29*B + 15*C*x)) + 3*a*b^3*x^6*(-448*A + x*(35*B + 16*C*x)) + a^4*(-105*A + x*(176*B + 105*C*x)) - 105*Sqrt[a]*B*x*(a + b*x^2)^(7/2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(105*a^5*x*(a + b*x^2)^(7/2))","A",1
57,1,178,219,0.5342391,"\int \frac{A+B x+C x^2}{x^3 \left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x + C*x^2)/(x^3*(a + b*x^2)^(9/2)),x]","\frac{\frac{a^5 \left(-105 A-210 B x+352 C x^2\right)}{x^2}-4 a^4 b (396 A+7 x (60 B-29 C x))+14 a^3 b^2 x^2 (10 x (5 C x-24 B)-261 A)+42 a^2 b^3 x^4 (x (5 C x-64 B)-75 A)-3 a b^4 x^6 (315 A+256 B x)+\frac{105 \left(a+b x^2\right)^4 (9 A b-2 a C) \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{\sqrt{\frac{b x^2}{a}+1}}}{210 a^6 \left(a+b x^2\right)^{7/2}}","\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,"(-3*a*b^4*x^6*(315*A + 256*B*x) + (a^5*(-105*A - 210*B*x + 352*C*x^2))/x^2 - 4*a^4*b*(396*A + 7*x*(60*B - 29*C*x)) + 42*a^2*b^3*x^4*(-75*A + x*(-64*B + 5*C*x)) + 14*a^3*b^2*x^2*(-261*A + 10*x*(-24*B + 5*C*x)) + (105*(9*A*b - 2*a*C)*(a + b*x^2)^4*ArcTanh[Sqrt[1 + (b*x^2)/a]])/Sqrt[1 + (b*x^2)/a])/(210*a^6*(a + b*x^2)^(7/2))","A",1
58,1,43,45,0.0092877,"\int \frac{A (c x)^m}{a+b x^2} \, dx","Integrate[(A*(c*x)^m)/(a + b*x^2),x]","\frac{A x (c x)^m \, _2F_1\left(1,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a (m+1)}","\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,"(A*x*(c*x)^m*Hypergeometric2F1[1, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a*(1 + m))","A",1
59,1,82,91,0.0407932,"\int \frac{(c x)^m (A+B x)}{a+b x^2} \, dx","Integrate[((c*x)^m*(A + B*x))/(a + b*x^2),x]","\frac{x (c x)^m \left(A (m+2) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+B (m+1) x \, _2F_1\left(1,\frac{m}{2}+1;\frac{m}{2}+2;-\frac{b x^2}{a}\right)\right)}{a (m+1) (m+2)}","\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,"(x*(c*x)^m*(B*(1 + m)*x*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, -((b*x^2)/a)] + A*(2 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a*(1 + m)*(2 + m))","A",1
60,1,56,76,0.0715031,"\int \frac{(c x)^m \left(A+C x^2\right)}{a+b x^2} \, dx","Integrate[((c*x)^m*(A + C*x^2))/(a + b*x^2),x]","\frac{x (c x)^m \left((A b-a C) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a C\right)}{a b (m+1)}","\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,"(x*(c*x)^m*(a*C + (A*b - a*C)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a*b*(1 + m))","A",1
61,1,99,121,0.0954101,"\int \frac{(c x)^m \left(A+B x+C x^2\right)}{a+b x^2} \, dx","Integrate[((c*x)^m*(A + B*x + C*x^2))/(a + b*x^2),x]","\frac{x (c x)^m \left((m+2) (A b-a C) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+b B (m+1) x \, _2F_1\left(1,\frac{m}{2}+1;\frac{m}{2}+2;-\frac{b x^2}{a}\right)+a C (m+2)\right)}{a b (m+1) (m+2)}","\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,"(x*(c*x)^m*(a*C*(2 + m) + b*B*(1 + m)*x*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, -((b*x^2)/a)] + (A*b - a*C)*(2 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a*b*(1 + m)*(2 + m))","A",1
62,1,65,65,0.0313446,"\int x^3 \left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[x^3*(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]","\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","\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,"(a*A*x^4)/4 + (a*B*x^5)/5 + ((A*b + a*C)*x^6)/6 + ((b*B + a*D)*x^7)/7 + (b*C*x^8)/8 + (b*D*x^9)/9","A",1
63,1,65,65,0.0158407,"\int x^2 \left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[x^2*(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]","\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","\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,"(a*A*x^3)/3 + (a*B*x^4)/4 + ((A*b + a*C)*x^5)/5 + ((b*B + a*D)*x^6)/6 + (b*C*x^7)/7 + (b*D*x^8)/8","A",1
64,1,65,65,0.0097681,"\int x \left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[x*(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]","\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","\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,"(a*A*x^2)/2 + (a*B*x^3)/3 + ((A*b + a*C)*x^4)/4 + ((b*B + a*D)*x^5)/5 + (b*C*x^6)/6 + (b*D*x^7)/7","A",1
65,1,60,60,0.0088373,"\int \left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]","\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","\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,"a*A*x + (a*B*x^2)/2 + ((A*b + a*C)*x^3)/3 + ((b*B + a*D)*x^4)/4 + (b*C*x^5)/5 + (b*D*x^6)/6","A",1
66,1,56,56,0.0158106,"\int \frac{\left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right)}{x} \, dx","Integrate[((a + b*x^2)*(A + B*x + C*x^2 + D*x^3))/x,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","\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,"a*B*x + ((A*b + a*C)*x^2)/2 + ((b*B + a*D)*x^3)/3 + (b*C*x^4)/4 + (b*D*x^5)/5 + a*A*Log[x]","A",1
67,1,54,54,0.0548019,"\int \frac{\left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right)}{x^2} \, dx","Integrate[((a + b*x^2)*(A + B*x + C*x^2 + D*x^3))/x^2,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","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,"-((a*A)/x) + (A*b + a*C)*x + ((b*B + a*D)*x^2)/2 + (b*C*x^3)/3 + (b*D*x^4)/4 + a*B*Log[x]","A",1
68,1,51,54,0.0370671,"\int \frac{\left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right)}{x^3} \, dx","Integrate[((a + b*x^2)*(A + B*x + C*x^2 + D*x^3))/x^3,x]","\log (x) (a C+A b)-\frac{a \left(A+2 B x-2 D x^3\right)}{2 x^2}+\frac{1}{6} b x \left(6 B+3 C x+2 D x^2\right)","\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,"(b*x*(6*B + 3*C*x + 2*D*x^2))/6 - (a*(A + 2*B*x - 2*D*x^3))/(2*x^2) + (A*b + a*C)*Log[x]","A",1
69,1,55,54,0.036723,"\int \frac{\left(a+b x^2\right) \left(A+B x+C x^2+D x^3\right)}{x^4} \, dx","Integrate[((a + b*x^2)*(A + B*x + C*x^2 + D*x^3))/x^4,x]","\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","-\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,"-1/3*(a*A)/x^3 - (a*B)/(2*x^2) + (-(A*b) - a*C)/x + b*C*x + (b*D*x^2)/2 + (b*B + a*D)*Log[x]","A",1
70,1,98,109,0.0567749,"\int x^3 \left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[x^3*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]","a^2 \left(\frac{A x^4}{4}+\frac{B x^5}{5}+\frac{1}{42} x^6 (7 C+6 D x)\right)+\frac{1}{252} a b x^6 (84 A+x (72 B+7 x (9 C+8 D x)))+\frac{b^2 x^8 \left(495 A+4 x \left(110 B+99 C x+90 D x^2\right)\right)}{3960}","\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,"a^2*((A*x^4)/4 + (B*x^5)/5 + (x^6*(7*C + 6*D*x))/42) + (b^2*x^8*(495*A + 4*x*(110*B + 99*C*x + 90*D*x^2)))/3960 + (a*b*x^6*(84*A + x*(72*B + 7*x*(9*C + 8*D*x))))/252","A",1
71,1,92,109,0.0695032,"\int x^2 \left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[x^2*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]","\frac{42 a^2 x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+6 a b x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))+b^2 x^7 (360 A+7 x (45 B+4 x (10 C+9 D x)))}{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,"(42*a^2*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))) + 6*a*b*x^5*(168*A + 5*x*(28*B + 3*x*(8*C + 7*D*x))) + b^2*x^7*(360*A + 7*x*(45*B + 4*x*(10*C + 9*D*x))))/2520","A",1
72,1,92,104,0.0442733,"\int x \left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[x*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]","\frac{42 a^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x)))+12 a b x^4 (105 A+2 x (42 B+5 x (7 C+6 D x)))+5 b^2 x^6 (84 A+x (72 B+7 x (9 C+8 D x)))}{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,"(42*a^2*x^2*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))) + 12*a*b*x^4*(105*A + 2*x*(42*B + 5*x*(7*C + 6*D*x))) + 5*b^2*x^6*(84*A + x*(72*B + 7*x*(9*C + 8*D*x))))/2520","A",1
73,1,88,99,0.0561931,"\int \left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]","\frac{1}{840} \left(70 a^2 x (12 A+x (6 B+x (4 C+3 D x)))+28 a b x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+b^2 x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))\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,"(70*a^2*x*(12*A + x*(6*B + x*(4*C + 3*D*x))) + 28*a*b*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))) + b^2*x^5*(168*A + 5*x*(28*B + 3*x*(8*C + 7*D*x))))/840","A",1
74,1,88,92,0.0794231,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right)}{x} \, dx","Integrate[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x,x]","\frac{1}{420} x \left(70 a^2 (6 B+x (3 C+2 D x))+14 a b x (30 A+x (20 B+3 x (5 C+4 D x)))+b^2 x^3 (105 A+2 x (42 B+5 x (7 C+6 D x)))\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,"(x*(70*a^2*(6*B + x*(3*C + 2*D*x)) + 14*a*b*x*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))) + b^2*x^3*(105*A + 2*x*(42*B + 5*x*(7*C + 6*D*x)))))/420 + a^2*A*Log[x]","A",1
75,1,88,90,0.0598641,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right)}{x^2} \, dx","Integrate[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^2,x]","a^2 \left(-\frac{A}{x}+C x+\frac{D x^2}{2}\right)+a^2 B \log (x)+\frac{1}{6} a b x (12 A+x (6 B+x (4 C+3 D x)))+\frac{1}{60} b^2 x^3 (20 A+x (15 B+2 x (6 C+5 D 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,"a^2*(-(A/x) + C*x + (D*x^2)/2) + (a*b*x*(12*A + x*(6*B + x*(4*C + 3*D*x))))/6 + (b^2*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))))/60 + a^2*B*Log[x]","A",1
76,1,87,98,0.0439518,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right)}{x^3} \, dx","Integrate[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^3,x]","-\frac{a^2 \left(A+2 B x-2 D x^3\right)}{2 x^2}+a \log (x) (a C+2 A b)+\frac{1}{3} a b x (6 B+x (3 C+2 D x))+\frac{1}{60} b^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x)))","-\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,"-1/2*(a^2*(A + 2*B*x - 2*D*x^3))/x^2 + (a*b*x*(6*B + x*(3*C + 2*D*x)))/3 + (b^2*x^2*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))))/60 + a*(2*A*b + a*C)*Log[x]","A",1
77,1,83,98,0.055815,"\int \frac{\left(a+b x^2\right)^2 \left(A+B x+C x^2+D x^3\right)}{x^4} \, dx","Integrate[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^4,x]","-\frac{a^2 (2 A+3 x (B+2 C x))}{6 x^3}-\frac{2 a A b}{x}+a \log (x) (a D+2 b B)+a b x (2 C+D x)+\frac{1}{12} b^2 x \left(12 A+x \left(6 B+4 C x+3 D x^2\right)\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,"(-2*a*A*b)/x + a*b*x*(2*C + D*x) - (a^2*(2*A + 3*x*(B + 2*C*x)))/(6*x^3) + (b^2*x*(12*A + x*(6*B + 4*C*x + 3*D*x^2)))/12 + a*(2*b*B + a*D)*Log[x]","A",1
78,1,149,149,0.0257175,"\int x^3 \left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[x^3*(a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3),x]","\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}","\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,"(a^3*A*x^4)/4 + (a^3*B*x^5)/5 + (a^2*(3*A*b + a*C)*x^6)/6 + (a^2*(3*b*B + a*D)*x^7)/7 + (3*a*b*(A*b + a*C)*x^8)/8 + (a*b*(b*B + a*D)*x^9)/3 + (b^2*(A*b + 3*a*C)*x^10)/10 + (b^2*(b*B + 3*a*D)*x^11)/11 + (b^3*C*x^12)/12 + (b^3*D*x^13)/13","A",1
79,1,125,149,0.0648676,"\int x^2 \left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[x^2*(a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3),x]","\frac{462 a^3 x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+99 a^2 b x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))+33 a b^2 x^7 (360 A+7 x (45 B+4 x (10 C+9 D x)))+14 b^3 x^9 \left(220 A+3 x \left(66 B+60 C x+55 D x^2\right)\right)}{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,"(14*b^3*x^9*(220*A + 3*x*(66*B + 60*C*x + 55*D*x^2)) + 462*a^3*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))) + 99*a^2*b*x^5*(168*A + 5*x*(28*B + 3*x*(8*C + 7*D*x))) + 33*a*b^2*x^7*(360*A + 7*x*(45*B + 4*x*(10*C + 9*D*x))))/27720","A",1
80,1,124,138,0.0629107,"\int x \left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[x*(a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3),x]","\frac{462 a^3 x^2 (30 A+x (20 B+3 x (5 C+4 D x)))+198 a^2 b x^4 (105 A+2 x (42 B+5 x (7 C+6 D x)))+165 a b^2 x^6 (84 A+x (72 B+7 x (9 C+8 D x)))+7 b^3 x^8 \left(495 A+4 x \left(110 B+99 C x+90 D x^2\right)\right)}{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,"(7*b^3*x^8*(495*A + 4*x*(110*B + 99*C*x + 90*D*x^2)) + 462*a^3*x^2*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))) + 198*a^2*b*x^4*(105*A + 2*x*(42*B + 5*x*(7*C + 6*D*x))) + 165*a*b^2*x^6*(84*A + x*(72*B + 7*x*(9*C + 8*D*x))))/27720","A",1
81,1,121,133,0.0495414,"\int \left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[(a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3),x]","\frac{210 a^3 x (12 A+x (6 B+x (4 C+3 D x)))+126 a^2 b x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+9 a b^2 x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))+b^3 x^7 (360 A+7 x (45 B+4 x (10 C+9 D x)))}{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,"(210*a^3*x*(12*A + x*(6*B + x*(4*C + 3*D*x))) + 126*a^2*b*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))) + 9*a*b^2*x^5*(168*A + 5*x*(28*B + 3*x*(8*C + 7*D*x))) + b^3*x^7*(360*A + 7*x*(45*B + 4*x*(10*C + 9*D*x))))/2520","A",1
82,1,121,129,0.0670551,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right)}{x} \, dx","Integrate[((a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3))/x,x]","a^3 A \log (x)+\frac{x \left(420 a^3 (6 B+x (3 C+2 D x))+126 a^2 b x (30 A+x (20 B+3 x (5 C+4 D x)))+18 a b^2 x^3 (105 A+2 x (42 B+5 x (7 C+6 D x)))+5 b^3 x^5 (84 A+x (72 B+7 x (9 C+8 D x)))\right)}{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,"(x*(420*a^3*(6*B + x*(3*C + 2*D*x)) + 126*a^2*b*x*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))) + 18*a*b^2*x^3*(105*A + 2*x*(42*B + 5*x*(7*C + 6*D*x))) + 5*b^3*x^5*(84*A + x*(72*B + 7*x*(9*C + 8*D*x)))))/2520 + a^3*A*Log[x]","A",1
83,1,123,124,0.0809787,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right)}{x^2} \, dx","Integrate[((a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3))/x^2,x]","a^3 \left(-\frac{A}{x}+C x+\frac{D x^2}{2}\right)+a^3 B \log (x)+\frac{1}{4} a^2 b x (12 A+x (6 B+x (4 C+3 D x)))+\frac{1}{20} a b^2 x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+\frac{1}{840} b^3 x^5 (168 A+5 x (28 B+3 x (8 C+7 D 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,"a^3*(-(A/x) + C*x + (D*x^2)/2) + (a^2*b*x*(12*A + x*(6*B + x*(4*C + 3*D*x))))/4 + (a*b^2*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))))/20 + (b^3*x^5*(168*A + 5*x*(28*B + 3*x*(8*C + 7*D*x))))/840 + a^3*B*Log[x]","A",1
84,1,124,135,0.0618959,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right)}{x^3} \, dx","Integrate[((a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3))/x^3,x]","-\frac{a^3 \left(A+2 B x-2 D x^3\right)}{2 x^2}+a^2 \log (x) (a C+3 A b)+\frac{1}{2} a^2 b x (6 B+x (3 C+2 D x))+\frac{1}{20} a b^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x)))+\frac{1}{420} b^3 x^4 (105 A+2 x (42 B+5 x (7 C+6 D x)))","-\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,"-1/2*(a^3*(A + 2*B*x - 2*D*x^3))/x^2 + (a^2*b*x*(6*B + x*(3*C + 2*D*x)))/2 + (a*b^2*x^2*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))))/20 + (b^3*x^4*(105*A + 2*x*(42*B + 5*x*(7*C + 6*D*x))))/420 + a^2*(3*A*b + a*C)*Log[x]","A",1
85,1,124,139,0.0504396,"\int \frac{\left(a+b x^2\right)^3 \left(A+B x+C x^2+D x^3\right)}{x^4} \, dx","Integrate[((a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3))/x^4,x]","-\frac{a^3 (2 A+3 x (B+2 C x))}{6 x^3}+\frac{3 a^2 b \left(x^2 (2 C+D x)-2 A\right)}{2 x}+a^2 \log (x) (a D+3 b B)+\frac{1}{4} a b^2 x (12 A+x (6 B+x (4 C+3 D x)))+\frac{1}{60} b^3 x^3 (20 A+x (15 B+2 x (6 C+5 D x)))","-\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,"-1/6*(a^3*(2*A + 3*x*(B + 2*C*x)))/x^3 + (3*a^2*b*(-2*A + x^2*(2*C + D*x)))/(2*x) + (a*b^2*x*(12*A + x*(6*B + x*(4*C + 3*D*x))))/4 + (b^3*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))))/60 + a^2*(3*b*B + a*D)*Log[x]","A",1
86,1,130,151,0.0823048,"\int \frac{x^4 \left(A+B x+C x^2+D x^3\right)}{a+b x^2} \, dx","Integrate[(x^4*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]","\frac{-60 a^{3/2} \sqrt{b} (a C-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+b x \left(30 a^2 (2 C+D x)-5 a b (12 A+x (6 B+x (4 C+3 D x)))+b^2 x^2 (20 A+x (15 B+2 x (6 C+5 D x)))\right)-30 a^2 (a D-b B) \log \left(a+b x^2\right)}{60 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,"(b*x*(30*a^2*(2*C + D*x) - 5*a*b*(12*A + x*(6*B + x*(4*C + 3*D*x))) + b^2*x^2*(20*A + x*(15*B + 2*x*(6*C + 5*D*x)))) - 60*a^(3/2)*Sqrt[b]*(-(A*b) + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] - 30*a^2*(-(b*B) + a*D)*Log[a + b*x^2])/(60*b^4)","A",1
87,1,114,130,0.1038429,"\int \frac{x^3 \left(A+B x+C x^2+D x^3\right)}{a+b x^2} \, dx","Integrate[(x^3*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]","\frac{x \left(60 a^2 D-10 a b (6 B+x (3 C+2 D x))+b^2 x (30 A+x (20 B+3 x (5 C+4 D x)))\right)+30 a (a C-A b) \log \left(a+b x^2\right)}{60 b^3}-\frac{a^{3/2} (a D-b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}","\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,"-((a^(3/2)*(-(b*B) + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)) + (x*(60*a^2*D - 10*a*b*(6*B + x*(3*C + 2*D*x)) + b^2*x*(30*A + x*(20*B + 3*x*(5*C + 4*D*x)))) + 30*a*(-(A*b) + a*C)*Log[a + b*x^2])/(60*b^3)","A",1
88,1,95,111,0.0573387,"\int \frac{x^2 \left(A+B x+C x^2+D x^3\right)}{a+b x^2} \, dx","Integrate[(x^2*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]","\frac{b x \left(-6 a (2 C+D x)+12 A b+b x \left(6 B+4 C x+3 D x^2\right)\right)+12 \sqrt{a} \sqrt{b} (a C-A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+6 a (a D-b B) \log \left(a+b x^2\right)}{12 b^3}","-\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,"(b*x*(12*A*b - 6*a*(2*C + D*x) + b*x*(6*B + 4*C*x + 3*D*x^2)) + 12*Sqrt[a]*Sqrt[b]*(-(A*b) + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + 6*a*(-(b*B) + a*D)*Log[a + b*x^2])/(12*b^3)","A",1
89,1,81,92,0.0706924,"\int \frac{x \left(A+B x+C x^2+D x^3\right)}{a+b x^2} \, dx","Integrate[(x*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]","\frac{3 (A b-a C) \log \left(a+b x^2\right)+x (-6 a D+6 b B+b x (3 C+2 D x))}{6 b^2}+\frac{\sqrt{a} (a D-b B) \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{\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,"(Sqrt[a]*(-(b*B) + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2) + (x*(6*b*B - 6*a*D + b*x*(3*C + 2*D*x)) + 3*(A*b - a*C)*Log[a + b*x^2])/(6*b^2)","A",1
90,1,68,73,0.0430012,"\int \frac{A+B x+C x^2+D x^3}{a+b x^2} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2),x]","\frac{\frac{2 \sqrt{b} (A b-a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a}}+(b B-a D) \log \left(a+b x^2\right)+b x (2 C+D x)}{2 b^2}","\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,"(b*x*(2*C + D*x) + (2*Sqrt[b]*(A*b - a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (b*B - a*D)*Log[a + b*x^2])/(2*b^2)","A",1
91,1,73,72,0.0564196,"\int \frac{A+B x+C x^2+D x^3}{x \left(a+b x^2\right)} \, dx","Integrate[(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{(a D-b B) \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
92,1,75,76,0.048246,"\int \frac{A+B x+C x^2+D x^3}{x^2 \left(a+b x^2\right)} \, dx","Integrate[(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
93,1,84,92,0.0836836,"\int \frac{A+B x+C x^2+D x^3}{x^3 \left(a+b x^2\right)} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)),x]","\frac{(A b-a C) \log \left(a+b x^2\right)+2 \log (x) (a C-A b)-\frac{a A}{x^2}+\frac{2 \sqrt{a} (a D-b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b}}-\frac{2 a B}{x}}{2 a^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*A)/x^2) - (2*a*B)/x + (2*Sqrt[a]*(-(b*B) + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + 2*(-(A*b) + a*C)*Log[x] + (A*b - a*C)*Log[a + b*x^2])/(2*a^2)","A",1
94,1,139,176,0.1340046,"\int \frac{x^4 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^4*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]","\frac{\frac{6 a \left(a^2 D-a b (B+C x)+A b^2 x\right)}{a+b x^2}+12 b x (A b-2 a C)+6 \sqrt{a} \sqrt{b} (5 a C-3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+6 b x^2 (b B-2 a D)+6 a (3 a D-2 b B) \log \left(a+b x^2\right)+4 b^2 C x^3+3 b^2 D x^4}{12 b^4}","-\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,"(12*b*(A*b - 2*a*C)*x + 6*b*(b*B - 2*a*D)*x^2 + 4*b^2*C*x^3 + 3*b^2*D*x^4 + (6*a*(a^2*D + A*b^2*x - a*b*(B + C*x)))/(a + b*x^2) + 6*Sqrt[a]*Sqrt[b]*(-3*A*b + 5*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + 6*a*(-2*b*B + 3*a*D)*Log[a + b*x^2])/(12*b^4)","A",1
95,1,128,154,0.0831342,"\int \frac{x^3 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^3*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]","\frac{a (-a (C+D x)+A b+b B x)}{2 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{\sqrt{a} (5 a D-3 b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{x (b B-2 a D)}{b^3}+\frac{C x^2}{2 b^2}+\frac{D x^3}{3 b^2}","\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,"((b*B - 2*a*D)*x)/b^3 + (C*x^2)/(2*b^2) + (D*x^3)/(3*b^2) + (a*(A*b + b*B*x - a*(C + D*x)))/(2*b^3*(a + b*x^2)) + (Sqrt[a]*(-3*b*B + 5*a*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
96,1,100,134,0.0817459,"\int \frac{x^2 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]","\frac{\frac{a^2 (-D)+a b (B+C x)-A b^2 x}{a+b x^2}+\frac{\sqrt{b} (A b-3 a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a}}+(b B-2 a D) \log \left(a+b x^2\right)+2 b C x+b D x^2}{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,"(2*b*C*x + b*D*x^2 + (-(a^2*D) - A*b^2*x + a*b*(B + C*x))/(a + b*x^2) + (Sqrt[b]*(A*b - 3*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (b*B - 2*a*D)*Log[a + b*x^2])/(2*b^3)","A",1
97,1,92,101,0.0513408,"\int \frac{x \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]","\frac{a C+a D x-A b-b B x}{2 b^2 \left(a+b x^2\right)}-\frac{(3 a D-b B) \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}","-\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,"(D*x)/b^2 + (-(A*b) + a*C - b*B*x + a*D*x)/(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
98,1,83,93,0.0906529,"\int \frac{A+B x+C x^2+D x^3}{\left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^2,x]","\frac{\frac{\sqrt{b} (a C+A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}+\frac{a^2 D-a b (B+C x)+A b^2 x}{a \left(a+b x^2\right)}+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^2*D + A*b^2*x - a*b*(B + C*x))/(a*(a + b*x^2)) + (Sqrt[b]*(A*b + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2) + D*Log[a + b*x^2])/(2*b^2)","A",1
99,1,85,95,0.0763027,"\int \frac{A+B x+C x^2+D x^3}{x \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/(x*(a + b*x^2)^2),x]","\frac{\frac{a (-a (C+D x)+A b+b B x)}{b \left(a+b x^2\right)}-A \log \left(a+b x^2\right)+\frac{\sqrt{a} (a D+b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}+2 A \log (x)}{2 a^2}","\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*(A*b + b*B*x - a*(C + D*x)))/(b*(a + b*x^2)) + (Sqrt[a]*(b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) + 2*A*Log[x] - A*Log[a + b*x^2])/(2*a^2)","A",1
100,1,110,110,0.0744015,"\int \frac{A+B x+C x^2+D x^3}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[(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)+a b B+a b C x-A b^2 x}{2 a^2 b \left(a+b x^2\right)}-\frac{A}{a^2 x}-\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,"-(A/(a^2*x)) + (a*b*B - a^2*D - A*b^2*x + a*b*C*x)/(2*a^2*b*(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
101,1,112,135,0.1034358,"\int \frac{A+B x+C x^2+D x^3}{x^3 \left(a+b x^2\right)^2} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)^2),x]","\frac{\frac{a (a (C+D x)-A b-b B x)}{a+b x^2}+(2 A b-a C) \log \left(a+b x^2\right)+2 \log (x) (a C-2 A b)-\frac{a A}{x^2}+\frac{\sqrt{a} (a D-3 b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b}}-\frac{2 a B}{x}}{2 a^3}","-\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)/x^2) - (2*a*B)/x + (a*(-(A*b) - b*B*x + a*(C + D*x)))/(a + b*x^2) + (Sqrt[a]*(-3*b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + 2*(-2*A*b + a*C)*Log[x] + (2*A*b - a*C)*Log[a + b*x^2])/(2*a^3)","A",1
102,1,139,185,0.1169706,"\int \frac{x^4 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^4*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^3,x]","\frac{\frac{2 a \left(a^2 D-a b (B+C x)+A b^2 x\right)}{\left(a+b x^2\right)^2}+\frac{-12 a^2 D+8 a b B+9 a b C x-5 A b^2 x}{a+b x^2}+\frac{3 \sqrt{b} (A b-5 a C) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a}}+4 (b B-3 a D) \log \left(a+b x^2\right)+8 b C x+4 b D x^2}{8 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,"(8*b*C*x + 4*b*D*x^2 + (8*a*b*B - 12*a^2*D - 5*A*b^2*x + 9*a*b*C*x)/(a + b*x^2) + (2*a*(a^2*D + A*b^2*x - a*b*(B + C*x)))/(a + b*x^2)^2 + (3*Sqrt[b]*(A*b - 5*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + 4*(b*B - 3*a*D)*Log[a + b*x^2])/(8*b^4)","A",1
103,1,126,155,0.076112,"\int \frac{x^3 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^3*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^3,x]","\frac{a (-a (C+D x)+A b+b B x)}{4 b^3 \left(a+b x^2\right)^2}+\frac{8 a C+9 a D x-4 A b-5 b B x}{8 b^3 \left(a+b x^2\right)}+\frac{3 (b B-5 a D) \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{D x}{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,"(D*x)/b^3 + (-4*A*b + 8*a*C - 5*b*B*x + 9*a*D*x)/(8*b^3*(a + b*x^2)) + (a*(A*b + b*B*x - a*(C + D*x)))/(4*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
104,1,122,136,0.0952706,"\int \frac{x^2 \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^2*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^3,x]","\frac{\frac{\sqrt{b} (3 a C+A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}+\frac{-2 a^2 D+2 a b (B+C x)-2 A b^2 x}{\left(a+b x^2\right)^2}+\frac{8 a^2 D-a b (4 B+5 C x)+A b^2 x}{a \left(a+b x^2\right)}+4 D \log \left(a+b x^2\right)}{8 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*D - 2*A*b^2*x + 2*a*b*(B + C*x))/(a + b*x^2)^2 + (8*a^2*D + A*b^2*x - a*b*(4*B + 5*C*x))/(a*(a + b*x^2)) + (Sqrt[b]*(A*b + 3*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2) + 4*D*Log[a + b*x^2])/(8*b^3)","A",1
105,1,99,119,0.1339164,"\int \frac{x \left(A+B x+C x^2+D x^3\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^3,x]","\frac{\frac{(3 a D+b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\sqrt{b} \left(-a^2 (2 C+3 D x)-a b \left(2 A+x \left(B+4 C x+5 D x^2\right)\right)+b^2 B x^3\right)}{a \left(a+b x^2\right)^2}}{8 b^{5/2}}","\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,"((Sqrt[b]*(b^2*B*x^3 - a^2*(2*C + 3*D*x) - a*b*(2*A + x*(B + 4*C*x + 5*D*x^2))))/(a*(a + b*x^2)^2) + ((b*B + 3*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2))/(8*b^(5/2))","A",1
106,1,104,116,0.114155,"\int \frac{A+B x+C x^2+D x^3}{\left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^3,x]","\frac{\frac{\sqrt{a} \left(-2 a^3 D-a^2 b (2 B+x (C+4 D x))+a b^2 x \left(5 A+C x^2\right)+3 A b^3 x^3\right)}{\left(a+b x^2\right)^2}+\sqrt{b} (a C+3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} b^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,"((Sqrt[a]*(-2*a^3*D + 3*A*b^3*x^3 + a*b^2*x*(5*A + C*x^2) - a^2*b*(2*B + x*(C + 4*D*x))))/(a + b*x^2)^2 + Sqrt[b]*(3*A*b + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^2)","A",1
107,1,117,130,0.1089699,"\int \frac{A+B x+C x^2+D x^3}{x \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/(x*(a + b*x^2)^3),x]","\frac{\frac{2 a^2 (-a (C+D x)+A b+b B x)}{b \left(a+b x^2\right)^2}+\frac{a (a D x+4 A b+3 b B x)}{b \left(a+b x^2\right)}-4 A \log \left(a+b x^2\right)+\frac{\sqrt{a} (a D+3 b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}+8 A \log (x)}{8 a^3}","\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,"((a*(4*A*b + 3*b*B*x + a*D*x))/(b*(a + b*x^2)) + (2*a^2*(A*b + b*B*x - a*(C + D*x)))/(b*(a + b*x^2)^2) + (Sqrt[a]*(3*b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) + 8*A*Log[x] - 4*A*Log[a + b*x^2])/(8*a^3)","A",1
108,1,141,144,0.0990788,"\int \frac{A+B x+C x^2+D x^3}{x^2 \left(a+b x^2\right)^3} \, dx","Integrate[(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{4 a B+3 a C x-7 A b x}{8 a^3 \left(a+b x^2\right)}-\frac{A}{a^3 x}-\frac{B \log \left(a+b x^2\right)}{2 a^3}+\frac{B \log (x)}{a^3}+\frac{a^2 (-D)+a b B+a b C x-A b^2 x}{4 a^2 b \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,"-(A/(a^3*x)) + (a*b*B - a^2*D - A*b^2*x + a*b*C*x)/(4*a^2*b*(a + b*x^2)^2) + (4*a*B - 7*A*b*x + 3*a*C*x)/(8*a^3*(a + b*x^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
109,1,147,174,0.1631948,"\int \frac{A+B x+C x^2+D x^3}{x^3 \left(a+b x^2\right)^3} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)^3),x]","\frac{\frac{2 a^2 (a (C+D x)-A b-b B x)}{\left(a+b x^2\right)^2}+\frac{a (4 a C+3 a D x-8 A b-7 b B x)}{a+b x^2}+4 (3 A b-a C) \log \left(a+b x^2\right)+8 \log (x) (a C-3 A b)-\frac{4 a A}{x^2}+\frac{3 \sqrt{a} (a D-5 b B) \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b}}-\frac{8 a B}{x}}{8 a^4}","-\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*A)/x^2 - (8*a*B)/x + (a*(-8*A*b + 4*a*C - 7*b*B*x + 3*a*D*x))/(a + b*x^2) + (2*a^2*(-(A*b) - b*B*x + a*(C + D*x)))/(a + b*x^2)^2 + (3*Sqrt[a]*(-5*b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + 8*(-3*A*b + a*C)*Log[x] + 4*(3*A*b - a*C)*Log[a + b*x^2])/(8*a^4)","A",1
110,1,20,20,0.0073599,"\int \frac{-x+4 x^3}{\left(5+x^2\right)^2} \, dx","Integrate[(-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
111,1,18,23,0.0088337,"\int \frac{-x+x^3}{\sqrt{-2+x^2}} \, dx","Integrate[(-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
112,1,25,25,0.0131775,"\int \frac{-x^2+2 x^4}{1+2 x^2} \, dx","Integrate[(-x^2 + 2*x^4)/(1 + 2*x^2),x]","\frac{x^3}{3}-x+\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 + x^3/3 + ArcTan[Sqrt[2]*x]/Sqrt[2]","A",1
113,1,30,30,0.0066512,"\int \frac{x^3+x^4}{1+x^2} \, dx","Integrate[(x^3 + x^4)/(1 + x^2),x]","\frac{x^3}{3}+\frac{x^2}{2}-\frac{1}{2} \log \left(x^2+1\right)-x+\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 + x^2/2 + x^3/3 + ArcTan[x] - Log[1 + x^2]/2","A",1
114,1,210,210,0.155538,"\int \frac{x^6 \left(c+d x^2+e x^4+f x^6\right)}{a+b x^2} \, dx","Integrate[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]","\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}","\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,"-((a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/b^6) + (a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^3)/(3*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^5)/(5*b^4) + ((b^2*d - a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^9)/(9*b^2) + (f*x^11)/(11*b) + (a^(5/2)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(13/2)","A",1
115,1,162,172,0.1206509,"\int \frac{x^4 \left(c+d x^2+e x^4+f x^6\right)}{a+b x^2} \, dx","Integrate[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]","\frac{x \left(315 a^4 f-105 a^3 b \left(3 e+f x^2\right)+21 a^2 b^2 \left(15 d+5 e x^2+3 f x^4\right)-3 a b^3 \left(105 c+35 d x^2+21 e x^4+15 f x^6\right)+b^4 x^2 \left(105 c+63 d x^2+45 e x^4+35 f x^6\right)\right)}{315 b^5}-\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^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^4*f - 105*a^3*b*(3*e + f*x^2) + 21*a^2*b^2*(15*d + 5*e*x^2 + 3*f*x^4) - 3*a*b^3*(105*c + 35*d*x^2 + 21*e*x^4 + 15*f*x^6) + b^4*x^2*(105*c + 63*d*x^2 + 45*e*x^4 + 35*f*x^6)))/(315*b^5) - (a^(3/2)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2)","A",1
116,1,128,136,0.0969326,"\int \frac{x^2 \left(c+d x^2+e x^4+f x^6\right)}{a+b x^2} \, dx","Integrate[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]","\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(-105 a^3 f+35 a^2 b \left(3 e+f x^2\right)-7 a b^2 \left(15 d+5 e x^2+3 f x^4\right)+b^3 \left(105 c+35 d x^2+21 e x^4+15 f x^6\right)\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*a^3*f + 35*a^2*b*(3*e + f*x^2) - 7*a*b^2*(15*d + 5*e*x^2 + 3*f*x^4) + b^3*(105*c + 35*d*x^2 + 21*e*x^4 + 15*f*x^6)))/(105*b^4) + (Sqrt[a]*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)","A",1
117,1,98,100,0.0828972,"\int \frac{c+d x^2+e x^4+f x^6}{a+b x^2} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(a + b*x^2),x]","\frac{x \left(15 a^2 f-5 a b \left(3 e+f x^2\right)+b^2 \left(15 d+5 e x^2+3 f x^4\right)\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*a^2*f - 5*a*b*(3*e + f*x^2) + b^2*(15*d + 5*e*x^2 + 3*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
118,1,83,84,0.0662773,"\int \frac{c+d x^2+e x^4+f x^6}{x^2 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^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{x (b e-a f)}{b^2}-\frac{c}{a x}+\frac{f x^3}{3 b}","-\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,"-(c/(a*x)) + ((b*e - a*f)*x)/b^2 + (f*x^3)/(3*b) + ((-(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
119,1,83,82,0.0865606,"\int \frac{c+d x^2+e x^4+f x^6}{x^4 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^4*(a + b*x^2)),x]","\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}","\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,"-1/3*c/(a*x^3) + (b*c - a*d)/(a^2*x) + (f*x)/b - ((-(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
120,1,103,104,0.0879941,"\int \frac{c+d x^2+e x^4+f x^6}{x^6 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^6*(a + b*x^2)),x]","\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}","\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,"-1/5*c/(a*x^5) + (b*c - a*d)/(3*a^2*x^3) + (-(b^2*c) + a*b*d - a^2*e)/(a^3*x) + ((-(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
121,1,139,137,0.1238205,"\int \frac{c+d x^2+e x^4+f x^6}{x^8 \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)),x]","\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}","\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,"-1/7*c/(a*x^7) + (b*c - a*d)/(5*a^2*x^5) + (-(b^2*c) + a*b*d - a^2*e)/(3*a^3*x^3) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(a^4*x) - (Sqrt[b]*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)","A",1
122,1,174,175,0.1440658,"\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)),x]","\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}","\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,"-1/9*c/(a*x^9) + (b*c - a*d)/(7*a^2*x^7) + (-(b^2*c) + a*b*d - a^2*e)/(5*a^3*x^5) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(3*a^4*x^3) + (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a^5*x) + (b^(3/2)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(11/2)","A",1
123,1,211,211,0.1709488,"\int \frac{c+d x^2+e x^4+f x^6}{x^{12} \left(a+b x^2\right)} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^12*(a + b*x^2)),x]","\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}}","\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,"-1/11*c/(a*x^11) + (b*c - a*d)/(9*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(7*a^3*x^7) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(5*a^4*x^5) + (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(3*a^5*x^3) + (b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(a^6*x) + (b^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(13/2)","A",1
124,1,227,240,0.1278253,"\int \frac{x^6 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^2,x]","\frac{x^5 \left(3 a^2 f-2 a b e+b^2 d\right)}{5 b^4}+\frac{a x \left(5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right)}{b^6}+\frac{x^3 \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{3 b^5}-\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 \left(a^5 (-f)+a^4 b e-a^3 b^2 d+a^2 b^3 c\right)}{2 b^6 \left(a+b x^2\right)}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^9}{9 b^2}","\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,"(a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x)/b^6 + ((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^3)/(3*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^5)/(5*b^4) + ((b*e - 2*a*f)*x^7)/(7*b^3) + (f*x^9)/(9*b^2) - ((a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x)/(2*b^6*(a + b*x^2)) - (a^(3/2)*(-5*b^3*c + 7*a*b^2*d - 9*a^2*b*e + 11*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))","A",1
125,1,187,202,0.1043365,"\int \frac{x^4 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^2,x]","\frac{x^3 \left(3 a^2 f-2 a b e+b^2 d\right)}{3 b^4}+\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(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{b^5}+\frac{x \left(a^4 (-f)+a^3 b e-a^2 b^2 d+a b^3 c\right)}{2 b^5 \left(a+b x^2\right)}+\frac{x^5 (b e-2 a f)}{5 b^3}+\frac{f x^7}{7 b^2}","\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,"((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x)/b^5 + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^3)/(3*b^4) + ((b*e - 2*a*f)*x^5)/(5*b^3) + (f*x^7)/(7*b^2) + ((a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x)/(2*b^5*(a + b*x^2)) + (Sqrt[a]*(-3*b^3*c + 5*a*b^2*d - 7*a^2*b*e + 9*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))","A",1
126,1,148,163,0.0886759,"\int \frac{x^2 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^2,x]","\frac{x \left(3 a^2 f-2 a b e+b^2 d\right)}{b^4}-\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(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{2 b^4 \left(a+b x^2\right)}+\frac{x^3 (b e-2 a f)}{3 b^3}+\frac{f x^5}{5 b^2}","\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,"((b^2*d - 2*a*b*e + 3*a^2*f)*x)/b^4 + ((b*e - 2*a*f)*x^3)/(3*b^3) + (f*x^5)/(5*b^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(2*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
127,1,122,118,0.0981378,"\int \frac{c+d x^2+e x^4+f x^6}{\left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(a + b*x^2)^2,x]","-\frac{x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{2 a b^3 \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}","\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,"((b*e - 2*a*f)*x)/b^3 + (f*x^3)/(3*b^2) - ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*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
128,1,115,112,0.069823,"\int \frac{c+d x^2+e x^4+f x^6}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^2),x]","-\frac{c}{a^2 x}+\frac{x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{2 a^2 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-a b^2 d+3 b^3 c\right)}{2 a^{5/2} b^{5/2}}+\frac{f x}{b^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,"-(c/(a^2*x)) + (f*x)/b^2 + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a^2*b^2*(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
129,1,125,121,0.0790445,"\int \frac{c+d x^2+e x^4+f x^6}{x^4 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^4*(a + b*x^2)^2),x]","\frac{2 b c-a d}{a^3 x}-\frac{c}{3 a^2 x^3}-\frac{x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{2 a^3 b \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,"-1/3*c/(a^2*x^3) + (2*b*c - a*d)/(a^3*x) - ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a^3*b*(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
130,1,151,152,0.0910693,"\int \frac{c+d x^2+e x^4+f x^6}{x^6 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^6*(a + b*x^2)^2),x]","\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)}","\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,"-1/5*c/(a^2*x^5) + (2*b*c - a*d)/(3*a^3*x^3) + (-3*b^2*c + 2*a*b*d - a^2*e)/(a^4*x) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a^4*(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
131,1,190,189,0.1087374,"\int \frac{c+d x^2+e x^4+f x^6}{x^8 \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^2),x]","\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}","\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,"-1/7*c/(a^2*x^7) + (2*b*c - a*d)/(5*a^3*x^5) + (-3*b^2*c + 2*a*b*d - a^2*e)/(3*a^4*x^3) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(a^5*x) - (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a^5*(a + b*x^2)) - (Sqrt[b]*(-9*b^3*c + 7*a*b^2*d - 5*a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(11/2))","A",1
132,1,230,230,0.1244027,"\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \left(a+b x^2\right)^2} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^2),x]","\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}","\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,"-1/9*c/(a^2*x^9) + (2*b*c - a*d)/(7*a^3*x^7) + (-3*b^2*c + 2*a*b*d - a^2*e)/(5*a^4*x^5) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(3*a^5*x^3) + (b*(-5*b^3*c + 4*a*b^2*d - 3*a^2*b*e + 2*a^3*f))/(a^6*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a^6*(a + b*x^2)) + (b^(3/2)*(-11*b^3*c + 9*a*b^2*d - 7*a^2*b*e + 5*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(13/2))","A",1
133,1,272,287,0.1857411,"\int \frac{x^8 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^8*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x]","\frac{x^5 \left(6 a^2 f-3 a b e+b^2 d\right)}{5 b^5}+\frac{a^2 x \left(25 a^3 f-21 a^2 b e+17 a b^2 d-13 b^3 c\right)}{8 b^7 \left(a+b x^2\right)}+\frac{a^3 x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{4 b^7 \left(a+b x^2\right)^2}+\frac{a x \left(15 a^3 f-10 a^2 b e+6 a b^2 d-3 b^3 c\right)}{b^7}+\frac{x^3 \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{3 b^6}-\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}","\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,"(a*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*x)/b^7 + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^3)/(3*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^5)/(5*b^5) + ((b*e - 3*a*f)*x^7)/(7*b^4) + (f*x^9)/(9*b^3) + (a^3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(4*b^7*(a + b*x^2)^2) + (a^2*(-13*b^3*c + 17*a*b^2*d - 21*a^2*b*e + 25*a^3*f)*x)/(8*b^7*(a + b*x^2)) - (a^(3/2)*(-35*b^3*c + 63*a*b^2*d - 99*a^2*b*e + 143*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(15/2))","A",1
134,1,232,247,0.1515707,"\int \frac{x^6 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x]","\frac{x^3 \left(6 a^2 f-3 a b e+b^2 d\right)}{3 b^5}+\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(-21 a^3 f+17 a^2 b e-13 a b^2 d+9 b^3 c\right)}{8 b^6 \left(a+b x^2\right)}+\frac{a^2 x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{4 b^6 \left(a+b x^2\right)^2}+\frac{x \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{b^6}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^7}{7 b^3}","\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,"((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6 + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^3)/(3*b^5) + ((b*e - 3*a*f)*x^5)/(5*b^4) + (f*x^7)/(7*b^3) + (a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*b^6*(a + b*x^2)^2) + (a*(9*b^3*c - 13*a*b^2*d + 17*a^2*b*e - 21*a^3*f)*x)/(8*b^6*(a + b*x^2)) + (Sqrt[a]*(-15*b^3*c + 35*a*b^2*d - 63*a^2*b*e + 99*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))","A",1
135,1,176,207,0.176962,"\int \frac{x^4 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(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 \left(e-3 f x^2\right)+a^2 b^2 \left(225 d-875 e x^2+504 f x^4\right)-a b^3 \left(45 c-375 d x^2+280 e x^4+72 f x^6\right)+b^4 x^2 \left(8 \left(15 d x^2+5 e x^4+3 f x^6\right)-75 c\right)\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*(945*a^4*f - 525*a^3*b*(e - 3*f*x^2) + a^2*b^2*(225*d - 875*e*x^2 + 504*f*x^4) - a*b^3*(45*c - 375*d*x^2 + 280*e*x^4 + 72*f*x^6) + b^4*x^2*(-75*c + 8*(15*d*x^2 + 5*e*x^4 + 3*f*x^6))))/(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
136,1,156,167,0.1338599,"\int \frac{x^2 \left(c+d x^2+e x^4+f x^6\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[(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+5 a^3 b \left(9 e-35 f x^2\right)+a^2 b^2 \left(-9 d+75 e x^2-56 f x^4\right)+a b^3 \left(-3 c-15 d x^2+24 e x^4+8 f x^6\right)+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,"(x*(-105*a^4*f + 3*b^4*c*x^2 + 5*a^3*b*(9*e - 35*f*x^2) + a^2*b^2*(-9*d + 75*e*x^2 - 56*f*x^4) + a*b^3*(-3*c - 15*d*x^2 + 24*e*x^4 + 8*f*x^6)))/(24*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
137,1,141,147,0.1224933,"\int \frac{c+d x^2+e x^4+f x^6}{\left(a+b x^2\right)^3} \, dx","Integrate[(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+a^3 b \left(25 f x^2-3 e\right)-a^2 b^2 \left(d+5 e x^2-8 f x^4\right)+a b^3 \left(5 c+d x^2\right)+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*(15*a^4*f + 3*b^4*c*x^2 + a*b^3*(5*c + d*x^2) + a^3*b*(-3*e + 25*f*x^2) - a^2*b^2*(d + 5*e*x^2 - 8*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
138,1,155,153,0.1315154,"\int \frac{c+d x^2+e x^4+f x^6}{x^2 \left(a+b x^2\right)^3} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^3),x]","-\frac{c}{a^3 x}-\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{x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{4 a^2 b^2 \left(a+b x^2\right)^2}+\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{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,"-(c/(a^3*x)) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*a^2*b^2*(a + b*x^2)^2) - ((7*b^3*c - 3*a*b^2*d - a^2*b*e + 5*a^3*f)*x)/(8*a^3*b^2*(a + b*x^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
139,1,169,168,0.158509,"\int \frac{c+d x^2+e x^4+f x^6}{x^4 \left(a+b x^2\right)^3} \, dx","Integrate[(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+a^3 b \left(3 x^2 \left(-8 d+5 e x^2+f x^4\right)-8 c\right)+a^2 b^2 x^2 \left(56 c-75 d x^2+9 e x^4\right)+5 a b^3 x^4 \left(35 c-9 d x^2\right)+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,"(-3*a^4*f*x^4 + 105*b^4*c*x^6 + 5*a*b^3*x^4*(35*c - 9*d*x^2) + a^2*b^2*x^2*(56*c - 75*d*x^2 + 9*e*x^4) + a^3*b*(-8*c + 3*x^2*(-8*d + 5*e*x^2 + f*x^4)))/(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
140,1,196,196,0.1241092,"\int \frac{c+d x^2+e x^4+f x^6}{x^6 \left(a+b x^2\right)^3} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^6*(a + b*x^2)^3),x]","\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}","\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,"-1/5*c/(a^3*x^5) + (3*b*c - a*d)/(3*a^4*x^3) + (-6*b^2*c + 3*a*b*d - a^2*e)/(a^5*x) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*a^4*(a + b*x^2)^2) + ((-15*b^3*c + 11*a*b^2*d - 7*a^2*b*e + 3*a^3*f)*x)/(8*a^5*(a + b*x^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
141,1,234,234,0.1469493,"\int \frac{c+d x^2+e x^4+f x^6}{x^8 \left(a+b x^2\right)^3} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^3),x]","\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}","\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,"-1/7*c/(a^3*x^7) + (3*b*c - a*d)/(5*a^4*x^5) - (6*b^2*c - 3*a*b*d + a^2*e)/(3*a^5*x^3) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(a^6*x) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(4*a^5*(a + b*x^2)^2) + (b*(19*b^3*c - 15*a*b^2*d + 11*a^2*b*e - 7*a^3*f)*x)/(8*a^6*(a + b*x^2)) + (Sqrt[b]*(99*b^3*c - 63*a*b^2*d + 35*a^2*b*e - 15*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(13/2))","A",1
142,1,276,277,0.1583618,"\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \left(a+b x^2\right)^3} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^3),x]","\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}","\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,"-1/9*c/(a^3*x^9) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(3*a^6*x^3) + (b*(-15*b^3*c + 10*a*b^2*d - 6*a^2*b*e + 3*a^3*f))/(a^7*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*a^6*(a + b*x^2)^2) + (b^2*(-23*b^3*c + 19*a*b^2*d - 15*a^2*b*e + 11*a^3*f)*x)/(8*a^7*(a + b*x^2)) + (b^(3/2)*(-143*b^3*c + 99*a*b^2*d - 63*a^2*b*e + 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(15/2))","A",1
143,1,158,214,0.174448,"\int \frac{x^5 \left(c+d x^2+e x^4+f x^6\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(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+128 a^4 b \left(11 e+5 f x^2\right)-16 a^3 b^2 \left(99 d+44 e x^2+30 f x^4\right)+8 a^2 b^3 \left(231 c+99 d x^2+66 e x^4+50 f x^6\right)-2 a b^4 x^2 \left(462 c+297 d x^2+220 e x^4+175 f x^6\right)+b^5 x^4 \left(693 c+5 \left(99 d x^2+77 e x^4+63 f x^6\right)\right)\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]*(-1280*a^5*f + 128*a^4*b*(11*e + 5*f*x^2) - 16*a^3*b^2*(99*d + 44*e*x^2 + 30*f*x^4) + 8*a^2*b^3*(231*c + 99*d*x^2 + 66*e*x^4 + 50*f*x^6) - 2*a*b^4*x^2*(462*c + 297*d*x^2 + 220*e*x^4 + 175*f*x^6) + b^5*x^4*(693*c + 5*(99*d*x^2 + 77*e*x^4 + 63*f*x^6))))/(3465*b^6)","A",1
144,1,122,167,0.1233683,"\int \frac{x^3 \left(c+d x^2+e x^4+f x^6\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(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-16 a^3 b \left(9 e+4 f x^2\right)+24 a^2 b^2 \left(7 d+3 e x^2+2 f x^4\right)-2 a b^3 \left(105 c+42 d x^2+27 e x^4+20 f x^6\right)+b^4 x^2 \left(105 c+63 d x^2+45 e x^4+35 f x^6\right)\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]*(128*a^4*f - 16*a^3*b*(9*e + 4*f*x^2) + 24*a^2*b^2*(7*d + 3*e*x^2 + 2*f*x^4) - 2*a*b^3*(105*c + 42*d*x^2 + 27*e*x^4 + 20*f*x^6) + b^4*x^2*(105*c + 63*d*x^2 + 45*e*x^4 + 35*f*x^6)))/(315*b^5)","A",1
145,1,89,121,0.0846194,"\int \frac{x \left(c+d x^2+e x^4+f x^6\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(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+8 a^2 b \left(7 e+3 f x^2\right)-2 a b^2 \left(35 d+14 e x^2+9 f x^4\right)+b^3 \left(105 c+35 d x^2+21 e x^4+15 f x^6\right)\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]*(-48*a^3*f + 8*a^2*b*(7*e + 3*f*x^2) - 2*a*b^2*(35*d + 14*e*x^2 + 9*f*x^4) + b^3*(105*c + 35*d*x^2 + 21*e*x^4 + 15*f*x^6)))/(105*b^4)","A",1
146,1,86,103,0.1301384,"\int \frac{c+d x^2+e x^4+f x^6}{x \sqrt{a+b x^2}} \, dx","Integrate[(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-2 a b \left(5 e+2 f x^2\right)+b^2 \left(15 d+5 e x^2+3 f x^4\right)\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]*(8*a^2*f - 2*a*b*(5*e + 2*f*x^2) + b^2*(15*d + 5*e*x^2 + 3*f*x^4)))/(15*b^3) - (c*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]","A",1
147,1,131,100,0.4120955,"\int \frac{c+d x^2+e x^4+f x^6}{x^3 \sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^3*Sqrt[a + b*x^2]),x]","\frac{3 b^3 c x^2 \sqrt{\frac{b x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)-\left(a+b x^2\right) \left(4 a^2 f x^2-2 a b x^2 \left(3 e+f x^2\right)+3 b^2 c\right)}{6 a b^2 x^2 \sqrt{a+b x^2}}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}","\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,"-((d*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]) + (-((a + b*x^2)*(3*b^2*c + 4*a^2*f*x^2 - 2*a*b*x^2*(3*e + f*x^2))) + 3*b^3*c*x^2*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]])/(6*a*b^2*x^2*Sqrt[a + b*x^2])","A",1
148,1,141,114,0.3738727,"\int \frac{c+d x^2+e x^4+f x^6}{x^5 \sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^5*Sqrt[a + b*x^2]),x]","-\frac{b^2 c \sqrt{a+b x^2} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{b x^2}{a}+1\right)}{a^3}-\frac{b d \sqrt{a+b x^2} \left(\frac{a}{b x^2}-\frac{\tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{2 a^2}-\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{f \sqrt{a+b x^2}}{b}","\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,"(f*Sqrt[a + b*x^2])/b - (e*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a] - (b*d*Sqrt[a + b*x^2]*(a/(b*x^2) - ArcTanh[Sqrt[1 + (b*x^2)/a]]/Sqrt[1 + (b*x^2)/a]))/(2*a^2) - (b^2*c*Sqrt[a + b*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (b*x^2)/a])/a^3","C",1
149,1,162,146,1.0240543,"\int \frac{c+d x^2+e x^4+f x^6}{x^7 \sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^7*Sqrt[a + b*x^2]),x]","\frac{b^3 c \sqrt{a+b x^2} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};\frac{b x^2}{a}+1\right)}{a^4}-\frac{b^2 d \sqrt{a+b x^2} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{b x^2}{a}+1\right)}{a^3}-\frac{b e \sqrt{a+b x^2} \left(\frac{a}{b x^2}-\frac{\tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{2 a^2}-\frac{f \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}","\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,"-((f*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]) - (b*e*Sqrt[a + b*x^2]*(a/(b*x^2) - ArcTanh[Sqrt[1 + (b*x^2)/a]]/Sqrt[1 + (b*x^2)/a]))/(2*a^2) - (b^2*d*Sqrt[a + b*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (b*x^2)/a])/a^3 + (b^3*c*Sqrt[a + b*x^2]*Hypergeometric2F1[1/2, 4, 3/2, 1 + (b*x^2)/a])/a^4","C",1
150,1,140,195,0.3361392,"\int \frac{c+d x^2+e x^4+f x^6}{x^9 \sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^9*Sqrt[a + b*x^2]),x]","\frac{b \sqrt{a+b x^2} \left(-\frac{a^4 f}{b x^2}+\frac{a^3 f \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{\sqrt{\frac{b x^2}{a}+1}}-2 a^2 b e \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{b x^2}{a}+1\right)-2 b^3 c \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};\frac{b x^2}{a}+1\right)+2 a b^2 d \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};\frac{b x^2}{a}+1\right)\right)}{2 a^5}","\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,"(b*Sqrt[a + b*x^2]*(-((a^4*f)/(b*x^2)) + (a^3*f*ArcTanh[Sqrt[1 + (b*x^2)/a]])/Sqrt[1 + (b*x^2)/a] - 2*a^2*b*e*Hypergeometric2F1[1/2, 3, 3/2, 1 + (b*x^2)/a] + 2*a*b^2*d*Hypergeometric2F1[1/2, 4, 3/2, 1 + (b*x^2)/a] - 2*b^3*c*Hypergeometric2F1[1/2, 5, 3/2, 1 + (b*x^2)/a]))/(2*a^5)","C",1
151,1,184,245,0.2379602,"\int \frac{x^4 \left(c+d x^2+e x^4+f x^6\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]","\frac{\sqrt{b} x \sqrt{a+b x^2} \left(945 a^4 f-210 a^3 b \left(5 e+3 f x^2\right)+4 a^2 b^2 \left(300 d+175 e x^2+126 f x^4\right)-16 a b^3 \left(90 c+50 d x^2+35 e x^4+27 f x^6\right)+32 b^4 x^2 \left(30 c+20 d x^2+15 e x^4+12 f x^6\right)\right)-15 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)}{3840 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[b]*x*Sqrt[a + b*x^2]*(945*a^4*f - 210*a^3*b*(5*e + 3*f*x^2) + 4*a^2*b^2*(300*d + 175*e*x^2 + 126*f*x^4) + 32*b^4*x^2*(30*c + 20*d*x^2 + 15*e*x^4 + 12*f*x^6) - 16*a*b^3*(90*c + 50*d*x^2 + 35*e*x^4 + 27*f*x^6)) - 15*a^2*(-96*b^3*c + 80*a*b^2*d - 70*a^2*b*e + 63*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(3840*b^(11/2))","A",1
152,1,149,194,0.1674366,"\int \frac{x^2 \left(c+d x^2+e x^4+f x^6\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]","\frac{3 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)+\sqrt{b} x \sqrt{a+b x^2} \left(-105 a^3 f+10 a^2 b \left(12 e+7 f x^2\right)-8 a b^2 \left(18 d+10 e x^2+7 f x^4\right)+16 b^3 \left(12 c+6 d x^2+4 e x^4+3 f x^6\right)\right)}{384 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[b]*x*Sqrt[a + b*x^2]*(-105*a^3*f + 10*a^2*b*(12*e + 7*f*x^2) - 8*a*b^2*(18*d + 10*e*x^2 + 7*f*x^4) + 16*b^3*(12*c + 6*d*x^2 + 4*e*x^4 + 3*f*x^6)) + 3*a*(-64*b^3*c + 48*a*b^2*d - 40*a^2*b*e + 35*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(384*b^(9/2))","A",1
153,1,118,145,0.1139018,"\int \frac{c+d x^2+e x^4+f x^6}{\sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/Sqrt[a + b*x^2],x]","\frac{\sqrt{b} x \sqrt{a+b x^2} \left(15 a^2 f-2 a b \left(9 e+5 f x^2\right)+4 b^2 \left(6 d+3 e x^2+2 f x^4\right)\right)+3 \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)}{48 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[b]*x*Sqrt[a + b*x^2]*(15*a^2*f - 2*a*b*(9*e + 5*f*x^2) + 4*b^2*(6*d + 3*e*x^2 + 2*f*x^4)) + 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]])/(48*b^(7/2))","A",1
154,1,103,117,0.1175992,"\int \frac{c+d x^2+e x^4+f x^6}{x^2 \sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*Sqrt[a + b*x^2]),x]","\frac{\frac{\sqrt{b} \sqrt{a+b x^2} \left(-3 a^2 f x^2+2 a b x^2 \left(2 e+f x^2\right)-8 b^2 c\right)}{a x}+\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[b]*Sqrt[a + b*x^2]*(-8*b^2*c - 3*a^2*f*x^2 + 2*a*b*x^2*(2*e + f*x^2)))/(a*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
155,1,93,110,0.1134496,"\int \frac{c+d x^2+e x^4+f x^6}{x^4 \sqrt{a+b x^2}} \, dx","Integrate[(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 \left(c+3 d x^2\right)+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]*(4*b^2*c*x^2 + 3*a^2*f*x^4 - 2*a*b*(c + 3*d*x^2)))/(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
156,1,95,118,0.112874,"\int \frac{c+d x^2+e x^4+f x^6}{x^6 \sqrt{a+b x^2}} \, dx","Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^6*Sqrt[a + b*x^2]),x]","\frac{f \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}-\frac{\sqrt{a+b x^2} \left(a^2 \left(3 c+5 d x^2+15 e x^4\right)-2 a b x^2 \left(2 c+5 d x^2\right)+8 b^2 c x^4\right)}{15 a^3 x^5}","\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,"-1/15*(Sqrt[a + b*x^2]*(8*b^2*c*x^4 - 2*a*b*x^2*(2*c + 5*d*x^2) + a^2*(3*c + 5*d*x^2 + 15*e*x^4)))/(a^3*x^5) + (f*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b]","A",1
157,1,103,140,0.0812081,"\int \frac{c+d x^2+e x^4+f x^6}{x^8 \sqrt{a+b x^2}} \, dx","Integrate[(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(-a^3 \left(15 c+21 d x^2+35 x^4 \left(e+3 f x^2\right)\right)+2 a^2 b x^2 \left(9 c+14 d x^2+35 e x^4\right)-8 a b^2 x^4 \left(3 c+7 d x^2\right)+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]*(48*b^3*c*x^6 - 8*a*b^2*x^4*(3*c + 7*d*x^2) + 2*a^2*b*x^2*(9*c + 14*d*x^2 + 35*e*x^4) - a^3*(15*c + 21*d*x^2 + 35*x^4*(e + 3*f*x^2))))/(105*a^4*x^7)","A",1
158,1,134,189,0.0955735,"\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \sqrt{a+b x^2}} \, dx","Integrate[(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(a^4 \left(35 c+45 d x^2+63 e x^4+105 f x^6\right)-2 a^3 b x^2 \left(20 c+27 d x^2+42 e x^4+105 f x^6\right)+24 a^2 b^2 x^4 \left(2 c+3 d x^2+7 e x^4\right)-16 a b^3 x^6 \left(4 c+9 d x^2\right)+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,"-1/315*(Sqrt[a + b*x^2]*(128*b^4*c*x^8 - 16*a*b^3*x^6*(4*c + 9*d*x^2) + 24*a^2*b^2*x^4*(2*c + 3*d*x^2 + 7*e*x^4) - 2*a^3*b*x^2*(20*c + 27*d*x^2 + 42*e*x^4 + 105*f*x^6) + a^4*(35*c + 45*d*x^2 + 63*e*x^4 + 105*f*x^6)))/(a^5*x^9)","A",1
159,1,273,381,0.5759603,"\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","Integrate[(x^8*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]","\frac{\sqrt{a+b x^2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(16 A b^3-3 a \left(143 a^2 D-66 a b C+24 b^2 B\right)\right)}{16 \sqrt{a} b^{15/2} \sqrt{\frac{b x^2}{a}+1}}+\frac{x \left(45045 a^6 D-2310 a^5 b \left(9 C-65 D x^2\right)+42 a^4 b^2 \left(180 B-1650 C x^2+4147 D x^4\right)-12 a^3 b^3 \left(140 A-2100 B x^2+6699 C x^4-6292 D x^6\right)+a^2 b^4 x^2 \left(-5600 A+29232 B x^2-34848 C x^4+5005 D x^6\right)-2 a b^5 x^4 \left(3248 A-6336 B x^2+1155 C x^4+455 D x^6\right)+4 b^6 x^6 \left(35 \left(6 B x^2+3 C x^4+2 D x^6\right)-704 A\right)\right)}{1680 b^7 \left(a+b x^2\right)^{7/2}}","-\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,"(x*(45045*a^6*D - 2310*a^5*b*(9*C - 65*D*x^2) + 42*a^4*b^2*(180*B - 1650*C*x^2 + 4147*D*x^4) - 12*a^3*b^3*(140*A - 2100*B*x^2 + 6699*C*x^4 - 6292*D*x^6) - 2*a*b^5*x^4*(3248*A - 6336*B*x^2 + 1155*C*x^4 + 455*D*x^6) + a^2*b^4*x^2*(-5600*A + 29232*B*x^2 - 34848*C*x^4 + 5005*D*x^6) + 4*b^6*x^6*(-704*A + 35*(6*B*x^2 + 3*C*x^4 + 2*D*x^6))))/(1680*b^7*(a + b*x^2)^(7/2)) + ((16*A*b^3 - 3*a*(24*b^2*B - 66*a*b*C + 143*a^2*D))*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(16*Sqrt[a]*b^(15/2)*Sqrt[1 + (b*x^2)/a])","A",1
160,1,229,279,0.4788025,"\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","Integrate[(x^6*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]","\frac{\sqrt{a+b x^2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right) \left(99 a^2 D-36 a b C+8 b^2 B\right)}{8 \sqrt{a} b^{13/2} \sqrt{\frac{b x^2}{a}+1}}-\frac{x \left(10395 a^6 D-630 a^5 b \left(6 C-55 D x^2\right)+42 a^4 b^2 \left(20 B-300 C x^2+957 D x^4\right)+8 a^3 b^3 x^2 \left(350 B-1827 C x^2+2178 D x^4\right)+a^2 b^4 x^4 \left(3248 B-6336 C x^2+1155 D x^4\right)+2 a b^5 x^6 \left(704 B-105 \left(2 C x^2+D x^4\right)\right)-120 A b^6 x^6\right)}{840 a b^6 \left(a+b x^2\right)^{7/2}}","\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,"-1/840*(x*(10395*a^6*D - 120*A*b^6*x^6 - 630*a^5*b*(6*C - 55*D*x^2) + 42*a^4*b^2*(20*B - 300*C*x^2 + 957*D*x^4) + a^2*b^4*x^4*(3248*B - 6336*C*x^2 + 1155*D*x^4) + 8*a^3*b^3*x^2*(350*B - 1827*C*x^2 + 2178*D*x^4) + 2*a*b^5*x^6*(704*B - 105*(2*C*x^2 + D*x^4))))/(a*b^6*(a + b*x^2)^(7/2)) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(13/2)*Sqrt[1 + (b*x^2)/a])","A",1
161,1,194,210,0.4972337,"\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","Integrate[(x^4*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]","\frac{105 a^{5/2} \sqrt{\frac{b x^2}{a}+1} \left(a+b x^2\right)^3 (2 b C-9 a D) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x \left(945 a^6 D-210 a^5 b \left(C-15 D x^2\right)+14 a^4 b^2 x^2 \left(261 D x^2-50 C\right)+4 a^3 b^3 x^4 \left(396 D x^2-203 C\right)+a^2 b^4 x^6 \left(105 D x^2-352 C\right)+6 a b^5 x^4 \left(7 A+5 B x^2\right)+12 A b^6 x^6\right)}{210 a^2 b^{11/2} \left(a+b x^2\right)^{7/2}}","\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,"(Sqrt[b]*x*(945*a^6*D + 12*A*b^6*x^6 + 6*a*b^5*x^4*(7*A + 5*B*x^2) - 210*a^5*b*(C - 15*D*x^2) + a^2*b^4*x^6*(-352*C + 105*D*x^2) + 14*a^4*b^2*x^2*(-50*C + 261*D*x^2) + 4*a^3*b^3*x^4*(-203*C + 396*D*x^2)) + 105*a^(5/2)*(2*b*C - 9*a*D)*(a + b*x^2)^3*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(210*a^2*b^(11/2)*(a + b*x^2)^(7/2))","A",1
162,1,168,179,0.4558205,"\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","Integrate[(x^2*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]","\frac{-105 a^6 D x-350 a^5 b D x^3-406 a^4 b^2 D x^5-176 a^3 b^3 D x^7+a^2 b^4 x^3 \left(35 A+21 B x^2+15 C x^4\right)+2 a b^5 x^5 \left(14 A+3 B x^2\right)+8 A b^6 x^7}{105 a^3 b^4 \left(a+b x^2\right)^{7/2}}+\frac{\sqrt{a} D \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2} \sqrt{a+b x^2}}","\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,"(-105*a^6*D*x - 350*a^5*b*D*x^3 - 406*a^4*b^2*D*x^5 + 8*A*b^6*x^7 - 176*a^3*b^3*D*x^7 + 2*a*b^5*x^5*(14*A + 3*B*x^2) + a^2*b^4*x^3*(35*A + 21*B*x^2 + 15*C*x^4))/(105*a^3*b^4*(a + b*x^2)^(7/2)) + (Sqrt[a]*D*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(b^(9/2)*Sqrt[a + b*x^2])","A",1
163,1,98,134,0.1074933,"\int \frac{A+B x^2+C x^4+D x^6}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2)^(9/2),x]","\frac{a^3 \left(105 A x+35 B x^3+21 C x^5+15 D x^7\right)+2 a^2 b x^3 \left(105 A+14 B x^2+3 C x^4\right)+8 a b^2 x^5 \left(21 A+B x^2\right)+48 A b^3 x^7}{105 a^4 \left(a+b x^2\right)^{7/2}}","\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,"(48*A*b^3*x^7 + 8*a*b^2*x^5*(21*A + B*x^2) + 2*a^2*b*x^3*(105*A + 14*B*x^2 + 3*C*x^4) + a^3*(105*A*x + 35*B*x^3 + 21*C*x^5 + 15*D*x^7))/(105*a^4*(a + b*x^2)^(7/2))","A",1
164,1,133,185,0.1838624,"\int \frac{A+B x^2+C x^4+D x^6}{x^2 \left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^2*(a + b*x^2)^(9/2)),x]","\frac{-7 a^4 \left(15 A-15 B x^2-5 C x^4-3 D x^6\right)+2 a^3 b x^2 \left(-420 A+105 B x^2+14 C x^4+3 D x^6\right)+8 a^2 b^2 x^4 \left(-210 A+21 B x^2+C x^4\right)+48 a b^3 x^6 \left(B x^2-28 A\right)-384 A b^4 x^8}{105 a^5 x \left(a+b x^2\right)^{7/2}}","-\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,"(-384*A*b^4*x^8 + 48*a*b^3*x^6*(-28*A + B*x^2) + 8*a^2*b^2*x^4*(-210*A + 21*B*x^2 + C*x^4) - 7*a^4*(15*A - 15*B*x^2 - 5*C*x^4 - 3*D*x^6) + 2*a^3*b*x^2*(-420*A + 105*B*x^2 + 14*C*x^4 + 3*D*x^6))/(105*a^5*x*(a + b*x^2)^(7/2))","A",1
165,1,165,242,0.1340323,"\int \frac{A+B x^2+C x^4+D x^6}{x^4 \left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^4*(a + b*x^2)^(9/2)),x]","\frac{-35 a^5 \left(A+3 B x^2-3 C x^4-D x^6\right)+14 a^4 b x^2 \left(25 A-60 B x^2+15 C x^4+2 D x^6\right)+8 a^3 b^2 x^4 \left(350 A-210 B x^2+21 C x^4+D x^6\right)+16 a^2 b^3 x^6 \left(350 A-84 B x^2+3 C x^4\right)+128 a b^4 x^8 \left(35 A-3 B x^2\right)+1280 A b^5 x^{10}}{105 a^6 x^3 \left(a+b x^2\right)^{7/2}}","\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,"(1280*A*b^5*x^10 + 128*a*b^4*x^8*(35*A - 3*B*x^2) + 16*a^2*b^3*x^6*(350*A - 84*B*x^2 + 3*C*x^4) - 35*a^5*(A + 3*B*x^2 - 3*C*x^4 - D*x^6) + 8*a^3*b^2*x^4*(350*A - 210*B*x^2 + 21*C*x^4 + D*x^6) + 14*a^4*b*x^2*(25*A - 60*B*x^2 + 15*C*x^4 + 2*D*x^6))/(105*a^6*x^3*(a + b*x^2)^(7/2))","A",1
166,1,202,281,0.158051,"\int \frac{A+B x^2+C x^4+D x^6}{x^6 \left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^6*(a + b*x^2)^(9/2)),x]","\frac{-7 a^6 \left(3 A+5 x^2 \left(B+3 C x^2-3 D x^4\right)\right)+14 a^5 b x^2 \left(6 A+25 B x^2-60 C x^4+15 D x^6\right)+56 a^4 b^2 x^4 \left(-15 A+50 B x^2-30 C x^4+3 D x^6\right)+16 a^3 b^3 x^6 \left(-420 A+350 B x^2-84 C x^4+3 D x^6\right)-128 a^2 b^4 x^8 \left(105 A-35 B x^2+3 C x^4\right)+256 a b^5 x^{10} \left(5 B x^2-42 A\right)-3072 A b^6 x^{12}}{105 a^7 x^5 \left(a+b x^2\right)^{7/2}}","-\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,"(-3072*A*b^6*x^12 + 256*a*b^5*x^10*(-42*A + 5*B*x^2) - 128*a^2*b^4*x^8*(105*A - 35*B*x^2 + 3*C*x^4) + 16*a^3*b^3*x^6*(-420*A + 350*B*x^2 - 84*C*x^4 + 3*D*x^6) + 56*a^4*b^2*x^4*(-15*A + 50*B*x^2 - 30*C*x^4 + 3*D*x^6) + 14*a^5*b*x^2*(6*A + 25*B*x^2 - 60*C*x^4 + 15*D*x^6) - 7*a^6*(3*A + 5*x^2*(B + 3*C*x^2 - 3*D*x^4)))/(105*a^7*x^5*(a + b*x^2)^(7/2))","A",1
167,1,234,334,0.1614834,"\int \frac{A+B x^2+C x^4+D x^6}{x^8 \left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^8*(a + b*x^2)^(9/2)),x]","\frac{-a^7 \left(15 A+21 B x^2+35 x^4 \left(C+3 D x^2\right)\right)+14 a^6 b x^2 \left(3 A+6 B x^2+25 C x^4-60 D x^6\right)-56 a^5 b^2 x^4 \left(3 A+15 B x^2-50 C x^4+30 D x^6\right)+112 a^4 b^3 x^6 \left(15 A-60 B x^2+50 C x^4-12 D x^6\right)+128 a^3 b^4 x^8 \left(105 A-105 B x^2+35 C x^4-3 D x^6\right)+256 a^2 b^5 x^{10} \left(105 A-42 B x^2+5 C x^4\right)-3072 a b^6 x^{12} \left(B x^2-7 A\right)+6144 A b^7 x^{14}}{105 a^8 x^7 \left(a+b x^2\right)^{7/2}}","-\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,"(6144*A*b^7*x^14 - 3072*a*b^6*x^12*(-7*A + B*x^2) + 256*a^2*b^5*x^10*(105*A - 42*B*x^2 + 5*C*x^4) + 14*a^6*b*x^2*(3*A + 6*B*x^2 + 25*C*x^4 - 60*D*x^6) + 112*a^4*b^3*x^6*(15*A - 60*B*x^2 + 50*C*x^4 - 12*D*x^6) + 128*a^3*b^4*x^8*(105*A - 105*B*x^2 + 35*C*x^4 - 3*D*x^6) - 56*a^5*b^2*x^4*(3*A + 15*B*x^2 - 50*C*x^4 + 30*D*x^6) - a^7*(15*A + 21*B*x^2 + 35*x^4*(C + 3*D*x^2)))/(105*a^8*x^7*(a + b*x^2)^(7/2))","A",1
168,1,270,392,0.1796707,"\int \frac{A+B x^2+C x^4+D x^6}{x^{10} \left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*(a + b*x^2)^(9/2)),x]","\frac{-a^8 \left(35 A+45 B x^2+63 C x^4+105 D x^6\right)+2 a^7 b x^2 \left(40 A+21 \left(3 B x^2+6 C x^4+25 D x^6\right)\right)-56 a^6 b^2 x^4 \left(4 A+9 B x^2+45 C x^4-150 D x^6\right)+112 a^5 b^3 x^6 \left(8 A+45 B x^2-180 C x^4+150 D x^6\right)+4480 a^4 b^4 x^8 \left(-2 A+9 B x^2-9 C x^4+3 D x^6\right)+256 a^3 b^5 x^{10} \left(-280 A+315 B x^2-126 C x^4+15 D x^6\right)-1024 a^2 b^6 x^{12} \left(140 A-63 B x^2+9 C x^4\right)+2048 a b^7 x^{14} \left(9 B x^2-56 A\right)-32768 A b^8 x^{16}}{315 a^9 x^9 \left(a+b x^2\right)^{7/2}}","-\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,"(-32768*A*b^8*x^16 + 2048*a*b^7*x^14*(-56*A + 9*B*x^2) - 1024*a^2*b^6*x^12*(140*A - 63*B*x^2 + 9*C*x^4) - 56*a^6*b^2*x^4*(4*A + 9*B*x^2 + 45*C*x^4 - 150*D*x^6) + 4480*a^4*b^4*x^8*(-2*A + 9*B*x^2 - 9*C*x^4 + 3*D*x^6) + 256*a^3*b^5*x^10*(-280*A + 315*B*x^2 - 126*C*x^4 + 15*D*x^6) - a^8*(35*A + 45*B*x^2 + 63*C*x^4 + 105*D*x^6) + 112*a^5*b^3*x^6*(8*A + 45*B*x^2 - 180*C*x^4 + 150*D*x^6) + 2*a^7*b*x^2*(40*A + 21*(3*B*x^2 + 6*C*x^4 + 25*D*x^6)))/(315*a^9*x^9*(a + b*x^2)^(7/2))","A",1
169,1,158,214,0.1850613,"\int \frac{c x^5+d x^7+e x^9+f x^{11}}{\sqrt{a+b x^2}} \, dx","Integrate[(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+128 a^4 b \left(11 e+5 f x^2\right)-16 a^3 b^2 \left(99 d+44 e x^2+30 f x^4\right)+8 a^2 b^3 \left(231 c+99 d x^2+66 e x^4+50 f x^6\right)-2 a b^4 x^2 \left(462 c+297 d x^2+220 e x^4+175 f x^6\right)+b^5 x^4 \left(693 c+5 \left(99 d x^2+77 e x^4+63 f x^6\right)\right)\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]*(-1280*a^5*f + 128*a^4*b*(11*e + 5*f*x^2) - 16*a^3*b^2*(99*d + 44*e*x^2 + 30*f*x^4) + 8*a^2*b^3*(231*c + 99*d*x^2 + 66*e*x^4 + 50*f*x^6) - 2*a*b^4*x^2*(462*c + 297*d*x^2 + 220*e*x^4 + 175*f*x^6) + b^5*x^4*(693*c + 5*(99*d*x^2 + 77*e*x^4 + 63*f*x^6))))/(3465*b^6)","A",1
170,1,122,167,0.1307309,"\int \frac{c x^3+d x^5+e x^7+f x^9}{\sqrt{a+b x^2}} \, dx","Integrate[(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-16 a^3 b \left(9 e+4 f x^2\right)+24 a^2 b^2 \left(7 d+3 e x^2+2 f x^4\right)-2 a b^3 \left(105 c+42 d x^2+27 e x^4+20 f x^6\right)+b^4 x^2 \left(105 c+63 d x^2+45 e x^4+35 f x^6\right)\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]*(128*a^4*f - 16*a^3*b*(9*e + 4*f*x^2) + 24*a^2*b^2*(7*d + 3*e*x^2 + 2*f*x^4) - 2*a*b^3*(105*c + 42*d*x^2 + 27*e*x^4 + 20*f*x^6) + b^4*x^2*(105*c + 63*d*x^2 + 45*e*x^4 + 35*f*x^6)))/(315*b^5)","A",1
171,1,89,121,0.0900783,"\int \frac{c x+d x^3+e x^5+f x^7}{\sqrt{a+b x^2}} \, dx","Integrate[(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+8 a^2 b \left(7 e+3 f x^2\right)-2 a b^2 \left(35 d+14 e x^2+9 f x^4\right)+b^3 \left(105 c+35 d x^2+21 e x^4+15 f x^6\right)\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]*(-48*a^3*f + 8*a^2*b*(7*e + 3*f*x^2) - 2*a*b^2*(35*d + 14*e*x^2 + 9*f*x^4) + b^3*(105*c + 35*d*x^2 + 21*e*x^4 + 15*f*x^6)))/(105*b^4)","A",1
172,1,221,261,0.5895597,"\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","Integrate[(x^2*(A + B*x^2 + C*x^4 + D*x^6 + F*x^8))/(a + b*x^2)^(9/2),x]","\frac{105 a^{7/2} \left(a+b x^2\right)^3 \sqrt{\frac{b x^2}{a}+1} (2 b D-9 a F) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x \left(945 a^7 F-210 a^6 b \left(D-15 F x^2\right)+14 a^5 b^2 x^2 \left(261 F x^2-50 D\right)+4 a^4 b^3 x^4 \left(396 F x^2-203 D\right)+a^3 b^4 x^6 \left(105 F x^2-352 D\right)+2 a^2 b^5 x^2 \left(35 A+21 B x^2+15 C x^4\right)+4 a b^6 x^4 \left(14 A+3 B x^2\right)+16 A b^7 x^6\right)}{210 a^3 b^{11/2} \left(a+b x^2\right)^{7/2}}","\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,"(Sqrt[b]*x*(945*a^7*F + 16*A*b^7*x^6 + 4*a*b^6*x^4*(14*A + 3*B*x^2) - 210*a^6*b*(D - 15*F*x^2) + a^3*b^4*x^6*(-352*D + 105*F*x^2) + 14*a^5*b^2*x^2*(-50*D + 261*F*x^2) + 4*a^4*b^3*x^4*(-203*D + 396*F*x^2) + 2*a^2*b^5*x^2*(35*A + 21*B*x^2 + 15*C*x^4)) + 105*a^(7/2)*(2*b*D - 9*a*F)*(a + b*x^2)^3*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(210*a^3*b^(11/2)*(a + b*x^2)^(7/2))","A",1
173,1,197,214,0.5144149,"\int \frac{A+B x^2+C x^4+D x^6+F x^8}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(A + B*x^2 + C*x^4 + D*x^6 + F*x^8)/(a + b*x^2)^(9/2),x]","\frac{x \left(-105 a^7 F-350 a^6 b F x^2-406 a^5 b^2 F x^4-176 a^4 b^3 F x^6+a^3 b^4 \left(105 A+35 B x^2+21 C x^4+15 D x^6\right)+2 a^2 b^5 x^2 \left(105 A+14 B x^2+3 C x^4\right)+8 a b^6 x^4 \left(21 A+B x^2\right)+48 A b^7 x^6\right)}{105 a^4 b^4 \left(a+b x^2\right)^{7/2}}+\frac{\sqrt{a} F \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2} \sqrt{a+b x^2}}","\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,"(x*(-105*a^7*F - 350*a^6*b*F*x^2 - 406*a^5*b^2*F*x^4 + 48*A*b^7*x^6 - 176*a^4*b^3*F*x^6 + 8*a*b^6*x^4*(21*A + B*x^2) + 2*a^2*b^5*x^2*(105*A + 14*B*x^2 + 3*C*x^4) + a^3*b^4*(105*A + 35*B*x^2 + 21*C*x^4 + 15*D*x^6)))/(105*a^4*b^4*(a + b*x^2)^(7/2)) + (Sqrt[a]*F*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(b^(9/2)*Sqrt[a + b*x^2])","A",1
174,1,138,193,0.1893382,"\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","Integrate[(A + B*x^2 + C*x^4 + D*x^6 + F*x^8)/(x^2*(a + b*x^2)^(9/2)),x]","\frac{a^4 \left(-105 A+105 B x^2+35 C x^4+21 D x^6+15 F x^8\right)+2 a^3 b x^2 \left(-420 A+105 B x^2+14 C x^4+3 D x^6\right)+8 a^2 b^2 x^4 \left(-210 A+21 B x^2+C x^4\right)+48 a b^3 x^6 \left(B x^2-28 A\right)-384 A b^4 x^8}{105 a^5 x \left(a+b x^2\right)^{7/2}}","-\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,"(-384*A*b^4*x^8 + 48*a*b^3*x^6*(-28*A + B*x^2) + 8*a^2*b^2*x^4*(-210*A + 21*B*x^2 + C*x^4) + 2*a^3*b*x^2*(-420*A + 105*B*x^2 + 14*C*x^4 + 3*D*x^6) + a^4*(-105*A + 105*B*x^2 + 35*C*x^4 + 21*D*x^6 + 15*F*x^8))/(105*a^5*x*(a + b*x^2)^(7/2))","A",1