1,1,73,0,0.586604," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4,x, algorithm=""fricas"")","-\frac{{\left(24 \, a^{4} x^{4} + 30 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 45 \, a x + 64\right)} \sqrt{-a^{2} x^{2} + 1} + 90 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{120 \, a^{5}}"," ",0,"-1/120*((24*a^4*x^4 + 30*a^3*x^3 + 32*a^2*x^2 + 45*a*x + 64)*sqrt(-a^2*x^2 + 1) + 90*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a^5","A",0
2,1,65,0,0.462982," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3,x, algorithm=""fricas"")","-\frac{{\left(6 \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 9 \, a x + 16\right)} \sqrt{-a^{2} x^{2} + 1} + 18 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{24 \, a^{4}}"," ",0,"-1/24*((6*a^3*x^3 + 8*a^2*x^2 + 9*a*x + 16)*sqrt(-a^2*x^2 + 1) + 18*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a^4","A",0
3,1,57,0,0.824487," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2,x, algorithm=""fricas"")","-\frac{{\left(2 \, a^{2} x^{2} + 3 \, a x + 4\right)} \sqrt{-a^{2} x^{2} + 1} + 6 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{6 \, a^{3}}"," ",0,"-1/6*((2*a^2*x^2 + 3*a*x + 4)*sqrt(-a^2*x^2 + 1) + 6*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a^3","A",0
4,1,48,0,0.543821," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x,x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 2\right)} + 2 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{2 \, a^{2}}"," ",0,"-1/2*(sqrt(-a^2*x^2 + 1)*(a*x + 2) + 2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a^2","A",0
5,1,42,0,0.658071," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} x^{2} + 1} + 2 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{a}"," ",0,"-(sqrt(-a^2*x^2 + 1) + 2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a","A",0
6,1,44,0,0.476876," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x,x, algorithm=""fricas"")","-2 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right)"," ",0,"-2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + log((sqrt(-a^2*x^2 + 1) - 1)/x)","B",0
7,1,41,0,0.524735," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2,x, algorithm=""fricas"")","\frac{a x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \sqrt{-a^{2} x^{2} + 1}}{x}"," ",0,"(a*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) - sqrt(-a^2*x^2 + 1))/x","A",0
8,1,52,0,0.425347," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3,x, algorithm=""fricas"")","\frac{a^{2} x^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \sqrt{-a^{2} x^{2} + 1} {\left(2 \, a x + 1\right)}}{2 \, x^{2}}"," ",0,"1/2*(a^2*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) - sqrt(-a^2*x^2 + 1)*(2*a*x + 1))/x^2","A",0
9,1,61,0,0.511419," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4,x, algorithm=""fricas"")","\frac{3 \, a^{3} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(4 \, a^{2} x^{2} + 3 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, x^{3}}"," ",0,"1/6*(3*a^3*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (4*a^2*x^2 + 3*a*x + 2)*sqrt(-a^2*x^2 + 1))/x^3","A",0
10,1,69,0,0.447718," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^5,x, algorithm=""fricas"")","\frac{9 \, a^{4} x^{4} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(16 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 8 \, a x + 6\right)} \sqrt{-a^{2} x^{2} + 1}}{24 \, x^{4}}"," ",0,"1/24*(9*a^4*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (16*a^3*x^3 + 9*a^2*x^2 + 8*a*x + 6)*sqrt(-a^2*x^2 + 1))/x^4","A",0
11,1,42,0,0.401118," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3,x, algorithm=""fricas"")","-\frac{3 \, a^{4} x^{4} + 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 24 \, a x + 24 \, \log\left(a x - 1\right)}{12 \, a^{4}}"," ",0,"-1/12*(3*a^4*x^4 + 8*a^3*x^3 + 12*a^2*x^2 + 24*a*x + 24*log(a*x - 1))/a^4","A",0
12,1,33,0,0.412837," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2,x, algorithm=""fricas"")","-\frac{a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x + 6 \, \log\left(a x - 1\right)}{3 \, a^{3}}"," ",0,"-1/3*(a^3*x^3 + 3*a^2*x^2 + 6*a*x + 6*log(a*x - 1))/a^3","A",0
13,1,25,0,0.611714," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x,x, algorithm=""fricas"")","-\frac{a^{2} x^{2} + 4 \, a x + 4 \, \log\left(a x - 1\right)}{2 \, a^{2}}"," ",0,"-1/2*(a^2*x^2 + 4*a*x + 4*log(a*x - 1))/a^2","A",0
14,1,17,0,0.424440," ","integrate((a*x+1)^2/(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{a x + 2 \, \log\left(a x - 1\right)}{a}"," ",0,"-(a*x + 2*log(a*x - 1))/a","A",0
15,1,11,0,0.682814," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x,x, algorithm=""fricas"")","-2 \, \log\left(a x - 1\right) + \log\left(x\right)"," ",0,"-2*log(a*x - 1) + log(x)","A",0
16,1,23,0,0.763465," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^2,x, algorithm=""fricas"")","-\frac{2 \, a x \log\left(a x - 1\right) - 2 \, a x \log\left(x\right) + 1}{x}"," ",0,"-(2*a*x*log(a*x - 1) - 2*a*x*log(x) + 1)/x","A",0
17,1,35,0,0.506817," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^3,x, algorithm=""fricas"")","-\frac{4 \, a^{2} x^{2} \log\left(a x - 1\right) - 4 \, a^{2} x^{2} \log\left(x\right) + 4 \, a x + 1}{2 \, x^{2}}"," ",0,"-1/2*(4*a^2*x^2*log(a*x - 1) - 4*a^2*x^2*log(x) + 4*a*x + 1)/x^2","A",0
18,1,43,0,0.469782," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^4,x, algorithm=""fricas"")","-\frac{6 \, a^{3} x^{3} \log\left(a x - 1\right) - 6 \, a^{3} x^{3} \log\left(x\right) + 6 \, a^{2} x^{2} + 3 \, a x + 1}{3 \, x^{3}}"," ",0,"-1/3*(6*a^3*x^3*log(a*x - 1) - 6*a^3*x^3*log(x) + 6*a^2*x^2 + 3*a*x + 1)/x^3","A",0
19,1,85,0,0.907101," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2,x, algorithm=""fricas"")","\frac{52 \, a x + 66 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{3} x^{3} + 7 \, a^{2} x^{2} + 19 \, a x - 52\right)} \sqrt{-a^{2} x^{2} + 1} - 52}{6 \, {\left(a^{4} x - a^{3}\right)}}"," ",0,"1/6*(52*a*x + 66*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*x^3 + 7*a^2*x^2 + 19*a*x - 52)*sqrt(-a^2*x^2 + 1) - 52)/(a^4*x - a^3)","A",0
20,1,76,0,1.130797," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x,x, algorithm=""fricas"")","\frac{14 \, a x + 18 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(a^{2} x^{2} + 5 \, a x - 14\right)} \sqrt{-a^{2} x^{2} + 1} - 14}{2 \, {\left(a^{3} x - a^{2}\right)}}"," ",0,"1/2*(14*a*x + 18*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a^2*x^2 + 5*a*x - 14)*sqrt(-a^2*x^2 + 1) - 14)/(a^3*x - a^2)","A",0
21,1,65,0,0.436866," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\frac{5 \, a x + 6 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a x - 5\right)} - 5}{a^{2} x - a}"," ",0,"(5*a*x + 6*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x - 5) - 5)/(a^2*x - a)","A",0
22,1,82,0,0.475397," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x,x, algorithm=""fricas"")","\frac{4 \, a x + 2 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(a x - 1\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 4 \, \sqrt{-a^{2} x^{2} + 1} - 4}{a x - 1}"," ",0,"(4*a*x + 2*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a*x - 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 4*sqrt(-a^2*x^2 + 1) - 4)/(a*x - 1)","A",0
23,1,78,0,0.428536," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^2,x, algorithm=""fricas"")","\frac{4 \, a^{2} x^{2} - 4 \, a x + 3 \, {\left(a^{2} x^{2} - a x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \sqrt{-a^{2} x^{2} + 1} {\left(5 \, a x - 1\right)}}{a x^{2} - x}"," ",0,"(4*a^2*x^2 - 4*a*x + 3*(a^2*x^2 - a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - sqrt(-a^2*x^2 + 1)*(5*a*x - 1))/(a*x^2 - x)","A",0
24,1,97,0,0.432174," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^3,x, algorithm=""fricas"")","\frac{8 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 9 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(14 \, a^{2} x^{2} - 5 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a x^{3} - x^{2}\right)}}"," ",0,"1/2*(8*a^3*x^3 - 8*a^2*x^2 + 9*(a^3*x^3 - a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (14*a^2*x^2 - 5*a*x - 1)*sqrt(-a^2*x^2 + 1))/(a*x^3 - x^2)","A",0
25,1,105,0,0.564050," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^4,x, algorithm=""fricas"")","\frac{24 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 33 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(52 \, a^{3} x^{3} - 19 \, a^{2} x^{2} - 7 \, a x - 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a x^{4} - x^{3}\right)}}"," ",0,"1/6*(24*a^4*x^4 - 24*a^3*x^3 + 33*(a^4*x^4 - a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (52*a^3*x^3 - 19*a^2*x^2 - 7*a*x - 2)*sqrt(-a^2*x^2 + 1))/(a*x^4 - x^3)","A",0
26,1,66,0,0.437466," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*x^3,x, algorithm=""fricas"")","\frac{3 \, a^{5} x^{5} + 13 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 96 \, a^{2} x^{2} - 144 \, a x + 192 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 48}{12 \, {\left(a^{5} x - a^{4}\right)}}"," ",0,"1/12*(3*a^5*x^5 + 13*a^4*x^4 + 32*a^3*x^3 + 96*a^2*x^2 - 144*a*x + 192*(a*x - 1)*log(a*x - 1) - 48)/(a^5*x - a^4)","A",0
27,1,57,0,0.410228," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*x^2,x, algorithm=""fricas"")","\frac{a^{4} x^{4} + 5 \, a^{3} x^{3} + 18 \, a^{2} x^{2} - 24 \, a x + 36 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 12}{3 \, {\left(a^{4} x - a^{3}\right)}}"," ",0,"1/3*(a^4*x^4 + 5*a^3*x^3 + 18*a^2*x^2 - 24*a*x + 36*(a*x - 1)*log(a*x - 1) - 12)/(a^4*x - a^3)","A",0
28,1,49,0,0.454879," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*x,x, algorithm=""fricas"")","\frac{a^{3} x^{3} + 7 \, a^{2} x^{2} - 8 \, a x + 16 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 8}{2 \, {\left(a^{3} x - a^{2}\right)}}"," ",0,"1/2*(a^3*x^3 + 7*a^2*x^2 - 8*a*x + 16*(a*x - 1)*log(a*x - 1) - 8)/(a^3*x - a^2)","A",0
29,1,38,0,0.434178," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2,x, algorithm=""fricas"")","\frac{a^{2} x^{2} - a x + 4 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 4}{a^{2} x - a}"," ",0,"(a^2*x^2 - a*x + 4*(a*x - 1)*log(a*x - 1) - 4)/(a^2*x - a)","A",0
30,1,18,0,0.440422," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/x,x, algorithm=""fricas"")","\frac{{\left(a x - 1\right)} \log\left(x\right) - 4}{a x - 1}"," ",0,"((a*x - 1)*log(x) - 4)/(a*x - 1)","A",0
31,1,55,0,0.402959," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/x^2,x, algorithm=""fricas"")","-\frac{5 \, a x + 4 \, {\left(a^{2} x^{2} - a x\right)} \log\left(a x - 1\right) - 4 \, {\left(a^{2} x^{2} - a x\right)} \log\left(x\right) - 1}{a x^{2} - x}"," ",0,"-(5*a*x + 4*(a^2*x^2 - a*x)*log(a*x - 1) - 4*(a^2*x^2 - a*x)*log(x) - 1)/(a*x^2 - x)","A",0
32,1,73,0,0.547372," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/x^3,x, algorithm=""fricas"")","-\frac{16 \, a^{2} x^{2} - 7 \, a x + 16 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(a x - 1\right) - 16 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(x\right) - 1}{2 \, {\left(a x^{3} - x^{2}\right)}}"," ",0,"-1/2*(16*a^2*x^2 - 7*a*x + 16*(a^3*x^3 - a^2*x^2)*log(a*x - 1) - 16*(a^3*x^3 - a^2*x^2)*log(x) - 1)/(a*x^3 - x^2)","A",0
33,1,81,0,0.448246," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/x^4,x, algorithm=""fricas"")","-\frac{36 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 5 \, a x + 36 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(a x - 1\right) - 36 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(x\right) - 1}{3 \, {\left(a x^{4} - x^{3}\right)}}"," ",0,"-1/3*(36*a^3*x^3 - 18*a^2*x^2 - 5*a*x + 36*(a^4*x^4 - a^3*x^3)*log(a*x - 1) - 36*(a^4*x^4 - a^3*x^3)*log(x) - 1)/(a*x^4 - x^3)","A",0
34,1,65,0,0.455575," ","integrate(x^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{{\left(6 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 9 \, a x - 16\right)} \sqrt{-a^{2} x^{2} + 1} + 18 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{24 \, a^{4}}"," ",0,"1/24*((6*a^3*x^3 - 8*a^2*x^2 + 9*a*x - 16)*sqrt(-a^2*x^2 + 1) + 18*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a^4","A",0
35,1,57,0,0.432605," ","integrate(x^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{{\left(2 \, a^{2} x^{2} - 3 \, a x + 4\right)} \sqrt{-a^{2} x^{2} + 1} - 6 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{6 \, a^{3}}"," ",0,"1/6*((2*a^2*x^2 - 3*a*x + 4)*sqrt(-a^2*x^2 + 1) - 6*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a^3","A",0
36,1,48,0,0.784675," ","integrate(x/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x - 2\right)} + 2 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{2 \, a^{2}}"," ",0,"1/2*(sqrt(-a^2*x^2 + 1)*(a*x - 2) + 2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a^2","A",0
37,1,41,0,0.572957," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} x^{2} + 1} - 2 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{a}"," ",0,"(sqrt(-a^2*x^2 + 1) - 2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a","A",0
38,1,44,0,0.489576," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x,x, algorithm=""fricas"")","2 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right)"," ",0,"2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + log((sqrt(-a^2*x^2 + 1) - 1)/x)","A",0
39,1,40,0,0.467715," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x, algorithm=""fricas"")","-\frac{a x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + \sqrt{-a^{2} x^{2} + 1}}{x}"," ",0,"-(a*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1))/x","A",0
40,1,51,0,0.625017," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^3,x, algorithm=""fricas"")","\frac{a^{2} x^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(2 \, a x - 1\right)}}{2 \, x^{2}}"," ",0,"1/2*(a^2*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*(2*a*x - 1))/x^2","A",0
41,1,60,0,0.515417," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^4,x, algorithm=""fricas"")","-\frac{3 \, a^{3} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(4 \, a^{2} x^{2} - 3 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, x^{3}}"," ",0,"-1/6*(3*a^3*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (4*a^2*x^2 - 3*a*x + 2)*sqrt(-a^2*x^2 + 1))/x^3","A",0
42,1,68,0,0.513459," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^5,x, algorithm=""fricas"")","\frac{9 \, a^{4} x^{4} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(16 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 8 \, a x - 6\right)} \sqrt{-a^{2} x^{2} + 1}}{24 \, x^{4}}"," ",0,"1/24*(9*a^4*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (16*a^3*x^3 - 9*a^2*x^2 + 8*a*x - 6)*sqrt(-a^2*x^2 + 1))/x^4","A",0
43,1,42,0,0.724395," ","integrate(x^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{3 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 24 \, a x + 24 \, \log\left(a x + 1\right)}{12 \, a^{4}}"," ",0,"-1/12*(3*a^4*x^4 - 8*a^3*x^3 + 12*a^2*x^2 - 24*a*x + 24*log(a*x + 1))/a^4","A",0
44,1,33,0,0.478204," ","integrate(x^2/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 6 \, a x - 6 \, \log\left(a x + 1\right)}{3 \, a^{3}}"," ",0,"-1/3*(a^3*x^3 - 3*a^2*x^2 + 6*a*x - 6*log(a*x + 1))/a^3","A",0
45,1,25,0,0.463173," ","integrate(x/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{a^{2} x^{2} - 4 \, a x + 4 \, \log\left(a x + 1\right)}{2 \, a^{2}}"," ",0,"-1/2*(a^2*x^2 - 4*a*x + 4*log(a*x + 1))/a^2","A",0
46,1,17,0,0.492850," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{a x - 2 \, \log\left(a x + 1\right)}{a}"," ",0,"-(a*x - 2*log(a*x + 1))/a","A",0
47,1,11,0,0.415637," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/x,x, algorithm=""fricas"")","-2 \, \log\left(a x + 1\right) + \log\left(x\right)"," ",0,"-2*log(a*x + 1) + log(x)","A",0
48,1,22,0,0.417880," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/x^2,x, algorithm=""fricas"")","\frac{2 \, a x \log\left(a x + 1\right) - 2 \, a x \log\left(x\right) - 1}{x}"," ",0,"(2*a*x*log(a*x + 1) - 2*a*x*log(x) - 1)/x","A",0
49,1,35,0,0.420198," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/x^3,x, algorithm=""fricas"")","-\frac{4 \, a^{2} x^{2} \log\left(a x + 1\right) - 4 \, a^{2} x^{2} \log\left(x\right) - 4 \, a x + 1}{2 \, x^{2}}"," ",0,"-1/2*(4*a^2*x^2*log(a*x + 1) - 4*a^2*x^2*log(x) - 4*a*x + 1)/x^2","A",0
50,1,43,0,0.516983," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm=""fricas"")","\frac{6 \, a^{3} x^{3} \log\left(a x + 1\right) - 6 \, a^{3} x^{3} \log\left(x\right) - 6 \, a^{2} x^{2} + 3 \, a x - 1}{3 \, x^{3}}"," ",0,"1/3*(6*a^3*x^3*log(a*x + 1) - 6*a^3*x^3*log(x) - 6*a^2*x^2 + 3*a*x - 1)/x^3","A",0
51,1,92,0,0.490015," ","integrate(x^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\frac{80 \, a x - 102 \, {\left(a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(2 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 11 \, a^{2} x^{2} - 29 \, a x - 80\right)} \sqrt{-a^{2} x^{2} + 1} + 80}{8 \, {\left(a^{5} x + a^{4}\right)}}"," ",0,"1/8*(80*a*x - 102*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^4*x^4 - 6*a^3*x^3 + 11*a^2*x^2 - 29*a*x - 80)*sqrt(-a^2*x^2 + 1) + 80)/(a^5*x + a^4)","A",0
52,1,83,0,0.443098," ","integrate(x^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{52 \, a x - 66 \, {\left(a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 19 \, a x + 52\right)} \sqrt{-a^{2} x^{2} + 1} + 52}{6 \, {\left(a^{4} x + a^{3}\right)}}"," ",0,"-1/6*(52*a*x - 66*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*x^3 - 7*a^2*x^2 + 19*a*x + 52)*sqrt(-a^2*x^2 + 1) + 52)/(a^4*x + a^3)","A",0
53,1,75,0,0.557953," ","integrate(x/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\frac{14 \, a x - 18 \, {\left(a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(a^{2} x^{2} - 5 \, a x - 14\right)} \sqrt{-a^{2} x^{2} + 1} + 14}{2 \, {\left(a^{3} x + a^{2}\right)}}"," ",0,"1/2*(14*a*x - 18*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (a^2*x^2 - 5*a*x - 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^3*x + a^2)","A",0
54,1,64,0,0.404601," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{5 \, a x - 6 \, {\left(a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a x + 5\right)} + 5}{a^{2} x + a}"," ",0,"-(5*a*x - 6*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x + 5) + 5)/(a^2*x + a)","A",0
55,1,82,0,0.557649," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x,x, algorithm=""fricas"")","\frac{4 \, a x - 2 \, {\left(a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(a x + 1\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} + 4}{a x + 1}"," ",0,"(4*a*x - 2*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a*x + 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 4*sqrt(-a^2*x^2 + 1) + 4)/(a*x + 1)","A",0
56,1,75,0,0.425583," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm=""fricas"")","-\frac{4 \, a^{2} x^{2} + 4 \, a x + 3 \, {\left(a^{2} x^{2} + a x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(5 \, a x + 1\right)}}{a x^{2} + x}"," ",0,"-(4*a^2*x^2 + 4*a*x + 3*(a^2*x^2 + a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*(5*a*x + 1))/(a*x^2 + x)","A",0
57,1,93,0,0.474698," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^3,x, algorithm=""fricas"")","\frac{8 \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 9 \, {\left(a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(14 \, a^{2} x^{2} + 5 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a x^{3} + x^{2}\right)}}"," ",0,"1/2*(8*a^3*x^3 + 8*a^2*x^2 + 9*(a^3*x^3 + a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (14*a^2*x^2 + 5*a*x - 1)*sqrt(-a^2*x^2 + 1))/(a*x^3 + x^2)","A",0
58,1,101,0,0.468488," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^4,x, algorithm=""fricas"")","-\frac{24 \, a^{4} x^{4} + 24 \, a^{3} x^{3} + 33 \, {\left(a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(52 \, a^{3} x^{3} + 19 \, a^{2} x^{2} - 7 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a x^{4} + x^{3}\right)}}"," ",0,"-1/6*(24*a^4*x^4 + 24*a^3*x^3 + 33*(a^4*x^4 + a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (52*a^3*x^3 + 19*a^2*x^2 - 7*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a*x^4 + x^3)","A",0
59,1,109,0,0.509858," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm=""fricas"")","\frac{32 \, a^{5} x^{5} + 32 \, a^{4} x^{4} + 51 \, {\left(a^{5} x^{5} + a^{4} x^{4}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(80 \, a^{4} x^{4} + 29 \, a^{3} x^{3} - 11 \, a^{2} x^{2} + 6 \, a x - 2\right)} \sqrt{-a^{2} x^{2} + 1}}{8 \, {\left(a x^{5} + x^{4}\right)}}"," ",0,"1/8*(32*a^5*x^5 + 32*a^4*x^4 + 51*(a^5*x^5 + a^4*x^4)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (80*a^4*x^4 + 29*a^3*x^3 - 11*a^2*x^2 + 6*a*x - 2)*sqrt(-a^2*x^2 + 1))/(a*x^5 + x^4)","A",0
60,0,0,0,0.622793," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x^m,x, algorithm=""fricas"")","{\rm integral}\left(x^{m} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}, x\right)"," ",0,"integral(x^m*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)), x)","F",0
61,1,544,0,0.677209," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x^2,x, algorithm=""fricas"")","-\frac{36 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{9} \sqrt{\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - \sqrt{2} a^{9} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - 1\right) + 36 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{9} \sqrt{-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - \sqrt{2} a^{9} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} + 1\right) - 9 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 9 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(8 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 11\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{96 \, a^{3}}"," ",0,"-1/96*(36*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*a^9*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(3/4) - sqrt(2)*a^9*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) - 1) + 36*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*a^9*sqrt(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(3/4) - sqrt(2)*a^9*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) + 1) - 9*sqrt(2)*a^3*(a^(-12))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 9*sqrt(2)*a^3*(a^(-12))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(8*a^3*x^3 + 2*a^2*x^2 + a*x - 11)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^3","B",0
62,1,535,0,0.499285," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x,x, algorithm=""fricas"")","-\frac{4 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{6} \sqrt{\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - \sqrt{2} a^{6} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - 1\right) + 4 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{6} \sqrt{-\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - \sqrt{2} a^{6} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + 1\right) - \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(2 \, a^{2} x^{2} + a x - 3\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, a^{2}}"," ",0,"-1/16*(4*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^6*sqrt((sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(3/4) - sqrt(2)*a^6*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) - 1) + 4*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^6*sqrt(-(sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(3/4) - sqrt(2)*a^6*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) + 1) - sqrt(2)*a^2*(a^(-8))^(1/4)*log((sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + sqrt(2)*a^2*(a^(-8))^(1/4)*log(-(sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(2*a^2*x^2 + a*x - 3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^2","B",0
63,1,511,0,0.627909," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x, algorithm=""fricas"")","-\frac{4 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - 1\right) + 4 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{-\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + 1\right) - \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{4 \, a}"," ",0,"-1/4*(4*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a^3*sqrt((sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(3/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - 1) + 4*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a^3*sqrt(-(sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(3/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + 1) - sqrt(2)*a*(a^(-4))^(1/4)*log((sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + sqrt(2)*a*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a","B",0
64,1,417,0,0.564460," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x,x, algorithm=""fricas"")","-2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\frac{a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\frac{a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + \frac{1}{2} \, \sqrt{2} \log\left(\frac{4 \, {\left(a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) - \frac{1}{2} \, \sqrt{2} \log\left(\frac{4 \, {\left(a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) - 2 \, \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right)"," ",0,"-2*sqrt(2)*arctan(sqrt(2)*sqrt((a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*sqrt(2)*arctan(sqrt(2)*sqrt((a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 1/2*sqrt(2)*log(4*(a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - 1/2*sqrt(2)*log(4*(a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - 2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1)","B",0
65,1,123,0,0.507722," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2,x, algorithm=""fricas"")","-\frac{2 \, a x \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + a x \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - a x \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, x}"," ",0,"-1/2*(2*a*x*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x","B",0
66,1,144,0,0.513571," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^3,x, algorithm=""fricas"")","-\frac{2 \, a^{2} x^{2} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + a^{2} x^{2} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - a^{2} x^{2} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(3 \, a^{2} x^{2} - a x - 2\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{2}}"," ",0,"-1/8*(2*a^2*x^2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(3*a^2*x^2 - a*x - 2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^2","A",0
67,1,153,0,0.560712," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^4,x, algorithm=""fricas"")","-\frac{18 \, a^{3} x^{3} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 9 \, a^{3} x^{3} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 9 \, a^{3} x^{3} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(11 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x - 8\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{3}}"," ",0,"-1/48*(18*a^3*x^3*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 9*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 9*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(11*a^3*x^3 - a^2*x^2 - 2*a*x - 8)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^3","A",0
68,1,161,0,0.482401," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^5,x, algorithm=""fricas"")","-\frac{66 \, a^{4} x^{4} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 33 \, a^{4} x^{4} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 33 \, a^{4} x^{4} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(83 \, a^{4} x^{4} - 25 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 8 \, a x - 48\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{384 \, x^{4}}"," ",0,"-1/384*(66*a^4*x^4*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 33*a^4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 33*a^4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(83*a^4*x^4 - 25*a^3*x^3 - 2*a^2*x^2 - 8*a*x - 48)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^4","A",0
69,1,169,0,0.611625," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^6,x, algorithm=""fricas"")","-\frac{930 \, a^{5} x^{5} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 465 \, a^{5} x^{5} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 465 \, a^{5} x^{5} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(611 \, a^{5} x^{5} - 73 \, a^{4} x^{4} - 98 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 48 \, a x - 384\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{3840 \, x^{5}}"," ",0,"-1/3840*(930*a^5*x^5*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 465*a^5*x^5*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 465*a^5*x^5*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(611*a^5*x^5 - 73*a^4*x^4 - 98*a^3*x^3 - 8*a^2*x^2 - 48*a*x - 384)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^5","A",0
70,0,0,0,0.600316," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)*x^m,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} x^{m} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*x^m*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))/(a*x - 1), x)","F",0
71,1,557,0,0.507362," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)*x^3,x, algorithm=""fricas"")","-\frac{492 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{4} \sqrt{\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} + {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - \sqrt{2} a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - 1\right) + 492 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{4} \sqrt{-\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - \sqrt{2} a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} + 1\right) + 123 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} + {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 123 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, {\left(16 \, a^{3} x^{3} + 24 \, a^{2} x^{2} + 30 \, a x + 63\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{256 \, a^{4}}"," ",0,"-1/256*(492*sqrt(2)*a^4*(a^(-16))^(1/4)*arctan(sqrt(2)*a^4*sqrt((sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(1/4) - sqrt(2)*a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) - 1) + 492*sqrt(2)*a^4*(a^(-16))^(1/4)*arctan(sqrt(2)*a^4*sqrt(-(sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) - (a^9*x - a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(1/4) - sqrt(2)*a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) + 1) + 123*sqrt(2)*a^4*(a^(-16))^(1/4)*log((sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 123*sqrt(2)*a^4*(a^(-16))^(1/4)*log(-(sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) - (a^9*x - a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*(16*a^3*x^3 + 24*a^2*x^2 + 30*a*x + 63)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^4","B",0
72,1,549,0,0.598194," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)*x^2,x, algorithm=""fricas"")","-\frac{204 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - 1\right) + 204 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{-\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} + 1\right) + 51 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 51 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, {\left(8 \, a^{2} x^{2} + 14 \, a x + 23\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{96 \, a^{3}}"," ",0,"-1/96*(204*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*a^3*sqrt((sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(1/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) - 1) + 204*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*a^3*sqrt(-(sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(1/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) + 1) + 51*sqrt(2)*a^3*(a^(-12))^(1/4)*log((sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 51*sqrt(2)*a^3*(a^(-12))^(1/4)*log(-(sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*(8*a^2*x^2 + 14*a*x + 23)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^3","B",0
73,1,541,0,0.532207," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)*x,x, algorithm=""fricas"")","-\frac{36 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{2} \sqrt{\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - 1\right) + 36 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{2} \sqrt{-\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} + 1\right) + 9 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 9 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} {\left(2 \, a x + 5\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, a^{2}}"," ",0,"-1/16*(36*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^2*sqrt((sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(1/4) - sqrt(2)*a^2*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - 1) + 36*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^2*sqrt(-(sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(1/4) - sqrt(2)*a^2*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) + 1) + 9*sqrt(2)*a^2*(a^(-8))^(1/4)*log((sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 9*sqrt(2)*a^2*(a^(-8))^(1/4)*log(-(sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*(2*a*x + 5)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^2","B",0
74,1,519,0,0.469336," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x, algorithm=""fricas"")","-\frac{12 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - 1\right) + 12 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + 1\right) + 3 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 3 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{4 \, a}"," ",0,"-1/4*(12*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - 1) + 12*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + 1) + 3*sqrt(2)*a*(a^(-4))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 3*sqrt(2)*a*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a","B",0
75,1,417,0,0.520703," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x,x, algorithm=""fricas"")","-2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\frac{a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\frac{a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - \frac{1}{2} \, \sqrt{2} \log\left(\frac{4 \, {\left(a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) + \frac{1}{2} \, \sqrt{2} \log\left(\frac{4 \, {\left(a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) + 2 \, \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right)"," ",0,"-2*sqrt(2)*arctan(sqrt(2)*sqrt((a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*sqrt(2)*arctan(sqrt(2)*sqrt((a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 1/2*sqrt(2)*log(4*(a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) + 1/2*sqrt(2)*log(4*(a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) + 2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1)","B",0
76,1,131,0,0.537208," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^2,x, algorithm=""fricas"")","\frac{6 \, a x \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 3 \, a x \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + 3 \, a x \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, x}"," ",0,"1/2*(6*a*x*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 3*a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 3*a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x","B",0
77,1,149,0,0.640522," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^3,x, algorithm=""fricas"")","\frac{18 \, a^{2} x^{2} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 9 \, a^{2} x^{2} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + 9 \, a^{2} x^{2} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} {\left(5 \, a x + 2\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{2}}"," ",0,"1/8*(18*a^2*x^2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 9*a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 9*a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*sqrt(-a^2*x^2 + 1)*(5*a*x + 2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^2","A",0
78,1,157,0,0.676831," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x, algorithm=""fricas"")","\frac{102 \, a^{3} x^{3} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 51 \, a^{3} x^{3} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + 51 \, a^{3} x^{3} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(23 \, a^{2} x^{2} + 14 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{3}}"," ",0,"1/48*(102*a^3*x^3*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 51*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 51*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(23*a^2*x^2 + 14*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^3","A",0
79,1,165,0,0.569569," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^5,x, algorithm=""fricas"")","\frac{246 \, a^{4} x^{4} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 123 \, a^{4} x^{4} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + 123 \, a^{4} x^{4} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(63 \, a^{3} x^{3} + 30 \, a^{2} x^{2} + 24 \, a x + 16\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{128 \, x^{4}}"," ",0,"1/128*(246*a^4*x^4*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 123*a^4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 123*a^4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(63*a^3*x^3 + 30*a^2*x^2 + 24*a*x + 16)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^4","A",0
80,0,0,0,0.577765," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)*x^m,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x + 1\right)} x^{m} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{a x - 1}, x\right)"," ",0,"integral(-(a*x + 1)*x^m*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))/(a*x - 1), x)","F",0
81,1,553,0,0.608954," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)*x^3,x, algorithm=""fricas"")","\frac{5700 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{12} \sqrt{\frac{\sqrt{2} {\left(a^{5} x - a^{4}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} + {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - \sqrt{2} a^{12} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - 1\right) + 5700 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{12} \sqrt{-\frac{\sqrt{2} {\left(a^{5} x - a^{4}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - \sqrt{2} a^{12} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} + 1\right) - 1425 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{5} x - a^{4}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} + {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 1425 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{5} x - a^{4}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(48 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 226 \, a^{2} x^{2} + 521 \, a x - 2467\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{768 \, a^{4}}"," ",0,"1/768*(5700*sqrt(2)*a^4*(a^(-16))^(1/4)*arctan(sqrt(2)*a^12*sqrt((sqrt(2)*(a^5*x - a^4)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(3/4) - sqrt(2)*a^12*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) - 1) + 5700*sqrt(2)*a^4*(a^(-16))^(1/4)*arctan(sqrt(2)*a^12*sqrt(-(sqrt(2)*(a^5*x - a^4)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) - (a^9*x - a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(3/4) - sqrt(2)*a^12*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) + 1) - 1425*sqrt(2)*a^4*(a^(-16))^(1/4)*log((sqrt(2)*(a^5*x - a^4)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 1425*sqrt(2)*a^4*(a^(-16))^(1/4)*log(-(sqrt(2)*(a^5*x - a^4)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) - (a^9*x - a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(48*a^4*x^4 + 136*a^3*x^3 + 226*a^2*x^2 + 521*a*x - 2467)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^4","B",0
82,1,545,0,0.567068," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)*x^2,x, algorithm=""fricas"")","\frac{660 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{9} \sqrt{\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - \sqrt{2} a^{9} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - 1\right) + 660 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{9} \sqrt{-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - \sqrt{2} a^{9} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} + 1\right) - 165 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 165 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(8 \, a^{3} x^{3} + 26 \, a^{2} x^{2} + 61 \, a x - 287\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{96 \, a^{3}}"," ",0,"1/96*(660*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*a^9*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(3/4) - sqrt(2)*a^9*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) - 1) + 660*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*a^9*sqrt(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(3/4) - sqrt(2)*a^9*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) + 1) - 165*sqrt(2)*a^3*(a^(-12))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 165*sqrt(2)*a^3*(a^(-12))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(8*a^3*x^3 + 26*a^2*x^2 + 61*a*x - 287)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^3","B",0
83,1,537,0,0.534400," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)*x,x, algorithm=""fricas"")","\frac{100 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{6} \sqrt{\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - \sqrt{2} a^{6} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - 1\right) + 100 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{6} \sqrt{-\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - \sqrt{2} a^{6} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + 1\right) - 25 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 25 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(2 \, a^{2} x^{2} + 9 \, a x - 43\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, a^{2}}"," ",0,"1/16*(100*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^6*sqrt((sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(3/4) - sqrt(2)*a^6*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) - 1) + 100*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^6*sqrt(-(sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(3/4) - sqrt(2)*a^6*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) + 1) - 25*sqrt(2)*a^2*(a^(-8))^(1/4)*log((sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 25*sqrt(2)*a^2*(a^(-8))^(1/4)*log(-(sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(2*a^2*x^2 + 9*a*x - 43)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^2","B",0
84,1,512,0,0.848311," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2),x, algorithm=""fricas"")","\frac{20 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - 1\right) + 20 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{-\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + 1\right) - 5 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 5 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(a x - 9\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{4 \, a}"," ",0,"1/4*(20*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a^3*sqrt((sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(3/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - 1) + 20*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a^3*sqrt(-(sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(3/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + 1) - 5*sqrt(2)*a*(a^(-4))^(1/4)*log((sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 5*sqrt(2)*a*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(a*x - 9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a","B",0
85,1,442,0,0.759214," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)/x,x, algorithm=""fricas"")","2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\frac{a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\frac{a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - \frac{1}{2} \, \sqrt{2} \log\left(\frac{4 \, {\left(a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) + \frac{1}{2} \, \sqrt{2} \log\left(\frac{4 \, {\left(a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) + 8 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 2 \, \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right)"," ",0,"2*sqrt(2)*arctan(sqrt(2)*sqrt((a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*sqrt(2)*arctan(sqrt(2)*sqrt((a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 1/2*sqrt(2)*log(4*(a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) + 1/2*sqrt(2)*log(4*(a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) + 8*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1)","B",0
86,1,125,0,0.431482," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)/x^2,x, algorithm=""fricas"")","-\frac{10 \, a x \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 5 \, a x \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 5 \, a x \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(9 \, a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, x}"," ",0,"-1/2*(10*a*x*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 5*a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 5*a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(9*a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x","A",0
87,1,145,0,0.568861," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)/x^3,x, algorithm=""fricas"")","-\frac{50 \, a^{2} x^{2} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 25 \, a^{2} x^{2} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 25 \, a^{2} x^{2} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(43 \, a^{2} x^{2} - 9 \, a x - 2\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{2}}"," ",0,"-1/8*(50*a^2*x^2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 25*a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 25*a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(43*a^2*x^2 - 9*a*x - 2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^2","A",0
88,1,153,0,0.615640," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)/x^4,x, algorithm=""fricas"")","-\frac{330 \, a^{3} x^{3} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 165 \, a^{3} x^{3} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 165 \, a^{3} x^{3} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(287 \, a^{3} x^{3} - 61 \, a^{2} x^{2} - 26 \, a x - 8\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{3}}"," ",0,"-1/48*(330*a^3*x^3*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 165*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 165*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(287*a^3*x^3 - 61*a^2*x^2 - 26*a*x - 8)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^3","A",0
89,1,161,0,0.655592," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)/x^5,x, algorithm=""fricas"")","-\frac{2850 \, a^{4} x^{4} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 1425 \, a^{4} x^{4} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 1425 \, a^{4} x^{4} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 2 \, {\left(2467 \, a^{4} x^{4} - 521 \, a^{3} x^{3} - 226 \, a^{2} x^{2} - 136 \, a x - 48\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{384 \, x^{4}}"," ",0,"-1/384*(2850*a^4*x^4*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 1425*a^4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 1425*a^4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(2467*a^4*x^4 - 521*a^3*x^3 - 226*a^2*x^2 - 136*a*x - 48)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^4","A",0
90,0,0,0,0.606087," ","integrate(x^m/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} x^{m} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*x^m*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))/(a*x + 1), x)","F",0
91,1,557,0,0.595308," ","integrate(x^3/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x, algorithm=""fricas"")","\frac{132 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{4} \sqrt{\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} + {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - \sqrt{2} a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - 1\right) + 132 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{4} \sqrt{-\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - \sqrt{2} a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} + 1\right) + 33 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} + {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 33 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, {\left(48 \, a^{3} x^{3} - 56 \, a^{2} x^{2} + 58 \, a x - 83\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{768 \, a^{4}}"," ",0,"1/768*(132*sqrt(2)*a^4*(a^(-16))^(1/4)*arctan(sqrt(2)*a^4*sqrt((sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(1/4) - sqrt(2)*a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) - 1) + 132*sqrt(2)*a^4*(a^(-16))^(1/4)*arctan(sqrt(2)*a^4*sqrt(-(sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) - (a^9*x - a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(1/4) - sqrt(2)*a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) + 1) + 33*sqrt(2)*a^4*(a^(-16))^(1/4)*log((sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 33*sqrt(2)*a^4*(a^(-16))^(1/4)*log(-(sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) - (a^9*x - a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*(48*a^3*x^3 - 56*a^2*x^2 + 58*a*x - 83)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^4","B",0
92,1,549,0,0.557863," ","integrate(x^2/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x, algorithm=""fricas"")","-\frac{36 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - 1\right) + 36 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{-\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} + 1\right) + 9 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 9 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(8 \, a^{2} x^{2} - 10 \, a x + 11\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{96 \, a^{3}}"," ",0,"-1/96*(36*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*a^3*sqrt((sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(1/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) - 1) + 36*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*a^3*sqrt(-(sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(1/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) + 1) + 9*sqrt(2)*a^3*(a^(-12))^(1/4)*log((sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 9*sqrt(2)*a^3*(a^(-12))^(1/4)*log(-(sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(8*a^2*x^2 - 10*a*x + 11)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^3","B",0
93,1,540,0,0.538384," ","integrate(x/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x, algorithm=""fricas"")","\frac{4 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{2} \sqrt{\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - 1\right) + 4 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{2} \sqrt{-\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} + 1\right) + \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} {\left(2 \, a x - 3\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, a^{2}}"," ",0,"1/16*(4*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^2*sqrt((sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(1/4) - sqrt(2)*a^2*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - 1) + 4*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^2*sqrt(-(sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(1/4) - sqrt(2)*a^2*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) + 1) + sqrt(2)*a^2*(a^(-8))^(1/4)*log((sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - sqrt(2)*a^2*(a^(-8))^(1/4)*log(-(sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*(2*a*x - 3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^2","B",0
94,1,518,0,1.103065," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x, algorithm=""fricas"")","-\frac{4 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - 1\right) + 4 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + 1\right) + \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{4 \, a}"," ",0,"-1/4*(4*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - 1) + 4*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + 1) + sqrt(2)*a*(a^(-4))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - sqrt(2)*a*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a","B",0
95,1,417,0,0.661410," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x,x, algorithm=""fricas"")","2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\frac{a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\frac{a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + \frac{1}{2} \, \sqrt{2} \log\left(\frac{4 \, {\left(a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) - \frac{1}{2} \, \sqrt{2} \log\left(\frac{4 \, {\left(a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) + 2 \, \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right)"," ",0,"2*sqrt(2)*arctan(sqrt(2)*sqrt((a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*sqrt(2)*arctan(sqrt(2)*sqrt((a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 1/2*sqrt(2)*log(4*(a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - 1/2*sqrt(2)*log(4*(a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) + 2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1)","B",0
96,1,130,0,0.507147," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2,x, algorithm=""fricas"")","-\frac{2 \, a x \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - a x \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + a x \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, x}"," ",0,"-1/2*(2*a*x*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x","B",0
97,1,148,0,0.515934," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^3,x, algorithm=""fricas"")","\frac{2 \, a^{2} x^{2} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - a^{2} x^{2} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + a^{2} x^{2} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} {\left(3 \, a x - 2\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{2}}"," ",0,"1/8*(2*a^2*x^2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*sqrt(-a^2*x^2 + 1)*(3*a*x - 2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^2","A",0
98,1,157,0,0.500612," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^4,x, algorithm=""fricas"")","-\frac{18 \, a^{3} x^{3} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 9 \, a^{3} x^{3} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + 9 \, a^{3} x^{3} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, {\left(11 \, a^{2} x^{2} - 10 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{3}}"," ",0,"-1/48*(18*a^3*x^3*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 9*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 9*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*(11*a^2*x^2 - 10*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^3","A",0
99,1,165,0,0.507522," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^5,x, algorithm=""fricas"")","\frac{66 \, a^{4} x^{4} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 33 \, a^{4} x^{4} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + 33 \, a^{4} x^{4} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, {\left(83 \, a^{3} x^{3} - 58 \, a^{2} x^{2} + 56 \, a x - 48\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{384 \, x^{4}}"," ",0,"1/384*(66*a^4*x^4*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 33*a^4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 33*a^4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*(83*a^3*x^3 - 58*a^2*x^2 + 56*a*x - 48)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^4","A",0
100,0,0,0,0.548032," ","integrate(x^m/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x - 1\right)} x^{m} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{a x + 1}, x\right)"," ",0,"integral(-(a*x - 1)*x^m*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))/(a*x + 1), x)","F",0
101,1,553,0,0.502797," ","integrate(x^3/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x, algorithm=""fricas"")","\frac{492 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{12} \sqrt{\frac{\sqrt{2} {\left(a^{5} x - a^{4}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} + {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - \sqrt{2} a^{12} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - 1\right) + 492 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{12} \sqrt{-\frac{\sqrt{2} {\left(a^{5} x - a^{4}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - \sqrt{2} a^{12} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} + 1\right) - 123 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{5} x - a^{4}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} + {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 123 \, \sqrt{2} a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{5} x - a^{4}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(16 \, a^{4} x^{4} - 40 \, a^{3} x^{3} + 54 \, a^{2} x^{2} - 93 \, a x + 63\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{256 \, a^{4}}"," ",0,"1/256*(492*sqrt(2)*a^4*(a^(-16))^(1/4)*arctan(sqrt(2)*a^12*sqrt((sqrt(2)*(a^5*x - a^4)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(3/4) - sqrt(2)*a^12*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) - 1) + 492*sqrt(2)*a^4*(a^(-16))^(1/4)*arctan(sqrt(2)*a^12*sqrt(-(sqrt(2)*(a^5*x - a^4)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) - (a^9*x - a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(3/4) - sqrt(2)*a^12*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) + 1) - 123*sqrt(2)*a^4*(a^(-16))^(1/4)*log((sqrt(2)*(a^5*x - a^4)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 123*sqrt(2)*a^4*(a^(-16))^(1/4)*log(-(sqrt(2)*(a^5*x - a^4)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) - (a^9*x - a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(16*a^4*x^4 - 40*a^3*x^3 + 54*a^2*x^2 - 93*a*x + 63)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^4","B",0
102,1,545,0,0.596607," ","integrate(x^2/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x, algorithm=""fricas"")","-\frac{204 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{9} \sqrt{\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - \sqrt{2} a^{9} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - 1\right) + 204 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{9} \sqrt{-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - \sqrt{2} a^{9} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} + 1\right) - 51 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 51 \, \sqrt{2} a^{3} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, {\left(8 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 37 \, a x - 23\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{96 \, a^{3}}"," ",0,"-1/96*(204*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*a^9*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(3/4) - sqrt(2)*a^9*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) - 1) + 204*sqrt(2)*a^3*(a^(-12))^(1/4)*arctan(sqrt(2)*a^9*sqrt(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(3/4) - sqrt(2)*a^9*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) + 1) - 51*sqrt(2)*a^3*(a^(-12))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 51*sqrt(2)*a^3*(a^(-12))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*(8*a^3*x^3 - 22*a^2*x^2 + 37*a*x - 23)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^3","B",0
103,1,537,0,1.181217," ","integrate(x/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x, algorithm=""fricas"")","\frac{36 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{6} \sqrt{\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - \sqrt{2} a^{6} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - 1\right) + 36 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{6} \sqrt{-\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - \sqrt{2} a^{6} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + 1\right) - 9 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 9 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{3} x - a^{2}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(2 \, a^{2} x^{2} - 7 \, a x + 5\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, a^{2}}"," ",0,"1/16*(36*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^6*sqrt((sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(3/4) - sqrt(2)*a^6*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) - 1) + 36*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^6*sqrt(-(sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(3/4) - sqrt(2)*a^6*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) + 1) - 9*sqrt(2)*a^2*(a^(-8))^(1/4)*log((sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 9*sqrt(2)*a^2*(a^(-8))^(1/4)*log(-(sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(2*a^2*x^2 - 7*a*x + 5)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^2","B",0
104,1,512,0,0.709131," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x, algorithm=""fricas"")","-\frac{12 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - 1\right) + 12 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{-\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + 1\right) - 3 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 3 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{2} x - a\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{4 \, a}"," ",0,"-1/4*(12*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a^3*sqrt((sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(3/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - 1) + 12*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a^3*sqrt(-(sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(3/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + 1) - 3*sqrt(2)*a*(a^(-4))^(1/4)*log((sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 3*sqrt(2)*a*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^2*x - a)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a","B",0
105,1,417,0,0.477901," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x,x, algorithm=""fricas"")","2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\frac{a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\frac{a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - \frac{1}{2} \, \sqrt{2} \log\left(\frac{4 \, {\left(a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) + \frac{1}{2} \, \sqrt{2} \log\left(\frac{4 \, {\left(a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) - 2 \, \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right)"," ",0,"2*sqrt(2)*arctan(sqrt(2)*sqrt((a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*sqrt(2)*arctan(sqrt(2)*sqrt((a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 1/2*sqrt(2)*log(4*(a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) + 1/2*sqrt(2)*log(4*(a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - 2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1)","B",0
106,1,124,0,0.514575," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^2,x, algorithm=""fricas"")","\frac{6 \, a x \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 3 \, a x \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 3 \, a x \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, x}"," ",0,"1/2*(6*a*x*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 3*a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 3*a*x*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x","B",0
107,1,145,0,0.516972," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^3,x, algorithm=""fricas"")","-\frac{18 \, a^{2} x^{2} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 9 \, a^{2} x^{2} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 9 \, a^{2} x^{2} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, {\left(5 \, a^{2} x^{2} - 7 \, a x + 2\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{8 \, x^{2}}"," ",0,"-1/8*(18*a^2*x^2*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 9*a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 9*a^2*x^2*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*(5*a^2*x^2 - 7*a*x + 2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^2","A",0
108,1,153,0,0.483714," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^4,x, algorithm=""fricas"")","\frac{102 \, a^{3} x^{3} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 51 \, a^{3} x^{3} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 51 \, a^{3} x^{3} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, {\left(23 \, a^{3} x^{3} - 37 \, a^{2} x^{2} + 22 \, a x - 8\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, x^{3}}"," ",0,"1/48*(102*a^3*x^3*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 51*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 51*a^3*x^3*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*(23*a^3*x^3 - 37*a^2*x^2 + 22*a*x - 8)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^3","A",0
109,1,161,0,0.500132," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2)/x^5,x, algorithm=""fricas"")","-\frac{246 \, a^{4} x^{4} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 123 \, a^{4} x^{4} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 123 \, a^{4} x^{4} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 2 \, {\left(63 \, a^{4} x^{4} - 93 \, a^{3} x^{3} + 54 \, a^{2} x^{2} - 40 \, a x + 16\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{128 \, x^{4}}"," ",0,"-1/128*(246*a^4*x^4*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 123*a^4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 123*a^4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 2*(63*a^4*x^4 - 93*a^3*x^3 + 54*a^2*x^2 - 40*a*x + 16)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^4","A",0
110,0,0,0,0.483774," ","integrate(x^m/((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)} x^{m} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{a^{2} x^{2} + 2 \, a x + 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(a*x - 1)*x^m*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))/(a^2*x^2 + 2*a*x + 1), x)","F",0
111,1,597,0,0.538677," ","integrate(x^3/((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2),x, algorithm=""fricas"")","-\frac{5700 \, \sqrt{2} {\left(a^{5} x + a^{4}\right)} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{4} \sqrt{\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} + {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - \sqrt{2} a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - 1\right) + 5700 \, \sqrt{2} {\left(a^{5} x + a^{4}\right)} \frac{1}{a^{16}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{4} \sqrt{-\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} - \sqrt{2} a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{1}{4}} + 1\right) + 1425 \, \sqrt{2} {\left(a^{5} x + a^{4}\right)} \frac{1}{a^{16}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} + {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 1425 \, \sqrt{2} {\left(a^{5} x + a^{4}\right)} \frac{1}{a^{16}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{13} x - a^{12}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{16}}^{\frac{3}{4}} - {\left(a^{9} x - a^{8}\right)} \sqrt{\frac{1}{a^{16}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, {\left(48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 226 \, a^{2} x^{2} - 521 \, a x - 2467\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{768 \, {\left(a^{5} x + a^{4}\right)}}"," ",0,"-1/768*(5700*sqrt(2)*(a^5*x + a^4)*(a^(-16))^(1/4)*arctan(sqrt(2)*a^4*sqrt((sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(1/4) - sqrt(2)*a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) - 1) + 5700*sqrt(2)*(a^5*x + a^4)*(a^(-16))^(1/4)*arctan(sqrt(2)*a^4*sqrt(-(sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) - (a^9*x - a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(1/4) - sqrt(2)*a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) + 1) + 1425*sqrt(2)*(a^5*x + a^4)*(a^(-16))^(1/4)*log((sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 1425*sqrt(2)*(a^5*x + a^4)*(a^(-16))^(1/4)*log(-(sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) - (a^9*x - a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*(48*a^4*x^4 - 136*a^3*x^3 + 226*a^2*x^2 - 521*a*x - 2467)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/(a^5*x + a^4)","B",0
112,1,589,0,0.513845," ","integrate(x^2/((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2),x, algorithm=""fricas"")","\frac{660 \, \sqrt{2} {\left(a^{4} x + a^{3}\right)} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - 1\right) + 660 \, \sqrt{2} {\left(a^{4} x + a^{3}\right)} \frac{1}{a^{12}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{3} \sqrt{-\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} - \sqrt{2} a^{3} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{1}{4}} + 1\right) + 165 \, \sqrt{2} {\left(a^{4} x + a^{3}\right)} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} + {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 165 \, \sqrt{2} {\left(a^{4} x + a^{3}\right)} \frac{1}{a^{12}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{10} x - a^{9}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{12}}^{\frac{3}{4}} - {\left(a^{7} x - a^{6}\right)} \sqrt{\frac{1}{a^{12}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 61 \, a x + 287\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{96 \, {\left(a^{4} x + a^{3}\right)}}"," ",0,"1/96*(660*sqrt(2)*(a^4*x + a^3)*(a^(-12))^(1/4)*arctan(sqrt(2)*a^3*sqrt((sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(1/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) - 1) + 660*sqrt(2)*(a^4*x + a^3)*(a^(-12))^(1/4)*arctan(sqrt(2)*a^3*sqrt(-(sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-12))^(1/4) - sqrt(2)*a^3*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(1/4) + 1) + 165*sqrt(2)*(a^4*x + a^3)*(a^(-12))^(1/4)*log((sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) + (a^7*x - a^6)*sqrt(a^(-12)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 165*sqrt(2)*(a^4*x + a^3)*(a^(-12))^(1/4)*log(-(sqrt(2)*(a^10*x - a^9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-12))^(3/4) - (a^7*x - a^6)*sqrt(a^(-12)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(8*a^3*x^3 - 26*a^2*x^2 + 61*a*x + 287)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/(a^4*x + a^3)","B",0
113,1,581,0,0.648320," ","integrate(x/((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2),x, algorithm=""fricas"")","-\frac{100 \, \sqrt{2} {\left(a^{3} x + a^{2}\right)} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{2} \sqrt{\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - 1\right) + 100 \, \sqrt{2} {\left(a^{3} x + a^{2}\right)} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a^{2} \sqrt{-\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} + 1\right) + 25 \, \sqrt{2} {\left(a^{3} x + a^{2}\right)} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 25 \, \sqrt{2} {\left(a^{3} x + a^{2}\right)} \frac{1}{a^{8}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{7} x - a^{6}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - {\left(a^{5} x - a^{4}\right)} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, {\left(2 \, a^{2} x^{2} - 9 \, a x - 43\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, {\left(a^{3} x + a^{2}\right)}}"," ",0,"-1/16*(100*sqrt(2)*(a^3*x + a^2)*(a^(-8))^(1/4)*arctan(sqrt(2)*a^2*sqrt((sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(1/4) - sqrt(2)*a^2*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - 1) + 100*sqrt(2)*(a^3*x + a^2)*(a^(-8))^(1/4)*arctan(sqrt(2)*a^2*sqrt(-(sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(1/4) - sqrt(2)*a^2*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) + 1) + 25*sqrt(2)*(a^3*x + a^2)*(a^(-8))^(1/4)*log((sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 25*sqrt(2)*(a^3*x + a^2)*(a^(-8))^(1/4)*log(-(sqrt(2)*(a^7*x - a^6)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*(2*a^2*x^2 - 9*a*x - 43)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/(a^3*x + a^2)","B",0
114,1,554,0,0.491218," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2),x, algorithm=""fricas"")","\frac{20 \, \sqrt{2} {\left(a^{2} x + a\right)} \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - 1\right) + 20 \, \sqrt{2} {\left(a^{2} x + a\right)} \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + 1\right) + 5 \, \sqrt{2} {\left(a^{2} x + a\right)} \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 5 \, \sqrt{2} {\left(a^{2} x + a\right)} \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, \sqrt{-a^{2} x^{2} + 1} {\left(a x + 9\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{4 \, {\left(a^{2} x + a\right)}}"," ",0,"1/4*(20*sqrt(2)*(a^2*x + a)*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - 1) + 20*sqrt(2)*(a^2*x + a)*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + 1) + 5*sqrt(2)*(a^2*x + a)*(a^(-4))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 5*sqrt(2)*(a^2*x + a)*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*sqrt(-a^2*x^2 + 1)*(a*x + 9)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/(a^2*x + a)","B",0
115,1,499,0,2.443495," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)/x,x, algorithm=""fricas"")","-\frac{4 \, \sqrt{2} {\left(a x + 1\right)} \arctan\left(\sqrt{2} \sqrt{\frac{a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) + 4 \, \sqrt{2} {\left(a x + 1\right)} \arctan\left(\sqrt{2} \sqrt{\frac{a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1}} - \sqrt{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + \sqrt{2} {\left(a x + 1\right)} \log\left(\frac{4 \, {\left(a x + \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) - \sqrt{2} {\left(a x + 1\right)} \log\left(\frac{4 \, {\left(a x - \sqrt{2} {\left(a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a x - 1}\right) - 4 \, {\left(a x + 1\right)} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 2 \, {\left(a x + 1\right)} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 2 \, {\left(a x + 1\right)} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right) - 16 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, {\left(a x + 1\right)}}"," ",0,"-1/2*(4*sqrt(2)*(a*x + 1)*arctan(sqrt(2)*sqrt((a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) + 4*sqrt(2)*(a*x + 1)*arctan(sqrt(2)*sqrt((a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + sqrt(2)*(a*x + 1)*log(4*(a*x + sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - sqrt(2)*(a*x + 1)*log(4*(a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - 4*(a*x + 1)*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 2*(a*x + 1)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 2*(a*x + 1)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 16*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/(a*x + 1)","B",0
116,1,170,0,0.574837," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)/x^2,x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-a^{2} x^{2} + 1} {\left(9 \, a x + 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 10 \, {\left(a^{2} x^{2} + a x\right)} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 5 \, {\left(a^{2} x^{2} + a x\right)} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + 5 \, {\left(a^{2} x^{2} + a x\right)} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right)}{2 \, {\left(a x^{2} + x\right)}}"," ",0,"-1/2*(2*sqrt(-a^2*x^2 + 1)*(9*a*x + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 10*(a^2*x^2 + a*x)*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 5*(a^2*x^2 + a*x)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 5*(a^2*x^2 + a*x)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1))/(a*x^2 + x)","B",0
117,1,192,0,0.935272," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)/x^3,x, algorithm=""fricas"")","\frac{2 \, {\left(43 \, a^{2} x^{2} + 9 \, a x - 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 50 \, {\left(a^{3} x^{3} + a^{2} x^{2}\right)} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 25 \, {\left(a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + 25 \, {\left(a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right)}{8 \, {\left(a x^{3} + x^{2}\right)}}"," ",0,"1/8*(2*(43*a^2*x^2 + 9*a*x - 2)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 50*(a^3*x^3 + a^2*x^2)*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 25*(a^3*x^3 + a^2*x^2)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 25*(a^3*x^3 + a^2*x^2)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1))/(a*x^3 + x^2)","A",0
118,1,200,0,0.591797," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)/x^4,x, algorithm=""fricas"")","-\frac{2 \, {\left(287 \, a^{3} x^{3} + 61 \, a^{2} x^{2} - 26 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 330 \, {\left(a^{4} x^{4} + a^{3} x^{3}\right)} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 165 \, {\left(a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + 165 \, {\left(a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right)}{48 \, {\left(a x^{4} + x^{3}\right)}}"," ",0,"-1/48*(2*(287*a^3*x^3 + 61*a^2*x^2 - 26*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 330*(a^4*x^4 + a^3*x^3)*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 165*(a^4*x^4 + a^3*x^3)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 165*(a^4*x^4 + a^3*x^3)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1))/(a*x^4 + x^3)","A",0
119,1,208,0,0.635134," ","integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)/x^5,x, algorithm=""fricas"")","\frac{2 \, {\left(2467 \, a^{4} x^{4} + 521 \, a^{3} x^{3} - 226 \, a^{2} x^{2} + 136 \, a x - 48\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 2850 \, {\left(a^{5} x^{5} + a^{4} x^{4}\right)} \arctan\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 1425 \, {\left(a^{5} x^{5} + a^{4} x^{4}\right)} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + 1425 \, {\left(a^{5} x^{5} + a^{4} x^{4}\right)} \log\left(\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right)}{384 \, {\left(a x^{5} + x^{4}\right)}}"," ",0,"1/384*(2*(2467*a^4*x^4 + 521*a^3*x^3 - 226*a^2*x^2 + 136*a*x - 48)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 2850*(a^5*x^5 + a^4*x^4)*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 1425*(a^5*x^5 + a^4*x^4)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 1425*(a^5*x^5 + a^4*x^4)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1))/(a*x^5 + x^4)","A",0
120,0,0,0,0.585207," ","integrate(((1+x)/(-x^2+1)^(1/2))^(1/3)*x^m,x, algorithm=""fricas"")","{\rm integral}\left(x^{m} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}, x\right)"," ",0,"integral(x^m*(-sqrt(-x^2 + 1)/(x - 1))^(1/3), x)","F",0
121,1,308,0,0.579524," ","integrate(((1+x)/(-x^2+1)^(1/2))^(1/3)*x^2,x, algorithm=""fricas"")","\frac{19}{324} \, \sqrt{3} \log\left(1444 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1444 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 1444\right) - \frac{19}{324} \, \sqrt{3} \log\left(-1444 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1444 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 1444\right) + \frac{1}{54} \, {\left(18 \, x^{3} + 3 \, x^{2} + x - 22\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - \frac{19}{81} \, \arctan\left(\sqrt{3} + \frac{1}{19} \, \sqrt{-1444 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1444 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 1444} - 2 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right) - \frac{19}{81} \, \arctan\left(-\sqrt{3} + 2 \, \sqrt{\sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 1} - 2 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right) + \frac{19}{81} \, \arctan\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right)"," ",0,"19/324*sqrt(3)*log(1444*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1444*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1444) - 19/324*sqrt(3)*log(-1444*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1444*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1444) + 1/54*(18*x^3 + 3*x^2 + x - 22)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 19/81*arctan(sqrt(3) + 1/19*sqrt(-1444*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1444*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1444) - 2*(-sqrt(-x^2 + 1)/(x - 1))^(1/3)) - 19/81*arctan(-sqrt(3) + 2*sqrt(sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + (-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1) - 2*(-sqrt(-x^2 + 1)/(x - 1))^(1/3)) + 19/81*arctan((-sqrt(-x^2 + 1)/(x - 1))^(1/3))","A",0
122,1,301,0,0.564203," ","integrate(((1+x)/(-x^2+1)^(1/2))^(1/3)*x,x, algorithm=""fricas"")","\frac{1}{36} \, \sqrt{3} \log\left(4 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 4 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 4\right) - \frac{1}{36} \, \sqrt{3} \log\left(-4 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 4 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 4\right) + \frac{1}{6} \, {\left(3 \, x^{2} + x - 4\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - \frac{1}{9} \, \arctan\left(\sqrt{3} + \sqrt{-4 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 4 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 4} - 2 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right) - \frac{1}{9} \, \arctan\left(-\sqrt{3} + 2 \, \sqrt{\sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 1} - 2 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right) + \frac{1}{9} \, \arctan\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right)"," ",0,"1/36*sqrt(3)*log(4*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 4*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 4) - 1/36*sqrt(3)*log(-4*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 4*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 4) + 1/6*(3*x^2 + x - 4)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 1/9*arctan(sqrt(3) + sqrt(-4*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 4*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 4) - 2*(-sqrt(-x^2 + 1)/(x - 1))^(1/3)) - 1/9*arctan(-sqrt(3) + 2*sqrt(sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + (-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1) - 2*(-sqrt(-x^2 + 1)/(x - 1))^(1/3)) + 1/9*arctan((-sqrt(-x^2 + 1)/(x - 1))^(1/3))","A",0
123,1,295,0,0.739827," ","integrate(((1+x)/(-x^2+1)^(1/2))^(1/3),x, algorithm=""fricas"")","\frac{1}{6} \, \sqrt{3} \log\left(4 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 4 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 4\right) - \frac{1}{6} \, \sqrt{3} \log\left(-4 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 4 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 4\right) + {\left(x - 1\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - \frac{2}{3} \, \arctan\left(\sqrt{3} + \sqrt{-4 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 4 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 4} - 2 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right) - \frac{2}{3} \, \arctan\left(-\sqrt{3} + 2 \, \sqrt{\sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 1} - 2 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right) + \frac{2}{3} \, \arctan\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right)"," ",0,"1/6*sqrt(3)*log(4*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 4*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 4) - 1/6*sqrt(3)*log(-4*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 4*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 4) + (x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 2/3*arctan(sqrt(3) + sqrt(-4*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 4*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 4) - 2*(-sqrt(-x^2 + 1)/(x - 1))^(1/3)) - 2/3*arctan(-sqrt(3) + 2*sqrt(sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + (-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1) - 2*(-sqrt(-x^2 + 1)/(x - 1))^(1/3)) + 2/3*arctan((-sqrt(-x^2 + 1)/(x - 1))^(1/3))","A",0
124,1,471,0,0.462063," ","integrate(((1+x)/(-x^2+1)^(1/2))^(1/3)/x,x, algorithm=""fricas"")","-\sqrt{3} \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right) - \sqrt{3} \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right) + \frac{1}{2} \, \sqrt{3} \log\left(4 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 4 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 4\right) - \frac{1}{2} \, \sqrt{3} \log\left(-4 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 4 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 4\right) - 2 \, \arctan\left(\sqrt{3} + \sqrt{-4 \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 4 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 4} - 2 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right) - 2 \, \arctan\left(-\sqrt{3} + 2 \, \sqrt{\sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 1} - 2 \, \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right) + 2 \, \arctan\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}\right) - \frac{1}{2} \, \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1\right) + \frac{1}{2} \, \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1\right) - \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1\right) + \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - 1\right)"," ",0,"-sqrt(3)*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1/3*sqrt(3)) - sqrt(3)*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 1/3*sqrt(3)) + 1/2*sqrt(3)*log(4*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 4*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 4) - 1/2*sqrt(3)*log(-4*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 4*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 4) - 2*arctan(sqrt(3) + sqrt(-4*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 4*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 4) - 2*(-sqrt(-x^2 + 1)/(x - 1))^(1/3)) - 2*arctan(-sqrt(3) + 2*sqrt(sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + (-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1) - 2*(-sqrt(-x^2 + 1)/(x - 1))^(1/3)) + 2*arctan((-sqrt(-x^2 + 1)/(x - 1))^(1/3)) - 1/2*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) + (-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1) + 1/2*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) - (-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1) - log((-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1) + log((-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 1)","A",0
125,1,234,0,0.501163," ","integrate(((1+x)/(-x^2+1)^(1/2))^(1/3)/x^2,x, algorithm=""fricas"")","-\frac{2 \, \sqrt{3} x \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right) + 2 \, \sqrt{3} x \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right) + x \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1\right) - x \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1\right) + 2 \, x \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1\right) - 2 \, x \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - 1\right) - 6 \, {\left(x - 1\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}}{6 \, x}"," ",0,"-1/6*(2*sqrt(3)*x*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1/3*sqrt(3)) + 2*sqrt(3)*x*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 1/3*sqrt(3)) + x*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) + (-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1) - x*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) - (-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1) + 2*x*log((-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1) - 2*x*log((-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 1) - 6*(x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3))/x","A",0
126,1,253,0,0.597149," ","integrate(((1+x)/(-x^2+1)^(1/2))^(1/3)/x^3,x, algorithm=""fricas"")","-\frac{2 \, \sqrt{3} x^{2} \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right) + 2 \, \sqrt{3} x^{2} \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right) + x^{2} \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1\right) - x^{2} \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1\right) + 2 \, x^{2} \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1\right) - 2 \, x^{2} \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - 1\right) - 6 \, {\left(4 \, x^{2} - x - 3\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}}{36 \, x^{2}}"," ",0,"-1/36*(2*sqrt(3)*x^2*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1/3*sqrt(3)) + 2*sqrt(3)*x^2*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 1/3*sqrt(3)) + x^2*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) + (-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1) - x^2*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) - (-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1) + 2*x^2*log((-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1) - 2*x^2*log((-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 1) - 6*(4*x^2 - x - 3)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3))/x^2","A",0
127,0,0,0,0.554981," ","integrate(((1+x)/(-x^2+1)^(1/2))^(2/3)*x^m,x, algorithm=""fricas"")","{\rm integral}\left(x^{m} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}}, x\right)"," ",0,"integral(x^m*(-sqrt(-x^2 + 1)/(x - 1))^(2/3), x)","F",0
128,1,159,0,0.501593," ","integrate(((1+x)/(-x^2+1)^(1/2))^(2/3)*x^2,x, algorithm=""fricas"")","\frac{1}{27} \, {\left(9 \, x^{3} + 3 \, x^{2} + 2 \, x - 14\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + \frac{22}{81} \, \sqrt{3} \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - \frac{1}{3} \, \sqrt{3}\right) + \frac{22}{81} \, \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 1\right) - \frac{11}{81} \, \log\left(-\frac{{\left(x - 1\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - x + \sqrt{-x^{2} + 1} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1}{x - 1}\right)"," ",0,"1/27*(9*x^3 + 3*x^2 + 2*x - 14)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 22/81*sqrt(3)*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) - 1/3*sqrt(3)) + 22/81*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1) - 11/81*log(-((x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) - x + sqrt(-x^2 + 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1)/(x - 1))","A",0
129,1,154,0,0.614470," ","integrate(((1+x)/(-x^2+1)^(1/2))^(2/3)*x,x, algorithm=""fricas"")","\frac{1}{6} \, {\left(3 \, x^{2} + 2 \, x - 5\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + \frac{2}{9} \, \sqrt{3} \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - \frac{1}{3} \, \sqrt{3}\right) + \frac{2}{9} \, \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 1\right) - \frac{1}{9} \, \log\left(-\frac{{\left(x - 1\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - x + \sqrt{-x^{2} + 1} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1}{x - 1}\right)"," ",0,"1/6*(3*x^2 + 2*x - 5)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 2/9*sqrt(3)*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) - 1/3*sqrt(3)) + 2/9*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1) - 1/9*log(-((x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) - x + sqrt(-x^2 + 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1)/(x - 1))","A",0
130,1,146,0,0.543515," ","integrate(((1+x)/(-x^2+1)^(1/2))^(2/3),x, algorithm=""fricas"")","{\left(x - 1\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + \frac{2}{3} \, \sqrt{3} \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - \frac{1}{3} \, \sqrt{3}\right) + \frac{2}{3} \, \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + 1\right) - \frac{1}{3} \, \log\left(-\frac{{\left(x - 1\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - x + \sqrt{-x^{2} + 1} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1}{x - 1}\right)"," ",0,"(x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 2/3*sqrt(3)*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) - 1/3*sqrt(3)) + 2/3*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1) - 1/3*log(-((x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) - x + sqrt(-x^2 + 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1)/(x - 1))","B",0
131,1,158,0,1.291085," ","integrate(((1+x)/(-x^2+1)^(1/2))^(2/3)/x,x, algorithm=""fricas"")","-\sqrt{3} \arctan\left(-\frac{\sqrt{3} {\left(x - 1\right)} - 2 \, \sqrt{3} \sqrt{-x^{2} + 1} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}}}{3 \, {\left(x - 1\right)}}\right) - \frac{1}{2} \, \log\left(-\frac{{\left(x + 1\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - x + \sqrt{-x^{2} + 1} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} + 1}{x - 1}\right) + \log\left(-\frac{x + \sqrt{-x^{2} + 1} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - 1}{x - 1}\right)"," ",0,"-sqrt(3)*arctan(-1/3*(sqrt(3)*(x - 1) - 2*sqrt(3)*sqrt(-x^2 + 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3))/(x - 1)) - 1/2*log(-((x + 1)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) - x + sqrt(-x^2 + 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1)/(x - 1)) + log(-(x + sqrt(-x^2 + 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 1)/(x - 1))","A",0
132,1,152,0,0.515001," ","integrate(((1+x)/(-x^2+1)^(1/2))^(2/3)/x^2,x, algorithm=""fricas"")","-\frac{2 \, \sqrt{3} x \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + \frac{1}{3} \, \sqrt{3}\right) - 2 \, x \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - 1\right) + x \log\left(\frac{{\left(x - 1\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + x - \sqrt{-x^{2} + 1} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - 1}{x - 1}\right) - 3 \, {\left(x - 1\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}}}{3 \, x}"," ",0,"-1/3*(2*sqrt(3)*x*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1/3*sqrt(3)) - 2*x*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) - 1) + x*log(((x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + x - sqrt(-x^2 + 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 1)/(x - 1)) - 3*(x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3))/x","B",0
133,1,166,0,0.705271," ","integrate(((1+x)/(-x^2+1)^(1/2))^(2/3)/x^3,x, algorithm=""fricas"")","-\frac{4 \, \sqrt{3} x^{2} \arctan\left(\frac{2}{3} \, \sqrt{3} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + \frac{1}{3} \, \sqrt{3}\right) - 4 \, x^{2} \log\left(\left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} - 1\right) + 2 \, x^{2} \log\left(\frac{{\left(x - 1\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}} + x - \sqrt{-x^{2} + 1} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{1}{3}} - 1}{x - 1}\right) - 3 \, {\left(5 \, x^{2} - 2 \, x - 3\right)} \left(-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right)^{\frac{2}{3}}}{18 \, x^{2}}"," ",0,"-1/18*(4*sqrt(3)*x^2*arctan(2/3*sqrt(3)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 1/3*sqrt(3)) - 4*x^2*log((-sqrt(-x^2 + 1)/(x - 1))^(2/3) - 1) + 2*x^2*log(((x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + x - sqrt(-x^2 + 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) - 1)/(x - 1)) - 3*(5*x^2 - 2*x - 3)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3))/x^2","A",0
134,0,0,0,0.607520," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^m,x, algorithm=""fricas"")","{\rm integral}\left(x^{m} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}, x\right)"," ",0,"integral(x^m*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4), x)","F",0
135,1,2486,0,0.753538," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x^2,x, algorithm=""fricas"")","-\frac{264 \, a^{3} \sqrt{\sqrt{2} + 2} \frac{1}{a^{24}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} + a^{3} \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{21} \frac{1}{a^{24}}^{\frac{7}{8}} - 2 \, a^{21} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{7}{8}} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right) + 264 \, a^{3} \sqrt{\sqrt{2} + 2} \frac{1}{a^{24}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} - a^{3} \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{21} \frac{1}{a^{24}}^{\frac{7}{8}} - 2 \, a^{21} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{7}{8}} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right) + 264 \, a^{3} \sqrt{-\sqrt{2} + 2} \frac{1}{a^{24}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} + a^{3} \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{21} \frac{1}{a^{24}}^{\frac{7}{8}} - 2 \, a^{21} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{7}{8}} - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right) + 264 \, a^{3} \sqrt{-\sqrt{2} + 2} \frac{1}{a^{24}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} - a^{3} \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{21} \frac{1}{a^{24}}^{\frac{7}{8}} - 2 \, a^{21} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{7}{8}} + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right) - 66 \, a^{3} \sqrt{\sqrt{2} + 2} \frac{1}{a^{24}}^{\frac{1}{8}} \log\left(a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} + a^{3} \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 66 \, a^{3} \sqrt{\sqrt{2} + 2} \frac{1}{a^{24}}^{\frac{1}{8}} \log\left(a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} - a^{3} \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 66 \, a^{3} \sqrt{-\sqrt{2} + 2} \frac{1}{a^{24}}^{\frac{1}{8}} \log\left(a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} + a^{3} \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 66 \, a^{3} \sqrt{-\sqrt{2} + 2} \frac{1}{a^{24}}^{\frac{1}{8}} \log\left(a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} - a^{3} \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 132 \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{24}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{21} \frac{1}{a^{24}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{21} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{7}{8}} - \sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}\right) + 132 \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{24}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{21} \frac{1}{a^{24}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{21} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{7}{8}} + \sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}\right) - 132 \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{24}}^{\frac{1}{8}} \arctan\left(-\frac{2 \, \sqrt{2} \sqrt{a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{21} \frac{1}{a^{24}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{21} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{7}{8}} - \sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}\right) - 132 \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{24}}^{\frac{1}{8}} \arctan\left(-\frac{2 \, \sqrt{2} \sqrt{a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{21} \frac{1}{a^{24}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{21} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{7}{8}} + \sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}\right) - 33 \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{24}}^{\frac{1}{8}} \log\left(a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 33 \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{24}}^{\frac{1}{8}} \log\left(a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 33 \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{24}}^{\frac{1}{8}} \log\left(a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 33 \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{24}}^{\frac{1}{8}} \log\left(a^{6} \frac{1}{a^{24}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a^{3} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{3} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{24}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 32 \, {\left(32 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x - 37\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}}{3072 \, a^{3}}"," ",0,"-1/3072*(264*a^3*sqrt(sqrt(2) + 2)*(a^(-24))^(1/8)*arctan((2*sqrt(a^6*(a^(-24))^(1/4) + a^3*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^21*(a^(-24))^(7/8) - 2*a^21*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(7/8) - sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 264*a^3*sqrt(sqrt(2) + 2)*(a^(-24))^(1/8)*arctan((2*sqrt(a^6*(a^(-24))^(1/4) - a^3*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^21*(a^(-24))^(7/8) - 2*a^21*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(7/8) + sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 264*a^3*sqrt(-sqrt(2) + 2)*(a^(-24))^(1/8)*arctan((2*sqrt(a^6*(a^(-24))^(1/4) + a^3*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^21*(a^(-24))^(7/8) - 2*a^21*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(7/8) - sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) + 264*a^3*sqrt(-sqrt(2) + 2)*(a^(-24))^(1/8)*arctan((2*sqrt(a^6*(a^(-24))^(1/4) - a^3*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^21*(a^(-24))^(7/8) - 2*a^21*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(7/8) + sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) - 66*a^3*sqrt(sqrt(2) + 2)*(a^(-24))^(1/8)*log(a^6*(a^(-24))^(1/4) + a^3*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 66*a^3*sqrt(sqrt(2) + 2)*(a^(-24))^(1/8)*log(a^6*(a^(-24))^(1/4) - a^3*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 66*a^3*sqrt(-sqrt(2) + 2)*(a^(-24))^(1/8)*log(a^6*(a^(-24))^(1/4) + a^3*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 66*a^3*sqrt(-sqrt(2) + 2)*(a^(-24))^(1/8)*log(a^6*(a^(-24))^(1/4) - a^3*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 132*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) + sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(a^(-24))^(1/8)*arctan((2*sqrt(2)*sqrt(a^6*(a^(-24))^(1/4) + 1/2*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) - sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^21*(a^(-24))^(7/8) - 2*sqrt(2)*a^21*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(7/8) - sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) + 132*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) + sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(a^(-24))^(1/8)*arctan((2*sqrt(2)*sqrt(a^6*(a^(-24))^(1/4) - 1/2*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) - sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^21*(a^(-24))^(7/8) - 2*sqrt(2)*a^21*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(7/8) + sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) - 132*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) - sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(a^(-24))^(1/8)*arctan(-(2*sqrt(2)*sqrt(a^6*(a^(-24))^(1/4) + 1/2*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) + sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^21*(a^(-24))^(7/8) - 2*sqrt(2)*a^21*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(7/8) - sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) - 132*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) - sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(a^(-24))^(1/8)*arctan(-(2*sqrt(2)*sqrt(a^6*(a^(-24))^(1/4) - 1/2*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) + sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^21*(a^(-24))^(7/8) - 2*sqrt(2)*a^21*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(7/8) + sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) - 33*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) + sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(a^(-24))^(1/8)*log(a^6*(a^(-24))^(1/4) + 1/2*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) + sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 33*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) + sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(a^(-24))^(1/8)*log(a^6*(a^(-24))^(1/4) - 1/2*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) + sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 33*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) - sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(a^(-24))^(1/8)*log(a^6*(a^(-24))^(1/4) + 1/2*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) - sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 33*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) - sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(a^(-24))^(1/8)*log(a^6*(a^(-24))^(1/4) - 1/2*(sqrt(2)*a^3*sqrt(sqrt(2) + 2) - sqrt(2)*a^3*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-24))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 32*(32*a^3*x^3 + 4*a^2*x^2 + a*x - 37)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/a^3","B",0
136,1,2476,0,0.608922," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)*x,x, algorithm=""fricas"")","-\frac{8 \, a^{2} \sqrt{\sqrt{2} + 2} \frac{1}{a^{16}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} + a^{2} \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{14} \frac{1}{a^{16}}^{\frac{7}{8}} - 2 \, a^{14} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{7}{8}} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right) + 8 \, a^{2} \sqrt{\sqrt{2} + 2} \frac{1}{a^{16}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} - a^{2} \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{14} \frac{1}{a^{16}}^{\frac{7}{8}} - 2 \, a^{14} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{7}{8}} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right) + 8 \, a^{2} \sqrt{-\sqrt{2} + 2} \frac{1}{a^{16}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} + a^{2} \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{14} \frac{1}{a^{16}}^{\frac{7}{8}} - 2 \, a^{14} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{7}{8}} - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right) + 8 \, a^{2} \sqrt{-\sqrt{2} + 2} \frac{1}{a^{16}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} - a^{2} \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{14} \frac{1}{a^{16}}^{\frac{7}{8}} - 2 \, a^{14} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{7}{8}} + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right) - 2 \, a^{2} \sqrt{\sqrt{2} + 2} \frac{1}{a^{16}}^{\frac{1}{8}} \log\left(a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} + a^{2} \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 2 \, a^{2} \sqrt{\sqrt{2} + 2} \frac{1}{a^{16}}^{\frac{1}{8}} \log\left(a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} - a^{2} \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 2 \, a^{2} \sqrt{-\sqrt{2} + 2} \frac{1}{a^{16}}^{\frac{1}{8}} \log\left(a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} + a^{2} \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 2 \, a^{2} \sqrt{-\sqrt{2} + 2} \frac{1}{a^{16}}^{\frac{1}{8}} \log\left(a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} - a^{2} \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 4 \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{16}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{14} \frac{1}{a^{16}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{14} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{7}{8}} - \sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}\right) + 4 \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{16}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{14} \frac{1}{a^{16}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{14} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{7}{8}} + \sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}\right) - 4 \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{16}}^{\frac{1}{8}} \arctan\left(-\frac{2 \, \sqrt{2} \sqrt{a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{14} \frac{1}{a^{16}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{14} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{7}{8}} - \sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}\right) - 4 \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{16}}^{\frac{1}{8}} \arctan\left(-\frac{2 \, \sqrt{2} \sqrt{a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{14} \frac{1}{a^{16}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{14} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{7}{8}} + \sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}\right) - {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{16}}^{\frac{1}{8}} \log\left(a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{16}}^{\frac{1}{8}} \log\left(a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{16}}^{\frac{1}{8}} \log\left(a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{16}}^{\frac{1}{8}} \log\left(a^{4} \frac{1}{a^{16}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a^{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} a^{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{16}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 32 \, {\left(4 \, a^{2} x^{2} + a x - 5\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}}{256 \, a^{2}}"," ",0,"-1/256*(8*a^2*sqrt(sqrt(2) + 2)*(a^(-16))^(1/8)*arctan((2*sqrt(a^4*(a^(-16))^(1/4) + a^2*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^14*(a^(-16))^(7/8) - 2*a^14*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(7/8) - sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 8*a^2*sqrt(sqrt(2) + 2)*(a^(-16))^(1/8)*arctan((2*sqrt(a^4*(a^(-16))^(1/4) - a^2*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^14*(a^(-16))^(7/8) - 2*a^14*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(7/8) + sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 8*a^2*sqrt(-sqrt(2) + 2)*(a^(-16))^(1/8)*arctan((2*sqrt(a^4*(a^(-16))^(1/4) + a^2*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^14*(a^(-16))^(7/8) - 2*a^14*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(7/8) - sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) + 8*a^2*sqrt(-sqrt(2) + 2)*(a^(-16))^(1/8)*arctan((2*sqrt(a^4*(a^(-16))^(1/4) - a^2*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^14*(a^(-16))^(7/8) - 2*a^14*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(7/8) + sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) - 2*a^2*sqrt(sqrt(2) + 2)*(a^(-16))^(1/8)*log(a^4*(a^(-16))^(1/4) + a^2*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 2*a^2*sqrt(sqrt(2) + 2)*(a^(-16))^(1/8)*log(a^4*(a^(-16))^(1/4) - a^2*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 2*a^2*sqrt(-sqrt(2) + 2)*(a^(-16))^(1/8)*log(a^4*(a^(-16))^(1/4) + a^2*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 2*a^2*sqrt(-sqrt(2) + 2)*(a^(-16))^(1/8)*log(a^4*(a^(-16))^(1/4) - a^2*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 4*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) + sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(a^(-16))^(1/8)*arctan((2*sqrt(2)*sqrt(a^4*(a^(-16))^(1/4) + 1/2*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) - sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^14*(a^(-16))^(7/8) - 2*sqrt(2)*a^14*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(7/8) - sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) + 4*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) + sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(a^(-16))^(1/8)*arctan((2*sqrt(2)*sqrt(a^4*(a^(-16))^(1/4) - 1/2*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) - sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^14*(a^(-16))^(7/8) - 2*sqrt(2)*a^14*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(7/8) + sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) - 4*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) - sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(a^(-16))^(1/8)*arctan(-(2*sqrt(2)*sqrt(a^4*(a^(-16))^(1/4) + 1/2*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) + sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^14*(a^(-16))^(7/8) - 2*sqrt(2)*a^14*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(7/8) - sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) - 4*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) - sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(a^(-16))^(1/8)*arctan(-(2*sqrt(2)*sqrt(a^4*(a^(-16))^(1/4) - 1/2*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) + sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^14*(a^(-16))^(7/8) - 2*sqrt(2)*a^14*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(7/8) + sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) - (sqrt(2)*a^2*sqrt(sqrt(2) + 2) + sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(a^(-16))^(1/8)*log(a^4*(a^(-16))^(1/4) + 1/2*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) + sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + (sqrt(2)*a^2*sqrt(sqrt(2) + 2) + sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(a^(-16))^(1/8)*log(a^4*(a^(-16))^(1/4) - 1/2*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) + sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - (sqrt(2)*a^2*sqrt(sqrt(2) + 2) - sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(a^(-16))^(1/8)*log(a^4*(a^(-16))^(1/4) + 1/2*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) - sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + (sqrt(2)*a^2*sqrt(sqrt(2) + 2) - sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(a^(-16))^(1/8)*log(a^4*(a^(-16))^(1/4) - 1/2*(sqrt(2)*a^2*sqrt(sqrt(2) + 2) - sqrt(2)*a^2*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-16))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 32*(4*a^2*x^2 + a*x - 5)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/a^2","B",0
137,1,2372,0,1.003886," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4),x, algorithm=""fricas"")","-\frac{8 \, a \sqrt{\sqrt{2} + 2} \frac{1}{a^{8}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} + a \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{7} \frac{1}{a^{8}}^{\frac{7}{8}} - 2 \, a^{7} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{7}{8}} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right) + 8 \, a \sqrt{\sqrt{2} + 2} \frac{1}{a^{8}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - a \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{7} \frac{1}{a^{8}}^{\frac{7}{8}} - 2 \, a^{7} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{7}{8}} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right) + 8 \, a \sqrt{-\sqrt{2} + 2} \frac{1}{a^{8}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} + a \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{7} \frac{1}{a^{8}}^{\frac{7}{8}} - 2 \, a^{7} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{7}{8}} - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right) + 8 \, a \sqrt{-\sqrt{2} + 2} \frac{1}{a^{8}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - a \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{7} \frac{1}{a^{8}}^{\frac{7}{8}} - 2 \, a^{7} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{7}{8}} + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right) - 2 \, a \sqrt{\sqrt{2} + 2} \frac{1}{a^{8}}^{\frac{1}{8}} \log\left(a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} + a \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 2 \, a \sqrt{\sqrt{2} + 2} \frac{1}{a^{8}}^{\frac{1}{8}} \log\left(a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - a \sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 2 \, a \sqrt{-\sqrt{2} + 2} \frac{1}{a^{8}}^{\frac{1}{8}} \log\left(a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} + a \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 2 \, a \sqrt{-\sqrt{2} + 2} \frac{1}{a^{8}}^{\frac{1}{8}} \log\left(a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - a \sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + 4 \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} + \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{8}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} - \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{7} \frac{1}{a^{8}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{7} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{7}{8}} - \sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}\right) + 4 \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} + \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{8}}^{\frac{1}{8}} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} - \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{7} \frac{1}{a^{8}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{7} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{7}{8}} + \sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}\right) - 4 \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} - \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{8}}^{\frac{1}{8}} \arctan\left(-\frac{2 \, \sqrt{2} \sqrt{a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} + \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{7} \frac{1}{a^{8}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{7} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{7}{8}} - \sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}\right) - 4 \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} - \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{8}}^{\frac{1}{8}} \arctan\left(-\frac{2 \, \sqrt{2} \sqrt{a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} + \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}} a^{7} \frac{1}{a^{8}}^{\frac{7}{8}} - 2 \, \sqrt{2} a^{7} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{7}{8}} + \sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}\right) - {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} + \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{8}}^{\frac{1}{8}} \log\left(a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} + \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} + \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{8}}^{\frac{1}{8}} \log\left(a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} + \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} - \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{8}}^{\frac{1}{8}} \log\left(a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} + \frac{1}{2} \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} - \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) + {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} - \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \frac{1}{a^{8}}^{\frac{1}{8}} \log\left(a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - \frac{1}{2} \, {\left(\sqrt{2} a \sqrt{\sqrt{2} + 2} - \sqrt{2} a \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} \frac{1}{a^{8}}^{\frac{1}{8}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right) - 32 \, {\left(a x - 1\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}}{32 \, a}"," ",0,"-1/32*(8*a*sqrt(sqrt(2) + 2)*(a^(-8))^(1/8)*arctan((2*sqrt(a^2*(a^(-8))^(1/4) + a*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^7*(a^(-8))^(7/8) - 2*a^7*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(7/8) - sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 8*a*sqrt(sqrt(2) + 2)*(a^(-8))^(1/8)*arctan((2*sqrt(a^2*(a^(-8))^(1/4) - a*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^7*(a^(-8))^(7/8) - 2*a^7*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(7/8) + sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 8*a*sqrt(-sqrt(2) + 2)*(a^(-8))^(1/8)*arctan((2*sqrt(a^2*(a^(-8))^(1/4) + a*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^7*(a^(-8))^(7/8) - 2*a^7*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(7/8) - sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) + 8*a*sqrt(-sqrt(2) + 2)*(a^(-8))^(1/8)*arctan((2*sqrt(a^2*(a^(-8))^(1/4) - a*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^7*(a^(-8))^(7/8) - 2*a^7*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(7/8) + sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) - 2*a*sqrt(sqrt(2) + 2)*(a^(-8))^(1/8)*log(a^2*(a^(-8))^(1/4) + a*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 2*a*sqrt(sqrt(2) + 2)*(a^(-8))^(1/8)*log(a^2*(a^(-8))^(1/4) - a*sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 2*a*sqrt(-sqrt(2) + 2)*(a^(-8))^(1/8)*log(a^2*(a^(-8))^(1/4) + a*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 2*a*sqrt(-sqrt(2) + 2)*(a^(-8))^(1/8)*log(a^2*(a^(-8))^(1/4) - a*sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 4*(sqrt(2)*a*sqrt(sqrt(2) + 2) + sqrt(2)*a*sqrt(-sqrt(2) + 2))*(a^(-8))^(1/8)*arctan((2*sqrt(2)*sqrt(a^2*(a^(-8))^(1/4) + 1/2*(sqrt(2)*a*sqrt(sqrt(2) + 2) - sqrt(2)*a*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^7*(a^(-8))^(7/8) - 2*sqrt(2)*a^7*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(7/8) - sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) + 4*(sqrt(2)*a*sqrt(sqrt(2) + 2) + sqrt(2)*a*sqrt(-sqrt(2) + 2))*(a^(-8))^(1/8)*arctan((2*sqrt(2)*sqrt(a^2*(a^(-8))^(1/4) - 1/2*(sqrt(2)*a*sqrt(sqrt(2) + 2) - sqrt(2)*a*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^7*(a^(-8))^(7/8) - 2*sqrt(2)*a^7*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(7/8) + sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) - 4*(sqrt(2)*a*sqrt(sqrt(2) + 2) - sqrt(2)*a*sqrt(-sqrt(2) + 2))*(a^(-8))^(1/8)*arctan(-(2*sqrt(2)*sqrt(a^2*(a^(-8))^(1/4) + 1/2*(sqrt(2)*a*sqrt(sqrt(2) + 2) + sqrt(2)*a*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^7*(a^(-8))^(7/8) - 2*sqrt(2)*a^7*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(7/8) - sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) - 4*(sqrt(2)*a*sqrt(sqrt(2) + 2) - sqrt(2)*a*sqrt(-sqrt(2) + 2))*(a^(-8))^(1/8)*arctan(-(2*sqrt(2)*sqrt(a^2*(a^(-8))^(1/4) - 1/2*(sqrt(2)*a*sqrt(sqrt(2) + 2) + sqrt(2)*a*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))*a^7*(a^(-8))^(7/8) - 2*sqrt(2)*a^7*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(7/8) + sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) - (sqrt(2)*a*sqrt(sqrt(2) + 2) + sqrt(2)*a*sqrt(-sqrt(2) + 2))*(a^(-8))^(1/8)*log(a^2*(a^(-8))^(1/4) + 1/2*(sqrt(2)*a*sqrt(sqrt(2) + 2) + sqrt(2)*a*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + (sqrt(2)*a*sqrt(sqrt(2) + 2) + sqrt(2)*a*sqrt(-sqrt(2) + 2))*(a^(-8))^(1/8)*log(a^2*(a^(-8))^(1/4) - 1/2*(sqrt(2)*a*sqrt(sqrt(2) + 2) + sqrt(2)*a*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - (sqrt(2)*a*sqrt(sqrt(2) + 2) - sqrt(2)*a*sqrt(-sqrt(2) + 2))*(a^(-8))^(1/8)*log(a^2*(a^(-8))^(1/4) + 1/2*(sqrt(2)*a*sqrt(sqrt(2) + 2) - sqrt(2)*a*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + (sqrt(2)*a*sqrt(sqrt(2) + 2) - sqrt(2)*a*sqrt(-sqrt(2) + 2))*(a^(-8))^(1/8)*log(a^2*(a^(-8))^(1/4) - 1/2*(sqrt(2)*a*sqrt(sqrt(2) + 2) - sqrt(2)*a*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)*(a^(-8))^(1/8) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 32*(a*x - 1)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/a","B",0
138,1,2281,0,0.542975," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)/x,x, algorithm=""fricas"")","-\frac{1}{2} \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \arctan\left(\frac{\sqrt{2} \sqrt{2 \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4} - 2 \, \sqrt{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} - \sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}\right) - \frac{1}{2} \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \arctan\left(\frac{\sqrt{2} \sqrt{-2 \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4} - 2 \, \sqrt{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}\right) + \frac{1}{2} \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \arctan\left(-\frac{\sqrt{2} \sqrt{2 \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4} - 2 \, \sqrt{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} - \sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}\right) + \frac{1}{2} \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \arctan\left(-\frac{\sqrt{2} \sqrt{-2 \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4} - 2 \, \sqrt{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{\sqrt{2} + 2} + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2} - \sqrt{-\sqrt{2} + 2}}\right) + \frac{1}{8} \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \log\left(2 \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4\right) - \frac{1}{8} \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \log\left(-2 \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} + \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4\right) + \frac{1}{8} \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \log\left(2 \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4\right) - \frac{1}{8} \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \log\left(-2 \, {\left(\sqrt{2} \sqrt{\sqrt{2} + 2} - \sqrt{2} \sqrt{-\sqrt{2} + 2}\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4\right) + 2 \, \sqrt{2} \arctan\left(\sqrt{2} \sqrt{\sqrt{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1} - \sqrt{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} - 1\right) + 2 \, \sqrt{2} \arctan\left(\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4} - \sqrt{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 1\right) - \sqrt{-\sqrt{2} + 2} \arctan\left(\frac{2 \, \sqrt{\sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1} - \sqrt{\sqrt{2} + 2} - 2 \, \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}}{\sqrt{-\sqrt{2} + 2}}\right) - \sqrt{-\sqrt{2} + 2} \arctan\left(\frac{2 \, \sqrt{-\sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1} + \sqrt{\sqrt{2} + 2} - 2 \, \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}}{\sqrt{-\sqrt{2} + 2}}\right) - \sqrt{\sqrt{2} + 2} \arctan\left(\frac{2 \, \sqrt{\sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1} - \sqrt{-\sqrt{2} + 2} - 2 \, \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}}{\sqrt{\sqrt{2} + 2}}\right) - \sqrt{\sqrt{2} + 2} \arctan\left(\frac{2 \, \sqrt{-\sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1} + \sqrt{-\sqrt{2} + 2} - 2 \, \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}}{\sqrt{\sqrt{2} + 2}}\right) - \frac{1}{2} \, \sqrt{2} \log\left(4 \, \sqrt{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4\right) + \frac{1}{2} \, \sqrt{2} \log\left(-4 \, \sqrt{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 4 \, \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 4\right) + \frac{1}{4} \, \sqrt{\sqrt{2} + 2} \log\left(\sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - \frac{1}{4} \, \sqrt{\sqrt{2} + 2} \log\left(-\sqrt{\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) + \frac{1}{4} \, \sqrt{-\sqrt{2} + 2} \log\left(\sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - \frac{1}{4} \, \sqrt{-\sqrt{2} + 2} \log\left(-\sqrt{-\sqrt{2} + 2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right) - 2 \, \arctan\left(\left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}\right) - \log\left(\left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 1\right) + \log\left(\left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} - 1\right)"," ",0,"-1/2*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*arctan((sqrt(2)*sqrt(2*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) - 2*sqrt(2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) - 1/2*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*arctan((sqrt(2)*sqrt(-2*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) - 2*sqrt(2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) + 1/2*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2))*arctan(-(sqrt(2)*sqrt(2*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) - 2*sqrt(2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) + 1/2*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2))*arctan(-(sqrt(2)*sqrt(-2*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) - 2*sqrt(2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) + 1/8*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*log(2*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) - 1/8*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*log(-2*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) + 1/8*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2))*log(2*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) - 1/8*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2))*log(-2*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2))*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) + 2*sqrt(2)*arctan(sqrt(2)*sqrt(sqrt(2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - sqrt(2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - 1) + 2*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-4*sqrt(2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) - sqrt(2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 1) - sqrt(-sqrt(2) + 2)*arctan((2*sqrt(sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - sqrt(sqrt(2) + 2) - 2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/sqrt(-sqrt(2) + 2)) - sqrt(-sqrt(2) + 2)*arctan((2*sqrt(-sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + sqrt(sqrt(2) + 2) - 2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/sqrt(-sqrt(2) + 2)) - sqrt(sqrt(2) + 2)*arctan((2*sqrt(sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - sqrt(-sqrt(2) + 2) - 2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/sqrt(sqrt(2) + 2)) - sqrt(sqrt(2) + 2)*arctan((2*sqrt(-sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + sqrt(-sqrt(2) + 2) - 2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/sqrt(sqrt(2) + 2)) - 1/2*sqrt(2)*log(4*sqrt(2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) + 1/2*sqrt(2)*log(-4*sqrt(2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 4) + 1/4*sqrt(sqrt(2) + 2)*log(sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 1/4*sqrt(sqrt(2) + 2)*log(-sqrt(sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 1/4*sqrt(-sqrt(2) + 2)*log(sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 1/4*sqrt(-sqrt(2) + 2)*log(-sqrt(-sqrt(2) + 2)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 2*arctan((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) - log((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 1) + log((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - 1)","B",0
139,1,544,0,1.236042," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)/x^2,x, algorithm=""fricas"")","-\frac{4 \, a x \arctan\left(\left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}\right) + 2 \, a x \log\left(\left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 1\right) - 2 \, a x \log\left(\left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} - 1\right) - 4 \, \sqrt{2} {\left(a^{4}\right)}^{\frac{1}{4}} x \arctan\left(-\frac{a^{4} + \sqrt{2} {\left(a^{4}\right)}^{\frac{3}{4}} a \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} - \sqrt{2} {\left(a^{4}\right)}^{\frac{3}{4}} \sqrt{a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt{2} {\left(a^{4}\right)}^{\frac{1}{4}} a \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{a^{4}}}}{a^{4}}\right) - 4 \, \sqrt{2} {\left(a^{4}\right)}^{\frac{1}{4}} x \arctan\left(\frac{a^{4} - \sqrt{2} {\left(a^{4}\right)}^{\frac{3}{4}} a \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{2} {\left(a^{4}\right)}^{\frac{3}{4}} \sqrt{a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{2} {\left(a^{4}\right)}^{\frac{1}{4}} a \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{a^{4}}}}{a^{4}}\right) + \sqrt{2} {\left(a^{4}\right)}^{\frac{1}{4}} x \log\left(a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt{2} {\left(a^{4}\right)}^{\frac{1}{4}} a \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{a^{4}}\right) - \sqrt{2} {\left(a^{4}\right)}^{\frac{1}{4}} x \log\left(a^{2} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{2} {\left(a^{4}\right)}^{\frac{1}{4}} a \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{a^{4}}\right) - 8 \, {\left(a x - 1\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}}{8 \, x}"," ",0,"-1/8*(4*a*x*arctan((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + 2*a*x*log((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 1) - 2*a*x*log((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - 1) - 4*sqrt(2)*(a^4)^(1/4)*x*arctan(-(a^4 + sqrt(2)*(a^4)^(3/4)*a*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - sqrt(2)*(a^4)^(3/4)*sqrt(a^2*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + sqrt(2)*(a^4)^(1/4)*a*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^4)))/a^4) - 4*sqrt(2)*(a^4)^(1/4)*x*arctan((a^4 - sqrt(2)*(a^4)^(3/4)*a*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(2)*(a^4)^(3/4)*sqrt(a^2*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(2)*(a^4)^(1/4)*a*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^4)))/a^4) + sqrt(2)*(a^4)^(1/4)*x*log(a^2*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + sqrt(2)*(a^4)^(1/4)*a*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^4)) - sqrt(2)*(a^4)^(1/4)*x*log(a^2*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(2)*(a^4)^(1/4)*a*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^4)) - 8*(a*x - 1)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/x","B",0
140,1,585,0,0.589660," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)/x^3,x, algorithm=""fricas"")","-\frac{4 \, a^{2} x^{2} \arctan\left(\left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}\right) + 2 \, a^{2} x^{2} \log\left(\left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + 1\right) - 2 \, a^{2} x^{2} \log\left(\left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} - 1\right) - 4 \, \sqrt{2} {\left(a^{8}\right)}^{\frac{1}{4}} x^{2} \arctan\left(-\frac{a^{8} + \sqrt{2} {\left(a^{8}\right)}^{\frac{3}{4}} a^{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} - \sqrt{2} {\left(a^{8}\right)}^{\frac{3}{4}} \sqrt{a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt{2} {\left(a^{8}\right)}^{\frac{1}{4}} a^{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{a^{8}}}}{a^{8}}\right) - 4 \, \sqrt{2} {\left(a^{8}\right)}^{\frac{1}{4}} x^{2} \arctan\left(\frac{a^{8} - \sqrt{2} {\left(a^{8}\right)}^{\frac{3}{4}} a^{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{2} {\left(a^{8}\right)}^{\frac{3}{4}} \sqrt{a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{2} {\left(a^{8}\right)}^{\frac{1}{4}} a^{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{a^{8}}}}{a^{8}}\right) + \sqrt{2} {\left(a^{8}\right)}^{\frac{1}{4}} x^{2} \log\left(a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt{2} {\left(a^{8}\right)}^{\frac{1}{4}} a^{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{a^{8}}\right) - \sqrt{2} {\left(a^{8}\right)}^{\frac{1}{4}} x^{2} \log\left(a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{2} {\left(a^{8}\right)}^{\frac{1}{4}} a^{2} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}} + \sqrt{a^{8}}\right) - 8 \, {\left(5 \, a^{2} x^{2} - a x - 4\right)} \left(-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)^{\frac{1}{4}}}{64 \, x^{2}}"," ",0,"-1/64*(4*a^2*x^2*arctan((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + 2*a^2*x^2*log((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + 1) - 2*a^2*x^2*log((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - 1) - 4*sqrt(2)*(a^8)^(1/4)*x^2*arctan(-(a^8 + sqrt(2)*(a^8)^(3/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - sqrt(2)*(a^8)^(3/4)*sqrt(a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + sqrt(2)*(a^8)^(1/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^8)))/a^8) - 4*sqrt(2)*(a^8)^(1/4)*x^2*arctan((a^8 - sqrt(2)*(a^8)^(3/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(2)*(a^8)^(3/4)*sqrt(a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(2)*(a^8)^(1/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^8)))/a^8) + sqrt(2)*(a^8)^(1/4)*x^2*log(a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + sqrt(2)*(a^8)^(1/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^8)) - sqrt(2)*(a^8)^(1/4)*x^2*log(a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(2)*(a^8)^(1/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^8)) - 8*(5*a^2*x^2 - a*x - 4)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/x^2","B",0
141,0,0,0,0.544960," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*x^m,x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} x^{2} + 2 \, a x + 1\right)} x^{m}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral((a^2*x^2 + 2*a*x + 1)*x^m/(a^2*x^2 - 2*a*x + 1), x)","F",0
142,0,0,0,0.490966," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} x^{m}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*x^m/(a^2*x^2 - 2*a*x + 1), x)","F",0
143,0,0,0,0.917628," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x + 1\right)} x^{m}}{a x - 1}, x\right)"," ",0,"integral(-(a*x + 1)*x^m/(a*x - 1), x)","F",0
144,0,0,0,0.544336," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} x^{m}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*x^m/(a*x - 1), x)","F",0
145,0,0,0,2.421284," ","integrate(x^m/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} x^{m}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*x^m/(a*x + 1), x)","F",0
146,0,0,0,0.458329," ","integrate(x^m/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x - 1\right)} x^{m}}{a x + 1}, x\right)"," ",0,"integral(-(a*x - 1)*x^m/(a*x + 1), x)","F",0
147,0,0,0,0.559114," ","integrate(x^m/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)} x^{m}}{a^{2} x^{2} + 2 \, a x + 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(a*x - 1)*x^m/(a^2*x^2 + 2*a*x + 1), x)","F",0
148,0,0,0,0.697364," ","integrate(exp(n*arctanh(a*x))*x^m,x, algorithm=""fricas"")","{\rm integral}\left(x^{m} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(x^m*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
149,0,0,0,0.825980," ","integrate(exp(n*arctanh(a*x))*x^3,x, algorithm=""fricas"")","{\rm integral}\left(x^{3} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(x^3*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
150,0,0,0,0.777480," ","integrate(exp(n*arctanh(a*x))*x^2,x, algorithm=""fricas"")","{\rm integral}\left(x^{2} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(x^2*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
151,0,0,0,0.698337," ","integrate(exp(n*arctanh(a*x))*x,x, algorithm=""fricas"")","{\rm integral}\left(x \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(x*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
152,0,0,0,0.722514," ","integrate(exp(n*arctanh(a*x)),x, algorithm=""fricas"")","{\rm integral}\left(\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
153,0,0,0,0.922344," ","integrate(exp(n*arctanh(a*x))/x,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{x}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n)/x, x)","F",0
154,0,0,0,0.900663," ","integrate(exp(n*arctanh(a*x))/x^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{x^{2}}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n)/x^2, x)","F",0
155,0,0,0,0.510777," ","integrate(exp(n*arctanh(a*x))/x^3,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{x^{3}}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n)/x^3, x)","F",0
156,0,0,0,0.511865," ","integrate(exp(n*arctanh(a*x))/x^4,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{x^{4}}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n)/x^4, x)","F",0
157,0,0,0,0.576078," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a c x + c\right)}^{p}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a*c*x + c)^p/(a*x - 1), x)","F",0
158,1,92,0,0.749266," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{210 \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(24 \, a^{4} c^{4} x^{4} - 90 \, a^{3} c^{4} x^{3} + 112 \, a^{2} c^{4} x^{2} - 15 \, a c^{4} x - 136 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, a}"," ",0,"-1/120*(210*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c^4*x^4 - 90*a^3*c^4*x^3 + 112*a^2*c^4*x^2 - 15*a*c^4*x - 136*c^4)*sqrt(-a^2*x^2 + 1))/a","A",0
159,1,82,0,0.681765," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3,x, algorithm=""fricas"")","-\frac{30 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(6 \, a^{3} c^{3} x^{3} - 16 \, a^{2} c^{3} x^{2} + 9 \, a c^{3} x + 16 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{24 \, a}"," ",0,"-1/24*(30*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (6*a^3*c^3*x^3 - 16*a^2*c^3*x^2 + 9*a*c^3*x + 16*c^3)*sqrt(-a^2*x^2 + 1))/a","A",0
160,1,70,0,1.171642," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{6 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{2} c^{2} x^{2} - 3 \, a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a}"," ",0,"-1/6*(6*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^2*c^2*x^2 - 3*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1))/a","A",0
161,1,47,0,0.727714," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} x^{2} + 1} a c x - 2 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{2 \, a}"," ",0,"1/2*(sqrt(-a^2*x^2 + 1)*a*c*x - 2*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a","A",0
162,1,62,0,0.787275," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c),x, algorithm=""fricas"")","\frac{2 \, {\left(a x + {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a^{2} c x - a c}"," ",0,"2*(a*x + (a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1) - 1)/(a^2*c*x - a*c)","A",0
163,1,60,0,0.423353," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{a^{2} x^{2} - 2 \, a x + \sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} + 1}{3 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"1/3*(a^2*x^2 - 2*a*x + sqrt(-a^2*x^2 + 1)*(a*x + 1) + 1)/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)","B",0
164,1,89,0,0.551048," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{4 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 12 \, a x + {\left(a^{2} x^{2} - 3 \, a x - 4\right)} \sqrt{-a^{2} x^{2} + 1} - 4}{15 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"1/15*(4*a^3*x^3 - 12*a^2*x^2 + 12*a*x + (a^2*x^2 - 3*a*x - 4)*sqrt(-a^2*x^2 + 1) - 4)/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)","A",0
165,1,116,0,0.440886," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{23 \, a^{4} x^{4} - 92 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 92 \, a x + {\left(2 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 13 \, a x + 23\right)} \sqrt{-a^{2} x^{2} + 1} + 23}{105 \, {\left(a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"1/105*(23*a^4*x^4 - 92*a^3*x^3 + 138*a^2*x^2 - 92*a*x + (2*a^3*x^3 - 8*a^2*x^2 + 13*a*x + 23)*sqrt(-a^2*x^2 + 1) + 23)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)","A",0
166,1,144,0,0.762287," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^5,x, algorithm=""fricas"")","\frac{58 \, a^{5} x^{5} - 290 \, a^{4} x^{4} + 580 \, a^{3} x^{3} - 580 \, a^{2} x^{2} + 290 \, a x + {\left(2 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 25 \, a x - 58\right)} \sqrt{-a^{2} x^{2} + 1} - 58}{315 \, {\left(a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\right)}}"," ",0,"1/315*(58*a^5*x^5 - 290*a^4*x^4 + 580*a^3*x^3 - 580*a^2*x^2 + 290*a*x + (2*a^4*x^4 - 10*a^3*x^3 + 21*a^2*x^2 - 25*a*x - 58)*sqrt(-a^2*x^2 + 1) - 58)/(a^6*c^5*x^5 - 5*a^5*c^5*x^4 + 10*a^4*c^5*x^3 - 10*a^3*c^5*x^2 + 5*a^2*c^5*x - a*c^5)","A",0
167,1,29,0,0.516115," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^p,x, algorithm=""fricas"")","-\frac{{\left(a p x + p + 2\right)} {\left(-a c x + c\right)}^{p}}{a p^{2} + a p}"," ",0,"-(a*p*x + p + 2)*(-a*c*x + c)^p/(a*p^2 + a*p)","A",0
168,1,59,0,0.622043," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^5,x, algorithm=""fricas"")","\frac{1}{6} \, a^{5} c^{5} x^{6} - \frac{3}{5} \, a^{4} c^{5} x^{5} + \frac{1}{2} \, a^{3} c^{5} x^{4} + \frac{2}{3} \, a^{2} c^{5} x^{3} - \frac{3}{2} \, a c^{5} x^{2} + c^{5} x"," ",0,"1/6*a^5*c^5*x^6 - 3/5*a^4*c^5*x^5 + 1/2*a^3*c^5*x^4 + 2/3*a^2*c^5*x^3 - 3/2*a*c^5*x^2 + c^5*x","A",0
169,1,37,0,0.773452," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{1}{5} \, a^{4} c^{4} x^{5} + \frac{1}{2} \, a^{3} c^{4} x^{4} - a c^{4} x^{2} + c^{4} x"," ",0,"-1/5*a^4*c^4*x^5 + 1/2*a^3*c^4*x^4 - a*c^4*x^2 + c^4*x","A",0
170,1,37,0,0.636540," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{1}{4} \, a^{3} c^{3} x^{4} - \frac{1}{3} \, a^{2} c^{3} x^{3} - \frac{1}{2} \, a c^{3} x^{2} + c^{3} x"," ",0,"1/4*a^3*c^3*x^4 - 1/3*a^2*c^3*x^3 - 1/2*a*c^3*x^2 + c^3*x","A",0
171,1,17,0,0.620823," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{1}{3} \, a^{2} c^{2} x^{3} + c^{2} x"," ",0,"-1/3*a^2*c^2*x^3 + c^2*x","A",0
172,1,11,0,0.497608," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c),x, algorithm=""fricas"")","\frac{1}{2} \, a c x^{2} + c x"," ",0,"1/2*a*c*x^2 + c*x","A",0
173,1,28,0,0.460413," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c),x, algorithm=""fricas"")","\frac{{\left(a x - 1\right)} \log\left(a x - 1\right) - 2}{a^{2} c x - a c}"," ",0,"((a*x - 1)*log(a*x - 1) - 2)/(a^2*c*x - a*c)","A",0
174,1,25,0,0.491808," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}"," ",0,"x/(a^2*c^2*x^2 - 2*a*c^2*x + c^2)","B",0
175,1,47,0,0.587757," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^3,x, algorithm=""fricas"")","-\frac{3 \, a x + 1}{6 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/6*(3*a*x + 1)/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)","A",0
176,1,57,0,0.457278," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{2 \, a x + 1}{6 \, {\left(a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"1/6*(2*a*x + 1)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)","A",0
177,0,0,0,0.589055," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} {\left(-a c x + c\right)}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*(-a*c*x + c)^p/(a^2*x^2 - 2*a*x + 1), x)","F",0
178,1,93,0,0.602772," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{30 \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(8 \, a^{4} c^{4} x^{4} - 10 \, a^{3} c^{4} x^{3} - 16 \, a^{2} c^{4} x^{2} + 25 \, a c^{4} x + 8 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{40 \, a}"," ",0,"-1/40*(30*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (8*a^4*c^4*x^4 - 10*a^3*c^4*x^3 - 16*a^2*c^4*x^2 + 25*a*c^4*x + 8*c^4)*sqrt(-a^2*x^2 + 1))/a","A",0
179,1,65,0,0.505593," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^3,x, algorithm=""fricas"")","-\frac{6 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{3} c^{3} x^{3} - 5 \, a c^{3} x\right)} \sqrt{-a^{2} x^{2} + 1}}{8 \, a}"," ",0,"-1/8*(6*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*c^3*x^3 - 5*a*c^3*x)*sqrt(-a^2*x^2 + 1))/a","A",0
180,1,71,0,0.418472," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{6 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(2 \, a^{2} c^{2} x^{2} + 3 \, a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a}"," ",0,"-1/6*(6*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^2*c^2*x^2 + 3*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1))/a","A",0
181,1,52,0,0.614201," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c),x, algorithm=""fricas"")","-\frac{6 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a c x + 4 \, c\right)}}{2 \, a}"," ",0,"-1/2*(6*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*c*x + 4*c))/a","A",0
182,1,94,0,0.435886," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c),x, algorithm=""fricas"")","-\frac{2 \, {\left(2 \, a^{2} x^{2} - 4 \, a x + 3 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} {\left(2 \, a x - 1\right)} + 2\right)}}{3 \, {\left(a^{3} c x^{2} - 2 \, a^{2} c x + a c\right)}}"," ",0,"-2/3*(2*a^2*x^2 - 4*a*x + 3*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 2*sqrt(-a^2*x^2 + 1)*(2*a*x - 1) + 2)/(a^3*c*x^2 - 2*a^2*c*x + a*c)","A",0
183,1,89,0,0.527532," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - {\left(a^{2} x^{2} + 2 \, a x + 1\right)} \sqrt{-a^{2} x^{2} + 1} - 1}{5 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}"," ",0,"1/5*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - (a^2*x^2 + 2*a*x + 1)*sqrt(-a^2*x^2 + 1) - 1)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)","B",0
184,1,116,0,0.772645," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{6 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 36 \, a^{2} x^{2} - 24 \, a x - {\left(a^{3} x^{3} - 4 \, a^{2} x^{2} - 11 \, a x - 6\right)} \sqrt{-a^{2} x^{2} + 1} + 6}{35 \, {\left(a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"1/35*(6*a^4*x^4 - 24*a^3*x^3 + 36*a^2*x^2 - 24*a*x - (a^3*x^3 - 4*a^2*x^2 - 11*a*x - 6)*sqrt(-a^2*x^2 + 1) + 6)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)","B",0
185,1,145,0,0.648004," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{47 \, a^{5} x^{5} - 235 \, a^{4} x^{4} + 470 \, a^{3} x^{3} - 470 \, a^{2} x^{2} + 235 \, a x - {\left(2 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 80 \, a x + 47\right)} \sqrt{-a^{2} x^{2} + 1} - 47}{315 \, {\left(a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"1/315*(47*a^5*x^5 - 235*a^4*x^4 + 470*a^3*x^3 - 470*a^2*x^2 + 235*a*x - (2*a^4*x^4 - 10*a^3*x^3 + 21*a^2*x^2 + 80*a*x + 47)*sqrt(-a^2*x^2 + 1) - 47)/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4)","A",0
186,1,171,0,0.690117," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^5,x, algorithm=""fricas"")","\frac{152 \, a^{6} x^{6} - 912 \, a^{5} x^{5} + 2280 \, a^{4} x^{4} - 3040 \, a^{3} x^{3} + 2280 \, a^{2} x^{2} - 912 \, a x - {\left(2 \, a^{5} x^{5} - 12 \, a^{4} x^{4} + 31 \, a^{3} x^{3} - 46 \, a^{2} x^{2} - 243 \, a x - 152\right)} \sqrt{-a^{2} x^{2} + 1} + 152}{1155 \, {\left(a^{7} c^{5} x^{6} - 6 \, a^{6} c^{5} x^{5} + 15 \, a^{5} c^{5} x^{4} - 20 \, a^{4} c^{5} x^{3} + 15 \, a^{3} c^{5} x^{2} - 6 \, a^{2} c^{5} x + a c^{5}\right)}}"," ",0,"1/1155*(152*a^6*x^6 - 912*a^5*x^5 + 2280*a^4*x^4 - 3040*a^3*x^3 + 2280*a^2*x^2 - 912*a*x - (2*a^5*x^5 - 12*a^4*x^4 + 31*a^3*x^3 - 46*a^2*x^2 - 243*a*x - 152)*sqrt(-a^2*x^2 + 1) + 152)/(a^7*c^5*x^6 - 6*a^6*c^5*x^5 + 15*a^5*c^5*x^4 - 20*a^4*c^5*x^3 + 15*a^3*c^5*x^2 - 6*a^2*c^5*x + a*c^5)","A",0
187,1,81,0,0.695383," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^p,x, algorithm=""fricas"")","-\frac{{\left({\left(a^{2} p^{2} - a^{2} p\right)} x^{2} + p^{2} + 2 \, {\left(a p^{2} + a p - 2 \, a\right)} x + 3 \, p + 4\right)} {\left(-a c x + c\right)}^{p}}{a p^{3} - a p - {\left(a^{2} p^{3} - a^{2} p\right)} x}"," ",0,"-((a^2*p^2 - a^2*p)*x^2 + p^2 + 2*(a*p^2 + a*p - 2*a)*x + 3*p + 4)*(-a*c*x + c)^p/(a*p^3 - a*p - (a^2*p^3 - a^2*p)*x)","A",0
188,1,59,0,0.518709," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^5,x, algorithm=""fricas"")","-\frac{1}{6} \, a^{5} c^{5} x^{6} + \frac{1}{5} \, a^{4} c^{5} x^{5} + \frac{1}{2} \, a^{3} c^{5} x^{4} - \frac{2}{3} \, a^{2} c^{5} x^{3} - \frac{1}{2} \, a c^{5} x^{2} + c^{5} x"," ",0,"-1/6*a^5*c^5*x^6 + 1/5*a^4*c^5*x^5 + 1/2*a^3*c^5*x^4 - 2/3*a^2*c^5*x^3 - 1/2*a*c^5*x^2 + c^5*x","A",0
189,1,28,0,0.514002," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{1}{5} \, a^{4} c^{4} x^{5} - \frac{2}{3} \, a^{2} c^{4} x^{3} + c^{4} x"," ",0,"1/5*a^4*c^4*x^5 - 2/3*a^2*c^4*x^3 + c^4*x","A",0
190,1,37,0,1.242738," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^3,x, algorithm=""fricas"")","-\frac{1}{4} \, a^{3} c^{3} x^{4} - \frac{1}{3} \, a^{2} c^{3} x^{3} + \frac{1}{2} \, a c^{3} x^{2} + c^{3} x"," ",0,"-1/4*a^3*c^3*x^4 - 1/3*a^2*c^3*x^3 + 1/2*a*c^3*x^2 + c^3*x","A",0
191,1,25,0,0.420745," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x"," ",0,"1/3*a^2*c^2*x^3 + a*c^2*x^2 + c^2*x","A",0
192,1,28,0,0.536018," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c),x, algorithm=""fricas"")","-\frac{a^{2} c x^{2} + 6 \, a c x + 8 \, c \log\left(a x - 1\right)}{2 \, a}"," ",0,"-1/2*(a^2*c*x^2 + 6*a*c*x + 8*c*log(a*x - 1))/a","A",0
193,1,49,0,0.554994," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a*c*x+c),x, algorithm=""fricas"")","\frac{4 \, a x - {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) - 2}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}"," ",0,"(4*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x - 1) - 2)/(a^3*c*x^2 - 2*a^2*c*x + a*c)","A",0
194,1,51,0,0.543641," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{3 \, a^{2} x^{2} + 1}{3 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}"," ",0,"-1/3*(3*a^2*x^2 + 1)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)","B",0
195,1,65,0,0.696203," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{3 \, a^{2} x^{2} + 2 \, a x + 1}{6 \, {\left(a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"1/6*(3*a^2*x^2 + 2*a*x + 1)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)","A",0
196,1,77,0,0.460400," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{5 \, a^{2} x^{2} + 5 \, a x + 2}{15 \, {\left(a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"-1/15*(5*a^2*x^2 + 5*a*x + 2)/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4)","A",0
197,0,0,0,0.679651," ","integrate((-a*c*x+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a c x + c\right)}^{p}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a*c*x + c)^p/(a*x + 1), x)","F",0
198,1,81,0,0.745227," ","integrate((-a*c*x+c)^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{210 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(6 \, a^{3} c^{3} x^{3} - 32 \, a^{2} c^{3} x^{2} + 81 \, a c^{3} x - 160 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{24 \, a}"," ",0,"-1/24*(210*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (6*a^3*c^3*x^3 - 32*a^2*c^3*x^2 + 81*a*c^3*x - 160*c^3)*sqrt(-a^2*x^2 + 1))/a","A",0
199,1,71,0,0.521054," ","integrate((-a*c*x+c)^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{30 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(2 \, a^{2} c^{2} x^{2} - 9 \, a c^{2} x + 22 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a}"," ",0,"-1/6*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^2*c^2*x^2 - 9*a*c^2*x + 22*c^2)*sqrt(-a^2*x^2 + 1))/a","A",0
200,1,52,0,1.429870," ","integrate((-a*c*x+c)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{6 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a c x - 4 \, c\right)}}{2 \, a}"," ",0,"-1/2*(6*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*c*x - 4*c))/a","A",0
201,1,30,0,1.319074," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c),x, algorithm=""fricas"")","-\frac{2 \, \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{a c}"," ",0,"-2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x))/(a*c)","B",0
202,1,37,0,0.505621," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{a x - \sqrt{-a^{2} x^{2} + 1} - 1}{a^{2} c^{2} x - a c^{2}}"," ",0,"(a*x - sqrt(-a^2*x^2 + 1) - 1)/(a^2*c^2*x - a*c^2)","A",0
203,1,62,0,0.422464," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{2 \, a^{2} x^{2} - 4 \, a x - \sqrt{-a^{2} x^{2} + 1} {\left(a x - 2\right)} + 2}{3 \, {\left(a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"1/3*(2*a^2*x^2 - 4*a*x - sqrt(-a^2*x^2 + 1)*(a*x - 2) + 2)/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3)","A",0
204,1,91,0,0.590185," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{7 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 21 \, a x - {\left(2 \, a^{2} x^{2} - 6 \, a x + 7\right)} \sqrt{-a^{2} x^{2} + 1} - 7}{15 \, {\left(a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"1/15*(7*a^3*x^3 - 21*a^2*x^2 + 21*a*x - (2*a^2*x^2 - 6*a*x + 7)*sqrt(-a^2*x^2 + 1) - 7)/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)","A",0
205,1,117,0,0.628590," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^5,x, algorithm=""fricas"")","\frac{12 \, a^{4} x^{4} - 48 \, a^{3} x^{3} + 72 \, a^{2} x^{2} - 48 \, a x - {\left(2 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 13 \, a x - 12\right)} \sqrt{-a^{2} x^{2} + 1} + 12}{35 \, {\left(a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 6 \, a^{3} c^{5} x^{2} - 4 \, a^{2} c^{5} x + a c^{5}\right)}}"," ",0,"1/35*(12*a^4*x^4 - 48*a^3*x^3 + 72*a^2*x^2 - 48*a*x - (2*a^3*x^3 - 8*a^2*x^2 + 13*a*x - 12)*sqrt(-a^2*x^2 + 1) + 12)/(a^5*c^5*x^4 - 4*a^4*c^5*x^3 + 6*a^3*c^5*x^2 - 4*a^2*c^5*x + a*c^5)","A",0
206,0,0,0,0.466721," ","integrate((-a*c*x+c)^p/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x - 1\right)} {\left(-a c x + c\right)}^{p}}{a x + 1}, x\right)"," ",0,"integral(-(a*x - 1)*(-a*c*x + c)^p/(a*x + 1), x)","F",0
207,1,68,0,0.575075," ","integrate((-a*c*x+c)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{6 \, a^{5} c^{4} x^{5} - 45 \, a^{4} c^{4} x^{4} + 160 \, a^{3} c^{4} x^{3} - 390 \, a^{2} c^{4} x^{2} + 930 \, a c^{4} x - 960 \, c^{4} \log\left(a x + 1\right)}{30 \, a}"," ",0,"-1/30*(6*a^5*c^4*x^5 - 45*a^4*c^4*x^4 + 160*a^3*c^4*x^3 - 390*a^2*c^4*x^2 + 930*a*c^4*x - 960*c^4*log(a*x + 1))/a","A",0
208,1,57,0,0.602194," ","integrate((-a*c*x+c)^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\frac{3 \, a^{4} c^{3} x^{4} - 20 \, a^{3} c^{3} x^{3} + 66 \, a^{2} c^{3} x^{2} - 180 \, a c^{3} x + 192 \, c^{3} \log\left(a x + 1\right)}{12 \, a}"," ",0,"1/12*(3*a^4*c^3*x^4 - 20*a^3*c^3*x^3 + 66*a^2*c^3*x^2 - 180*a*c^3*x + 192*c^3*log(a*x + 1))/a","A",0
209,1,45,0,0.441721," ","integrate((-a*c*x+c)^2/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{a^{3} c^{2} x^{3} - 6 \, a^{2} c^{2} x^{2} + 21 \, a c^{2} x - 24 \, c^{2} \log\left(a x + 1\right)}{3 \, a}"," ",0,"-1/3*(a^3*c^2*x^3 - 6*a^2*c^2*x^2 + 21*a*c^2*x - 24*c^2*log(a*x + 1))/a","A",0
210,1,28,0,0.466666," ","integrate((-a*c*x+c)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\frac{a^{2} c x^{2} - 6 \, a c x + 8 \, c \log\left(a x + 1\right)}{2 \, a}"," ",0,"1/2*(a^2*c*x^2 - 6*a*c*x + 8*c*log(a*x + 1))/a","A",0
211,1,13,0,0.503159," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c),x, algorithm=""fricas"")","\frac{\log\left(a x + 1\right)}{a c}"," ",0,"log(a*x + 1)/(a*c)","A",0
212,1,23,0,0.487286," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{\log\left(a x + 1\right) - \log\left(a x - 1\right)}{2 \, a c^{2}}"," ",0,"1/2*(log(a*x + 1) - log(a*x - 1))/(a*c^2)","B",0
213,1,46,0,0.460736," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{{\left(a x - 1\right)} \log\left(a x + 1\right) - {\left(a x - 1\right)} \log\left(a x - 1\right) - 2}{4 \, {\left(a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"1/4*((a*x - 1)*log(a*x + 1) - (a*x - 1)*log(a*x - 1) - 2)/(a^2*c^3*x - a*c^3)","A",0
214,1,76,0,0.520140," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{2 \, a x - {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) + {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) - 4}{8 \, {\left(a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/8*(2*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + (a^2*x^2 - 2*a*x + 1)*log(a*x - 1) - 4)/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)","A",0
215,1,113,0,0.514897," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^5,x, algorithm=""fricas"")","-\frac{6 \, a^{2} x^{2} - 18 \, a x - 3 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \log\left(a x + 1\right) + 3 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \log\left(a x - 1\right) + 20}{48 \, {\left(a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right)}}"," ",0,"-1/48*(6*a^2*x^2 - 18*a*x - 3*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(a*x + 1) + 3*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(a*x - 1) + 20)/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5)","A",0
216,0,0,0,0.562624," ","integrate((-a*c*x+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)} {\left(-a c x + c\right)}^{p}}{a^{2} x^{2} + 2 \, a x + 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(a*x - 1)*(-a*c*x + c)^p/(a^2*x^2 + 2*a*x + 1), x)","F",0
217,1,118,0,0.512652," ","integrate((-a*c*x+c)^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{496 \, a c^{3} x + 496 \, c^{3} - 630 \, {\left(a c^{3} x + c^{3}\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(2 \, a^{4} c^{3} x^{4} - 14 \, a^{3} c^{3} x^{3} + 51 \, a^{2} c^{3} x^{2} - 173 \, a c^{3} x - 496 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{8 \, {\left(a^{2} x + a\right)}}"," ",0,"-1/8*(496*a*c^3*x + 496*c^3 - 630*(a*c^3*x + c^3)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^4*c^3*x^4 - 14*a^3*c^3*x^3 + 51*a^2*c^3*x^2 - 173*a*c^3*x - 496*c^3)*sqrt(-a^2*x^2 + 1))/(a^2*x + a)","A",0
218,1,106,0,0.499598," ","integrate((-a*c*x+c)^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{166 \, a c^{2} x + 166 \, c^{2} - 210 \, {\left(a c^{2} x + c^{2}\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{3} c^{2} x^{3} - 13 \, a^{2} c^{2} x^{2} + 55 \, a c^{2} x + 166 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{2} x + a\right)}}"," ",0,"-1/6*(166*a*c^2*x + 166*c^2 - 210*(a*c^2*x + c^2)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*c^2*x^3 - 13*a^2*c^2*x^2 + 55*a*c^2*x + 166*c^2)*sqrt(-a^2*x^2 + 1))/(a^2*x + a)","A",0
219,1,81,0,0.452710," ","integrate((-a*c*x+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{24 \, a c x - 30 \, {\left(a c x + c\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(a^{2} c x^{2} - 7 \, a c x - 24 \, c\right)} \sqrt{-a^{2} x^{2} + 1} + 24 \, c}{2 \, {\left(a^{2} x + a\right)}}"," ",0,"-1/2*(24*a*c*x - 30*(a*c*x + c)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (a^2*c*x^2 - 7*a*c*x - 24*c)*sqrt(-a^2*x^2 + 1) + 24*c)/(a^2*x + a)","A",0
220,1,60,0,0.460675," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c),x, algorithm=""fricas"")","-\frac{2 \, {\left(a x - {\left(a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} + 1\right)}}{a^{2} c x + a c}"," ",0,"-2*(a*x - (a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1) + 1)/(a^2*c*x + a*c)","A",0
221,1,35,0,0.436552," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{a x + \sqrt{-a^{2} x^{2} + 1} + 1}{a^{2} c^{2} x + a c^{2}}"," ",0,"-(a*x + sqrt(-a^2*x^2 + 1) + 1)/(a^2*c^2*x + a*c^2)","A",0
222,1,33,0,0.435220," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^3,x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} x^{2} + 1} x}{a^{2} c^{3} x^{2} - c^{3}}"," ",0,"-sqrt(-a^2*x^2 + 1)*x/(a^2*c^3*x^2 - c^3)","A",0
223,1,89,0,0.534036," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{a^{3} x^{3} - a^{2} x^{2} - a x - {\left(2 \, a^{2} x^{2} - 2 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1} + 1}{3 \, {\left(a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"1/3*(a^3*x^3 - a^2*x^2 - a*x - (2*a^2*x^2 - 2*a*x - 1)*sqrt(-a^2*x^2 + 1) + 1)/(a^4*c^4*x^3 - a^3*c^4*x^2 - a^2*c^4*x + a*c^4)","A",0
224,1,98,0,0.451522," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^5,x, algorithm=""fricas"")","\frac{2 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 4 \, a x - {\left(2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} - 2}{5 \, {\left(a^{5} c^{5} x^{4} - 2 \, a^{4} c^{5} x^{3} + 2 \, a^{2} c^{5} x - a c^{5}\right)}}"," ",0,"1/5*(2*a^4*x^4 - 4*a^3*x^3 + 4*a*x - (2*a^3*x^3 - 4*a^2*x^2 + a*x + 2)*sqrt(-a^2*x^2 + 1) - 2)/(a^5*c^5*x^4 - 2*a^4*c^5*x^3 + 2*a^2*c^5*x - a*c^5)","A",0
225,1,144,0,0.515122," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^6,x, algorithm=""fricas"")","\frac{13 \, a^{5} x^{5} - 39 \, a^{4} x^{4} + 26 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 39 \, a x - {\left(8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right)} \sqrt{-a^{2} x^{2} + 1} + 13}{35 \, {\left(a^{6} c^{6} x^{5} - 3 \, a^{5} c^{6} x^{4} + 2 \, a^{4} c^{6} x^{3} + 2 \, a^{3} c^{6} x^{2} - 3 \, a^{2} c^{6} x + a c^{6}\right)}}"," ",0,"1/35*(13*a^5*x^5 - 39*a^4*x^4 + 26*a^3*x^3 + 26*a^2*x^2 - 39*a*x - (8*a^4*x^4 - 24*a^3*x^3 + 20*a^2*x^2 + 4*a*x - 13)*sqrt(-a^2*x^2 + 1) + 13)/(a^6*c^6*x^5 - 3*a^5*c^6*x^4 + 2*a^4*c^6*x^3 + 2*a^3*c^6*x^2 - 3*a^2*c^6*x + a*c^6)","A",0
226,1,91,0,0.477831," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(9/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(315 \, a^{5} c^{4} x^{5} - 1505 \, a^{4} c^{4} x^{4} + 2710 \, a^{3} c^{4} x^{3} - 1866 \, a^{2} c^{4} x^{2} - 977 \, a c^{4} x + 5419 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{3465 \, {\left(a^{2} x - a\right)}}"," ",0,"-2/3465*(315*a^5*c^4*x^5 - 1505*a^4*c^4*x^4 + 2710*a^3*c^4*x^3 - 1866*a^2*c^4*x^2 - 977*a*c^4*x + 5419*c^4)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
227,1,80,0,0.446825," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","\frac{2 \, {\left(35 \, a^{4} c^{3} x^{4} - 130 \, a^{3} c^{3} x^{3} + 156 \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x - 319 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{315 \, {\left(a^{2} x - a\right)}}"," ",0,"2/315*(35*a^4*c^3*x^4 - 130*a^3*c^3*x^3 + 156*a^2*c^3*x^2 + 2*a*c^3*x - 319*c^3)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
228,1,69,0,0.625741," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(15 \, a^{3} c^{2} x^{3} - 39 \, a^{2} c^{2} x^{2} + 17 \, a c^{2} x + 71 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{105 \, {\left(a^{2} x - a\right)}}"," ",0,"-2/105*(15*a^3*c^2*x^3 - 39*a^2*c^2*x^2 + 17*a*c^2*x + 71*c^2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
229,1,52,0,0.453391," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, a^{2} c x^{2} - 4 \, a c x - 7 \, c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{15 \, {\left(a^{2} x - a\right)}}"," ",0,"2/15*(3*a^2*c*x^2 - 4*a*c*x - 7*c)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
230,1,39,0,0.728041," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 1\right)}}{3 \, {\left(a^{2} x - a\right)}}"," ",0,"-2/3*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 1)/(a^2*x - a)","A",0
231,1,215,0,0.497838," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{\frac{\sqrt{2} {\left(a c x - c\right)} \log\left(-\frac{a^{2} x^{2} + 2 \, a x - \frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{\sqrt{c}} - 3}{a^{2} x^{2} - 2 \, a x + 1}\right)}{\sqrt{c}} + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a^{2} c x - a c}, \frac{2 \, {\left(\sqrt{2} {\left(a c x - c\right)} \sqrt{-\frac{1}{c}} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-\frac{1}{c}}}{a^{2} x^{2} - 1}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right)}}{a^{2} c x - a c}\right]"," ",0,"[(sqrt(2)*(a*c*x - c)*log(-(a^2*x^2 + 2*a*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/sqrt(c) - 3)/(a^2*x^2 - 2*a*x + 1))/sqrt(c) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*c*x - a*c), 2*(sqrt(2)*(a*c*x - c)*sqrt(-1/c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-1/c)/(a^2*x^2 - 1)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*c*x - a*c)]","A",0
232,1,258,0,0.674667," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{4 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}, -\frac{\sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{2 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}\right]"," ",0,"[1/4*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), -1/2*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]","A",0
233,1,308,0,0.479036," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) - 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 3\right)}}{32 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}, -\frac{\sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 3\right)}}{16 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}\right]"," ",0,"[1/32*(sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) - 4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 3))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3), -1/16*(sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 3))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)]","A",0
234,1,364,0,0.706886," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{2} {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) - 4 \, {\left(3 \, a^{2} x^{2} - 10 \, a x - 25\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{384 \, {\left(a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right)}}, -\frac{3 \, \sqrt{2} {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 2 \, {\left(3 \, a^{2} x^{2} - 10 \, a x - 25\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{192 \, {\left(a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right)}}\right]"," ",0,"[1/384*(3*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) - 4*(3*a^2*x^2 - 10*a*x - 25)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4), -1/192*(3*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 2*(3*a^2*x^2 - 10*a*x - 25)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)]","A",0
235,1,60,0,0.743926," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","\frac{2 \, {\left(7 \, a^{4} c^{3} x^{4} - 10 \, a^{3} c^{3} x^{3} - 12 \, a^{2} c^{3} x^{2} + 26 \, a c^{3} x - 11 \, c^{3}\right)} \sqrt{-a c x + c}}{63 \, a}"," ",0,"2/63*(7*a^4*c^3*x^4 - 10*a^3*c^3*x^3 - 12*a^2*c^3*x^2 + 26*a*c^3*x - 11*c^3)*sqrt(-a*c*x + c)/a","A",0
236,1,49,0,0.586475," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(5 \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - 13 \, a c^{2} x + 9 \, c^{2}\right)} \sqrt{-a c x + c}}{35 \, a}"," ",0,"-2/35*(5*a^3*c^2*x^3 - a^2*c^2*x^2 - 13*a*c^2*x + 9*c^2)*sqrt(-a*c*x + c)/a","A",0
237,1,32,0,0.742954," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, a^{2} c x^{2} + 4 \, a c x - 7 \, c\right)} \sqrt{-a c x + c}}{15 \, a}"," ",0,"2/15*(3*a^2*c*x^2 + 4*a*c*x - 7*c)*sqrt(-a*c*x + c)/a","A",0
238,1,19,0,0.475246," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-a c x + c} {\left(a x + 5\right)}}{3 \, a}"," ",0,"-2/3*sqrt(-a*c*x + c)*(a*x + 5)/a","A",0
239,1,29,0,0.510346," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","\frac{2 \, \sqrt{-a c x + c} {\left(a x - 3\right)}}{a^{2} c x - a c}"," ",0,"2*sqrt(-a*c*x + c)*(a*x - 3)/(a^2*c*x - a*c)","A",0
240,1,44,0,0.560532," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","\frac{2 \, \sqrt{-a c x + c} {\left(3 \, a x - 1\right)}}{3 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"2/3*sqrt(-a*c*x + c)*(3*a*x - 1)/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)","A",0
241,1,56,0,0.415538," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-a c x + c} {\left(5 \, a x + 1\right)}}{15 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-2/15*sqrt(-a*c*x + c)*(5*a*x + 1)/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)","A",0
242,1,66,0,0.417207," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","\frac{2 \, \sqrt{-a c x + c} {\left(7 \, a x + 3\right)}}{35 \, {\left(a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"2/35*sqrt(-a*c*x + c)*(7*a*x + 3)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)","B",0
243,1,91,0,0.419038," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(9/2),x, algorithm=""fricas"")","\frac{2 \, {\left(105 \, a^{5} c^{4} x^{5} - 245 \, a^{4} c^{4} x^{4} - 50 \, a^{3} c^{4} x^{3} + 522 \, a^{2} c^{4} x^{2} - 311 \, a c^{4} x - 533 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{1155 \, {\left(a^{2} x - a\right)}}"," ",0,"2/1155*(105*a^5*c^4*x^5 - 245*a^4*c^4*x^4 - 50*a^3*c^4*x^3 + 522*a^2*c^4*x^2 - 311*a*c^4*x - 533*c^4)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
244,1,80,0,0.475396," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(35 \, a^{4} c^{3} x^{4} - 40 \, a^{3} c^{3} x^{3} - 78 \, a^{2} c^{3} x^{2} + 104 \, a c^{3} x + 107 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{315 \, {\left(a^{2} x - a\right)}}"," ",0,"-2/315*(35*a^4*c^3*x^4 - 40*a^3*c^3*x^3 - 78*a^2*c^3*x^2 + 104*a*c^3*x + 107*c^3)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
245,1,68,0,0.421107," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","\frac{2 \, {\left(5 \, a^{3} c^{2} x^{3} + a^{2} c^{2} x^{2} - 13 \, a c^{2} x - 9 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{35 \, {\left(a^{2} x - a\right)}}"," ",0,"2/35*(5*a^3*c^2*x^3 + a^2*c^2*x^2 - 13*a*c^2*x - 9*c^2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
246,1,49,0,0.545526," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a^{2} c x^{2} + 2 \, a c x + c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{5 \, {\left(a^{2} x - a\right)}}"," ",0,"-2/5*(a^2*c*x^2 + 2*a*c*x + c)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
247,1,220,0,0.562180," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(3 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 7\right)}\right)}}{3 \, {\left(a^{2} x - a\right)}}, \frac{2 \, {\left(6 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 7\right)}\right)}}{3 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[2/3*(3*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 7))/(a^2*x - a), 2/3*(6*sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 7))/(a^2*x - a)]","A",0
248,1,259,0,0.494512," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x - 2\right)} - \frac{3 \, \sqrt{2} {\left(a^{2} c x^{2} - 2 \, a c x + c\right)} \log\left(-\frac{a^{2} x^{2} + 2 \, a x + \frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{\sqrt{c}} - 3}{a^{2} x^{2} - 2 \, a x + 1}\right)}{\sqrt{c}}}{2 \, {\left(a^{3} c x^{2} - 2 \, a^{2} c x + a c\right)}}, -\frac{3 \, \sqrt{2} {\left(a^{2} c x^{2} - 2 \, a c x + c\right)} \sqrt{-\frac{1}{c}} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-\frac{1}{c}}}{a^{2} x^{2} - 1}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x - 2\right)}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}\right]"," ",0,"[-1/2*(4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x - 2) - 3*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)*log(-(a^2*x^2 + 2*a*x + 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/sqrt(c) - 3)/(a^2*x^2 - 2*a*x + 1))/sqrt(c))/(a^3*c*x^2 - 2*a^2*c*x + a*c), -(3*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)*sqrt(-1/c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-1/c)/(a^2*x^2 - 1)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x - 2))/(a^3*c*x^2 - 2*a^2*c*x + a*c)]","A",0
249,1,312,0,0.484607," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) - 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(5 \, a x - 1\right)}}{16 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}, \frac{3 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(5 \, a x - 1\right)}}{8 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}\right]"," ",0,"[1/16*(3*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) - 4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(5*a*x - 1))/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2), 1/8*(3*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(5*a*x - 1))/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)]","A",0
250,1,364,0,0.523677," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{2} {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 4 \, {\left(3 \, a^{2} x^{2} + 22 \, a x + 7\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{192 \, {\left(a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right)}}, \frac{3 \, \sqrt{2} {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 2 \, {\left(3 \, a^{2} x^{2} + 22 \, a x + 7\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{96 \, {\left(a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right)}}\right]"," ",0,"[1/192*(3*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 4*(3*a^2*x^2 + 22*a*x + 7)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3), 1/96*(3*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 2*(3*a^2*x^2 + 22*a*x + 7)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)]","A",0
251,1,420,0,0.529667," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{2} {\left(a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 4 \, {\left(3 \, a^{3} x^{3} - 13 \, a^{2} x^{2} - 79 \, a x - 39\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{1024 \, {\left(a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right)}}, \frac{3 \, \sqrt{2} {\left(a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 2 \, {\left(3 \, a^{3} x^{3} - 13 \, a^{2} x^{2} - 79 \, a x - 39\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{512 \, {\left(a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right)}}\right]"," ",0,"[1/1024*(3*sqrt(2)*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 4*(3*a^3*x^3 - 13*a^2*x^2 - 79*a*x - 39)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4), 1/512*(3*sqrt(2)*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 2*(3*a^3*x^3 - 13*a^2*x^2 - 79*a*x - 39)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4)]","A",0
252,1,91,0,0.603828," ","integrate((-a*c*x+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{2 \, {\left(63 \, a^{5} c^{4} x^{5} - 455 \, a^{4} c^{4} x^{4} + 1510 \, a^{3} c^{4} x^{3} - 3198 \, a^{2} c^{4} x^{2} + 5419 \, a c^{4} x - 11531 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{693 \, {\left(a^{2} x - a\right)}}"," ",0,"2/693*(63*a^5*c^4*x^5 - 455*a^4*c^4*x^4 + 1510*a^3*c^4*x^3 - 3198*a^2*c^4*x^2 + 5419*a*c^4*x - 11531*c^4)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
253,1,80,0,0.408220," ","integrate((-a*c*x+c)^(7/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(35 \, a^{4} c^{3} x^{4} - 220 \, a^{3} c^{3} x^{3} + 642 \, a^{2} c^{3} x^{2} - 1276 \, a c^{3} x + 2867 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{315 \, {\left(a^{2} x - a\right)}}"," ",0,"-2/315*(35*a^4*c^3*x^4 - 220*a^3*c^3*x^3 + 642*a^2*c^3*x^2 - 1276*a*c^3*x + 2867*c^3)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
254,1,69,0,0.413636," ","integrate((-a*c*x+c)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{2 \, {\left(5 \, a^{3} c^{2} x^{3} - 27 \, a^{2} c^{2} x^{2} + 71 \, a c^{2} x - 177 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{35 \, {\left(a^{2} x - a\right)}}"," ",0,"2/35*(5*a^3*c^2*x^3 - 27*a^2*c^2*x^2 + 71*a*c^2*x - 177*c^2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
255,1,52,0,0.512069," ","integrate((-a*c*x+c)^(3/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(3 \, a^{2} c x^{2} - 14 \, a c x + 43 \, c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{15 \, {\left(a^{2} x - a\right)}}"," ",0,"-2/15*(3*a^2*c*x^2 - 14*a*c*x + 43*c)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)","A",0
256,1,39,0,0.452701," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x - 5\right)}}{3 \, {\left(a^{2} x - a\right)}}"," ",0,"2/3*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x - 5)/(a^2*x - a)","A",0
257,1,36,0,0.590683," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a^{2} c x - a c}"," ",0,"-2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*c*x - a*c)","A",0
258,1,133,0,0.685580," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} \log\left(-\frac{a^{2} x^{2} + 2 \, a x - \frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{\sqrt{c}} - 3}{a^{2} x^{2} - 2 \, a x + 1}\right)}{2 \, a c^{\frac{3}{2}}}, \frac{\sqrt{2} \sqrt{-\frac{1}{c}} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-\frac{1}{c}}}{a^{2} x^{2} - 1}\right)}{a c}\right]"," ",0,"[1/2*sqrt(2)*log(-(a^2*x^2 + 2*a*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/sqrt(c) - 3)/(a^2*x^2 - 2*a*x + 1))/(a*c^(3/2)), sqrt(2)*sqrt(-1/c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-1/c)/(a^2*x^2 - 1))/(a*c)]","A",0
259,1,258,0,0.558032," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{8 \, {\left(a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right)}}, \frac{\sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{4 \, {\left(a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right)}}\right]"," ",0,"[1/8*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3), 1/4*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3)]","A",0
260,1,312,0,0.444797," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(3 \, a x - 7\right)}}{64 \, {\left(a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right)}}, \frac{3 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(3 \, a x - 7\right)}}{32 \, {\left(a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right)}}\right]"," ",0,"[1/64*(3*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(3*a*x - 7))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4), 1/32*(3*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(3*a*x - 7))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)]","A",0
261,1,203,0,0.509167," ","integrate((-a*c*x+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(2520 \, \sqrt{2} c^{\frac{7}{2}} \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) + {\left(35 \, a^{4} c^{3} x^{4} - 230 \, a^{3} c^{3} x^{3} + 732 \, a^{2} c^{3} x^{2} - 1754 \, a c^{3} x + 6257 \, c^{3}\right)} \sqrt{-a c x + c}\right)}}{315 \, a}, \frac{2 \, {\left(5040 \, \sqrt{2} \sqrt{-c} c^{3} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) + {\left(35 \, a^{4} c^{3} x^{4} - 230 \, a^{3} c^{3} x^{3} + 732 \, a^{2} c^{3} x^{2} - 1754 \, a c^{3} x + 6257 \, c^{3}\right)} \sqrt{-a c x + c}\right)}}{315 \, a}\right]"," ",0,"[2/315*(2520*sqrt(2)*c^(7/2)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + (35*a^4*c^3*x^4 - 230*a^3*c^3*x^3 + 732*a^2*c^3*x^2 - 1754*a*c^3*x + 6257*c^3)*sqrt(-a*c*x + c))/a, 2/315*(5040*sqrt(2)*sqrt(-c)*c^3*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) + (35*a^4*c^3*x^4 - 230*a^3*c^3*x^3 + 732*a^2*c^3*x^2 - 1754*a*c^3*x + 6257*c^3)*sqrt(-a*c*x + c))/a]","A",0
262,1,183,0,0.471869," ","integrate((-a*c*x+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(420 \, \sqrt{2} c^{\frac{5}{2}} \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) - {\left(15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right)} \sqrt{-a c x + c}\right)}}{105 \, a}, \frac{2 \, {\left(840 \, \sqrt{2} \sqrt{-c} c^{2} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - {\left(15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right)} \sqrt{-a c x + c}\right)}}{105 \, a}\right]"," ",0,"[2/105*(420*sqrt(2)*c^(5/2)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - (15*a^3*c^2*x^3 - 87*a^2*c^2*x^2 + 269*a*c^2*x - 1037*c^2)*sqrt(-a*c*x + c))/a, 2/105*(840*sqrt(2)*sqrt(-c)*c^2*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - (15*a^3*c^2*x^3 - 87*a^2*c^2*x^2 + 269*a*c^2*x - 1037*c^2)*sqrt(-a*c*x + c))/a]","A",0
263,1,145,0,0.494021," ","integrate((-a*c*x+c)^(3/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(30 \, \sqrt{2} c^{\frac{3}{2}} \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) + {\left(3 \, a^{2} c x^{2} - 16 \, a c x + 73 \, c\right)} \sqrt{-a c x + c}\right)}}{15 \, a}, \frac{2 \, {\left(60 \, \sqrt{2} \sqrt{-c} c \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) + {\left(3 \, a^{2} c x^{2} - 16 \, a c x + 73 \, c\right)} \sqrt{-a c x + c}\right)}}{15 \, a}\right]"," ",0,"[2/15*(30*sqrt(2)*c^(3/2)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + (3*a^2*c*x^2 - 16*a*c*x + 73*c)*sqrt(-a*c*x + c))/a, 2/15*(60*sqrt(2)*sqrt(-c)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) + (3*a^2*c*x^2 - 16*a*c*x + 73*c)*sqrt(-a*c*x + c))/a]","A",0
264,1,120,0,0.558815," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(3 \, \sqrt{2} \sqrt{c} \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) - \sqrt{-a c x + c} {\left(a x - 7\right)}\right)}}{3 \, a}, \frac{2 \, {\left(6 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - \sqrt{-a c x + c} {\left(a x - 7\right)}\right)}}{3 \, a}\right]"," ",0,"[2/3*(3*sqrt(2)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - sqrt(-a*c*x + c)*(a*x - 7))/a, 2/3*(6*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - sqrt(-a*c*x + c)*(a*x - 7))/a]","A",0
265,1,118,0,0.540582," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} \sqrt{c} \log\left(\frac{a x + \frac{2 \, \sqrt{2} \sqrt{-a c x + c}}{\sqrt{c}} - 3}{a x + 1}\right) + 2 \, \sqrt{-a c x + c}}{a c}, -\frac{2 \, {\left(\sqrt{2} c \sqrt{-\frac{1}{c}} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-\frac{1}{c}}}{a x - 1}\right) - \sqrt{-a c x + c}\right)}}{a c}\right]"," ",0,"[(sqrt(2)*sqrt(c)*log((a*x + 2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(c) - 3)/(a*x + 1)) + 2*sqrt(-a*c*x + c))/(a*c), -2*(sqrt(2)*c*sqrt(-1/c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(-1/c)/(a*x - 1)) - sqrt(-a*c*x + c))/(a*c)]","A",0
266,1,89,0,0.459667," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} \log\left(\frac{a x + \frac{2 \, \sqrt{2} \sqrt{-a c x + c}}{\sqrt{c}} - 3}{a x + 1}\right)}{2 \, a c^{\frac{3}{2}}}, -\frac{\sqrt{2} \sqrt{-\frac{1}{c}} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-\frac{1}{c}}}{a x - 1}\right)}{a c}\right]"," ",0,"[1/2*sqrt(2)*log((a*x + 2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(c) - 3)/(a*x + 1))/(a*c^(3/2)), -sqrt(2)*sqrt(-1/c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(-1/c)/(a*x - 1))/(a*c)]","A",0
267,1,146,0,0.417218," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} {\left(a x - 1\right)} \sqrt{c} \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) - 4 \, \sqrt{-a c x + c}}{4 \, {\left(a^{2} c^{3} x - a c^{3}\right)}}, \frac{\sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - 2 \, \sqrt{-a c x + c}}{2 \, {\left(a^{2} c^{3} x - a c^{3}\right)}}\right]"," ",0,"[1/4*(sqrt(2)*(a*x - 1)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - 4*sqrt(-a*c*x + c))/(a^2*c^3*x - a*c^3), 1/2*(sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 2*sqrt(-a*c*x + c))/(a^2*c^3*x - a*c^3)]","A",0
268,1,196,0,0.624804," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) - 4 \, \sqrt{-a c x + c} {\left(3 \, a x - 5\right)}}{24 \, {\left(a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}, \frac{3 \, \sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - 2 \, \sqrt{-a c x + c} {\left(3 \, a x - 5\right)}}{12 \, {\left(a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}\right]"," ",0,"[1/24*(3*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - 4*sqrt(-a*c*x + c)*(3*a*x - 5))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4), 1/12*(3*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 2*sqrt(-a*c*x + c)*(3*a*x - 5))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)]","A",0
269,1,252,0,0.586366," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(9/2),x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) - 4 \, {\left(15 \, a^{2} x^{2} - 40 \, a x + 37\right)} \sqrt{-a c x + c}}{240 \, {\left(a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right)}}, \frac{15 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - 2 \, {\left(15 \, a^{2} x^{2} - 40 \, a x + 37\right)} \sqrt{-a c x + c}}{120 \, {\left(a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right)}}\right]"," ",0,"[1/240*(15*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - 4*(15*a^2*x^2 - 40*a*x + 37)*sqrt(-a*c*x + c))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5), 1/120*(15*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 2*(15*a^2*x^2 - 40*a*x + 37)*sqrt(-a*c*x + c))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5)]","A",0
270,1,82,0,0.448014," ","integrate((-a*c*x+c)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(5 \, a^{4} c^{2} x^{4} - 36 \, a^{3} c^{2} x^{3} + 142 \, a^{2} c^{2} x^{2} - 708 \, a c^{2} x - 1451 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{35 \, {\left(a^{3} x^{2} - a\right)}}"," ",0,"-2/35*(5*a^4*c^2*x^4 - 36*a^3*c^2*x^3 + 142*a^2*c^2*x^2 - 708*a*c^2*x - 1451*c^2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^3*x^2 - a)","A",0
271,1,62,0,0.536085," ","integrate((-a*c*x+c)^(3/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\frac{2 \, {\left(a^{3} c x^{3} - 7 \, a^{2} c x^{2} + 43 \, a c x + 91 \, c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{5 \, {\left(a^{3} x^{2} - a\right)}}"," ",0,"2/5*(a^3*c*x^3 - 7*a^2*c*x^2 + 43*a*c*x + 91*c)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^3*x^2 - a)","A",0
272,1,49,0,0.473691," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a^{2} x^{2} - 10 \, a x - 23\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{3 \, {\left(a^{3} x^{2} - a\right)}}"," ",0,"-2/3*(a^2*x^2 - 10*a*x - 23)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^3*x^2 - a)","A",0
273,1,43,0,0.481266," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 3\right)}}{a^{3} c x^{2} - a c}"," ",0,"2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 3)/(a^3*c*x^2 - a*c)","A",0
274,1,42,0,0.484507," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a^{3} c^{2} x^{2} - a c^{2}}"," ",0,"2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^3*c^2*x^2 - a*c^2)","A",0
275,1,234,0,0.573601," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{4 \, {\left(a^{3} c^{3} x^{2} - a c^{3}\right)}}, \frac{\sqrt{2} {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{2 \, {\left(a^{3} c^{3} x^{2} - a c^{3}\right)}}\right]"," ",0,"[1/4*(sqrt(2)*(a^2*x^2 - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^3*x^2 - a*c^3), 1/2*(sqrt(2)*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*c^3*x^2 - a*c^3)]","A",0
276,1,310,0,0.455018," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{2} {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(3 \, a x - 1\right)}}{16 \, {\left(a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right)}}, \frac{3 \, \sqrt{2} {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(3 \, a x - 1\right)}}{8 \, {\left(a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right)}}\right]"," ",0,"[1/16*(3*sqrt(2)*(a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(3*a*x - 1))/(a^4*c^4*x^3 - a^3*c^4*x^2 - a^2*c^4*x + a*c^4), 1/8*(3*sqrt(2)*(a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(3*a*x - 1))/(a^4*c^4*x^3 - a^3*c^4*x^2 - a^2*c^4*x + a*c^4)]","A",0
277,1,328,0,0.434448," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(9/2),x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{2} {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 4 \, {\left(15 \, a^{2} x^{2} - 20 \, a x - 3\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{128 \, {\left(a^{5} c^{5} x^{4} - 2 \, a^{4} c^{5} x^{3} + 2 \, a^{2} c^{5} x - a c^{5}\right)}}, \frac{15 \, \sqrt{2} {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 2 \, {\left(15 \, a^{2} x^{2} - 20 \, a x - 3\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{64 \, {\left(a^{5} c^{5} x^{4} - 2 \, a^{4} c^{5} x^{3} + 2 \, a^{2} c^{5} x - a c^{5}\right)}}\right]"," ",0,"[1/128*(15*sqrt(2)*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 4*(15*a^2*x^2 - 20*a*x - 3)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^5*c^5*x^4 - 2*a^4*c^5*x^3 + 2*a^2*c^5*x - a*c^5), 1/64*(15*sqrt(2)*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 2*(15*a^2*x^2 - 20*a*x - 3)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^5*c^5*x^4 - 2*a^4*c^5*x^3 + 2*a^2*c^5*x - a*c^5)]","A",0
278,0,0,0,0.460391," ","integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} + 3 \, a c^{3} x - c^{3}\right)} \sqrt{-a c x + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(-(a^3*c^3*x^3 - 3*a^2*c^3*x^2 + 3*a*c^3*x - c^3)*sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
279,0,0,0,0.541338," ","integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right)} \sqrt{-a c x + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral((a^2*c^2*x^2 - 2*a*c^2*x + c^2)*sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
280,0,0,0,0.531272," ","integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a c x - c\right)} \sqrt{-a c x + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(-(a*c*x - c)*sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
281,0,0,0,0.616937," ","integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-a c x + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
282,0,0,0,0.485544," ","integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a c x + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c x - c}, x\right)"," ",0,"integral(-sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c), x)","F",0
283,0,0,0,0.484323," ","integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a c x + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, x\right)"," ",0,"integral(sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c^2*x^2 - 2*a*c^2*x + c^2), x)","F",0
284,0,0,0,0.445245," ","integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a c x + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} + 3 \, a c^{3} x - c^{3}}, x\right)"," ",0,"integral(-sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^3*c^3*x^3 - 3*a^2*c^3*x^2 + 3*a*c^3*x - c^3), x)","F",0
285,0,0,0,0.439606," ","integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a c x + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{4} x^{4} - 4 \, a^{3} c^{4} x^{3} + 6 \, a^{2} c^{4} x^{2} - 4 \, a c^{4} x + c^{4}}, x\right)"," ",0,"integral(sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^4*x^4 - 4*a^3*c^4*x^3 + 6*a^2*c^4*x^2 - 4*a*c^4*x + c^4), x)","F",0
286,1,69,0,0.505576," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4*(-a*c*x+c),x, algorithm=""fricas"")","-\frac{6 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(8 \, a^{5} c x^{5} - 2 \, a^{3} c x^{3} - 3 \, a c x\right)} \sqrt{-a^{2} x^{2} + 1}}{48 \, a^{5}}"," ",0,"-1/48*(6*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (8*a^5*c*x^5 - 2*a^3*c*x^3 - 3*a*c*x)*sqrt(-a^2*x^2 + 1))/a^5","A",0
287,1,39,0,0.506733," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a*c*x+c),x, algorithm=""fricas"")","\frac{{\left(3 \, a^{4} c x^{4} - a^{2} c x^{2} - 2 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{15 \, a^{4}}"," ",0,"1/15*(3*a^4*c*x^4 - a^2*c*x^2 - 2*c)*sqrt(-a^2*x^2 + 1)/a^4","A",0
288,1,60,0,0.424818," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c),x, algorithm=""fricas"")","-\frac{2 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(2 \, a^{3} c x^{3} - a c x\right)} \sqrt{-a^{2} x^{2} + 1}}{8 \, a^{3}}"," ",0,"-1/8*(2*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^3*c*x^3 - a*c*x)*sqrt(-a^2*x^2 + 1))/a^3","A",0
289,1,29,0,0.428104," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a*c*x+c),x, algorithm=""fricas"")","\frac{{\left(a^{2} c x^{2} - c\right)} \sqrt{-a^{2} x^{2} + 1}}{3 \, a^{2}}"," ",0,"1/3*(a^2*c*x^2 - c)*sqrt(-a^2*x^2 + 1)/a^2","A",0
290,1,47,0,0.410068," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} x^{2} + 1} a c x - 2 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right)}{2 \, a}"," ",0,"1/2*(sqrt(-a^2*x^2 + 1)*a*c*x - 2*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a","A",0
291,1,36,0,0.491584," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)/x,x, algorithm=""fricas"")","c \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + \sqrt{-a^{2} x^{2} + 1} c"," ",0,"c*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*c","A",0
292,1,47,0,0.480026," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)/x^2,x, algorithm=""fricas"")","\frac{2 \, a c x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - \sqrt{-a^{2} x^{2} + 1} c}{x}"," ",0,"(2*a*c*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1)*c)/x","A",0
293,1,47,0,0.470110," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)/x^3,x, algorithm=""fricas"")","-\frac{a^{2} c x^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + \sqrt{-a^{2} x^{2} + 1} c}{2 \, x^{2}}"," ",0,"-1/2*(a^2*c*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*c)/x^2","A",0
294,1,29,0,0.414430," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)/x^4,x, algorithm=""fricas"")","\frac{{\left(a^{2} c x^{2} - c\right)} \sqrt{-a^{2} x^{2} + 1}}{3 \, x^{3}}"," ",0,"1/3*(a^2*c*x^2 - c)*sqrt(-a^2*x^2 + 1)/x^3","A",0
295,1,104,0,0.449059," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{30 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(40 \, a^{5} c^{2} x^{5} - 48 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} + 16 \, a^{2} c^{2} x^{2} - 15 \, a c^{2} x + 32 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{240 \, a^{4}}"," ",0,"1/240*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (40*a^5*c^2*x^5 - 48*a^4*c^2*x^4 - 10*a^3*c^2*x^3 + 16*a^2*c^2*x^2 - 15*a*c^2*x + 32*c^2)*sqrt(-a^2*x^2 + 1))/a^4","A",0
296,1,92,0,0.436410," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{30 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(24 \, a^{4} c^{2} x^{4} - 30 \, a^{3} c^{2} x^{3} - 8 \, a^{2} c^{2} x^{2} + 15 \, a c^{2} x - 16 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, a^{3}}"," ",0,"-1/120*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c^2*x^4 - 30*a^3*c^2*x^3 - 8*a^2*c^2*x^2 + 15*a*c^2*x - 16*c^2)*sqrt(-a^2*x^2 + 1))/a^3","A",0
297,1,82,0,0.416852," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{6 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(6 \, a^{3} c^{2} x^{3} - 8 \, a^{2} c^{2} x^{2} - 3 \, a c^{2} x + 8 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{24 \, a^{2}}"," ",0,"1/24*(6*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (6*a^3*c^2*x^3 - 8*a^2*c^2*x^2 - 3*a*c^2*x + 8*c^2)*sqrt(-a^2*x^2 + 1))/a^2","A",0
298,1,70,0,0.485990," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{6 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{2} c^{2} x^{2} - 3 \, a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a}"," ",0,"-1/6*(6*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^2*c^2*x^2 - 3*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1))/a","A",0
299,1,76,0,0.457995," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x,x, algorithm=""fricas"")","c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + c^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \frac{1}{2} \, {\left(a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}"," ",0,"c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + c^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 1/2*(a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1)","A",0
300,1,91,0,0.520776," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^2,x, algorithm=""fricas"")","\frac{2 \, a c^{2} x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - a c^{2} x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - a c^{2} x - {\left(a c^{2} x + c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{x}"," ",0,"(2*a*c^2*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - a*c^2*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) - a*c^2*x - (a*c^2*x + c^2)*sqrt(-a^2*x^2 + 1))/x","A",0
301,1,95,0,1.009231," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^3,x, algorithm=""fricas"")","-\frac{4 \, a^{2} c^{2} x^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + a^{2} c^{2} x^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(2 \, a c^{2} x - c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, x^{2}}"," ",0,"-1/2*(4*a^2*c^2*x^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + a^2*c^2*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (2*a*c^2*x - c^2)*sqrt(-a^2*x^2 + 1))/x^2","A",0
302,1,73,0,0.477222," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^4,x, algorithm=""fricas"")","\frac{3 \, a^{3} c^{2} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(2 \, a^{2} c^{2} x^{2} + 3 \, a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, x^{3}}"," ",0,"1/6*(3*a^3*c^2*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (2*a^2*c^2*x^2 + 3*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1))/x^3","A",0
303,1,84,0,0.483703," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^5,x, algorithm=""fricas"")","-\frac{3 \, a^{4} c^{2} x^{4} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(8 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 8 \, a c^{2} x + 6 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{24 \, x^{4}}"," ",0,"-1/24*(3*a^4*c^2*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (8*a^3*c^2*x^3 - 3*a^2*c^2*x^2 - 8*a*c^2*x + 6*c^2)*sqrt(-a^2*x^2 + 1))/x^4","A",0
304,1,95,0,0.435546," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^6,x, algorithm=""fricas"")","\frac{15 \, a^{5} c^{2} x^{5} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(16 \, a^{4} c^{2} x^{4} - 15 \, a^{3} c^{2} x^{3} + 8 \, a^{2} c^{2} x^{2} + 30 \, a c^{2} x - 24 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, x^{5}}"," ",0,"1/120*(15*a^5*c^2*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (16*a^4*c^2*x^4 - 15*a^3*c^2*x^3 + 8*a^2*c^2*x^2 + 30*a*c^2*x - 24*c^2)*sqrt(-a^2*x^2 + 1))/x^5","A",0
305,1,106,0,0.469812," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^7,x, algorithm=""fricas"")","-\frac{15 \, a^{6} c^{2} x^{6} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(32 \, a^{5} c^{2} x^{5} - 15 \, a^{4} c^{2} x^{4} + 16 \, a^{3} c^{2} x^{3} - 10 \, a^{2} c^{2} x^{2} - 48 \, a c^{2} x + 40 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{240 \, x^{6}}"," ",0,"-1/240*(15*a^6*c^2*x^6*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (32*a^5*c^2*x^5 - 15*a^4*c^2*x^4 + 16*a^3*c^2*x^3 - 10*a^2*c^2*x^2 - 48*a*c^2*x + 40*c^2)*sqrt(-a^2*x^2 + 1))/x^6","A",0
306,1,114,0,0.414908," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{210 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(120 \, a^{6} c^{3} x^{6} - 280 \, a^{5} c^{3} x^{5} + 144 \, a^{4} c^{3} x^{4} + 70 \, a^{3} c^{3} x^{3} - 88 \, a^{2} c^{3} x^{2} + 105 \, a c^{3} x - 176 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{840 \, a^{4}}"," ",0,"1/840*(210*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (120*a^6*c^3*x^6 - 280*a^5*c^3*x^5 + 144*a^4*c^3*x^4 + 70*a^3*c^3*x^3 - 88*a^2*c^3*x^2 + 105*a*c^3*x - 176*c^3)*sqrt(-a^2*x^2 + 1))/a^4","A",0
307,1,104,0,0.606175," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c)^3,x, algorithm=""fricas"")","-\frac{90 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(40 \, a^{5} c^{3} x^{5} - 96 \, a^{4} c^{3} x^{4} + 50 \, a^{3} c^{3} x^{3} + 32 \, a^{2} c^{3} x^{2} - 45 \, a c^{3} x + 64 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{240 \, a^{3}}"," ",0,"-1/240*(90*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (40*a^5*c^3*x^5 - 96*a^4*c^3*x^4 + 50*a^3*c^3*x^3 + 32*a^2*c^3*x^2 - 45*a*c^3*x + 64*c^3)*sqrt(-a^2*x^2 + 1))/a^3","A",0
308,1,92,0,0.483492," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{30 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(12 \, a^{4} c^{3} x^{4} - 30 \, a^{3} c^{3} x^{3} + 16 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 28 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{60 \, a^{2}}"," ",0,"1/60*(30*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (12*a^4*c^3*x^4 - 30*a^3*c^3*x^3 + 16*a^2*c^3*x^2 + 15*a*c^3*x - 28*c^3)*sqrt(-a^2*x^2 + 1))/a^2","A",0
309,1,82,0,0.505063," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3,x, algorithm=""fricas"")","-\frac{30 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(6 \, a^{3} c^{3} x^{3} - 16 \, a^{2} c^{3} x^{2} + 9 \, a c^{3} x + 16 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{24 \, a}"," ",0,"-1/24*(30*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (6*a^3*c^3*x^3 - 16*a^2*c^3*x^2 + 9*a*c^3*x + 16*c^3)*sqrt(-a^2*x^2 + 1))/a","A",0
310,1,88,0,0.554377," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x,x, algorithm=""fricas"")","2 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + c^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + \frac{1}{3} \, {\left(a^{2} c^{3} x^{2} - 3 \, a c^{3} x + 2 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}"," ",0,"2*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + c^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 1/3*(a^2*c^3*x^2 - 3*a*c^3*x + 2*c^3)*sqrt(-a^2*x^2 + 1)","A",0
311,1,104,0,0.550259," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^2,x, algorithm=""fricas"")","\frac{2 \, a c^{3} x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 4 \, a c^{3} x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 4 \, a c^{3} x + {\left(a^{2} c^{3} x^{2} - 4 \, a c^{3} x - 2 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, x}"," ",0,"1/2*(2*a*c^3*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 4*a*c^3*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 4*a*c^3*x + (a^2*c^3*x^2 - 4*a*c^3*x - 2*c^3)*sqrt(-a^2*x^2 + 1))/x","A",0
312,1,118,0,0.525632," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^3,x, algorithm=""fricas"")","-\frac{8 \, a^{2} c^{3} x^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - a^{2} c^{3} x^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 2 \, a^{2} c^{3} x^{2} - {\left(2 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, x^{2}}"," ",0,"-1/2*(8*a^2*c^3*x^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - a^2*c^3*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 2*a^2*c^3*x^2 - (2*a^2*c^3*x^2 + 4*a*c^3*x - c^3)*sqrt(-a^2*x^2 + 1))/x^2","A",0
313,1,105,0,0.495695," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^4,x, algorithm=""fricas"")","\frac{6 \, a^{3} c^{3} x^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 3 \, a^{3} c^{3} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(2 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x + c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{3 \, x^{3}}"," ",0,"1/3*(6*a^3*c^3*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*a^3*c^3*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (2*a^2*c^3*x^2 - 3*a*c^3*x + c^3)*sqrt(-a^2*x^2 + 1))/x^3","A",0
314,1,84,0,0.448465," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^5,x, algorithm=""fricas"")","-\frac{15 \, a^{4} c^{3} x^{4} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(16 \, a^{3} c^{3} x^{3} + 9 \, a^{2} c^{3} x^{2} - 16 \, a c^{3} x + 6 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{24 \, x^{4}}"," ",0,"-1/24*(15*a^4*c^3*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (16*a^3*c^3*x^3 + 9*a^2*c^3*x^2 - 16*a*c^3*x + 6*c^3)*sqrt(-a^2*x^2 + 1))/x^4","A",0
315,1,95,0,0.422801," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^6,x, algorithm=""fricas"")","\frac{15 \, a^{5} c^{3} x^{5} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(28 \, a^{4} c^{3} x^{4} - 15 \, a^{3} c^{3} x^{3} - 16 \, a^{2} c^{3} x^{2} + 30 \, a c^{3} x - 12 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{60 \, x^{5}}"," ",0,"1/60*(15*a^5*c^3*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (28*a^4*c^3*x^4 - 15*a^3*c^3*x^3 - 16*a^2*c^3*x^2 + 30*a*c^3*x - 12*c^3)*sqrt(-a^2*x^2 + 1))/x^5","A",0
316,1,126,0,0.444128," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{6090 \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(1680 \, a^{7} c^{4} x^{7} - 5760 \, a^{6} c^{4} x^{6} + 6440 \, a^{5} c^{4} x^{5} - 1536 \, a^{4} c^{4} x^{4} - 2030 \, a^{3} c^{4} x^{3} + 2432 \, a^{2} c^{4} x^{2} - 3045 \, a c^{4} x + 4864 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{13440 \, a^{4}}"," ",0,"1/13440*(6090*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (1680*a^7*c^4*x^7 - 5760*a^6*c^4*x^6 + 6440*a^5*c^4*x^5 - 1536*a^4*c^4*x^4 - 2030*a^3*c^4*x^3 + 2432*a^2*c^4*x^2 - 3045*a*c^4*x + 4864*c^4)*sqrt(-a^2*x^2 + 1))/a^4","A",0
317,1,114,0,0.405906," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{210 \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(48 \, a^{6} c^{4} x^{6} - 168 \, a^{5} c^{4} x^{5} + 192 \, a^{4} c^{4} x^{4} - 42 \, a^{3} c^{4} x^{3} - 80 \, a^{2} c^{4} x^{2} + 105 \, a c^{4} x - 160 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{336 \, a^{3}}"," ",0,"-1/336*(210*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (48*a^6*c^4*x^6 - 168*a^5*c^4*x^5 + 192*a^4*c^4*x^4 - 42*a^3*c^4*x^3 - 80*a^2*c^4*x^2 + 105*a*c^4*x - 160*c^4)*sqrt(-a^2*x^2 + 1))/a^3","A",0
318,1,104,0,0.481886," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{210 \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(40 \, a^{5} c^{4} x^{5} - 144 \, a^{4} c^{4} x^{4} + 170 \, a^{3} c^{4} x^{3} - 32 \, a^{2} c^{4} x^{2} - 105 \, a c^{4} x + 176 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{240 \, a^{2}}"," ",0,"1/240*(210*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (40*a^5*c^4*x^5 - 144*a^4*c^4*x^4 + 170*a^3*c^4*x^3 - 32*a^2*c^4*x^2 - 105*a*c^4*x + 176*c^4)*sqrt(-a^2*x^2 + 1))/a^2","A",0
319,1,92,0,0.465669," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{210 \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(24 \, a^{4} c^{4} x^{4} - 90 \, a^{3} c^{4} x^{3} + 112 \, a^{2} c^{4} x^{2} - 15 \, a c^{4} x - 136 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, a}"," ",0,"-1/120*(210*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c^4*x^4 - 90*a^3*c^4*x^3 + 112*a^2*c^4*x^2 - 15*a*c^4*x - 136*c^4)*sqrt(-a^2*x^2 + 1))/a","A",0
320,1,95,0,0.479711," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x,x, algorithm=""fricas"")","\frac{13}{4} \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + c^{4} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \frac{1}{8} \, {\left(2 \, a^{3} c^{4} x^{3} - 8 \, a^{2} c^{4} x^{2} + 11 \, a c^{4} x\right)} \sqrt{-a^{2} x^{2} + 1}"," ",0,"13/4*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + c^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 1/8*(2*a^3*c^4*x^3 - 8*a^2*c^4*x^2 + 11*a*c^4*x)*sqrt(-a^2*x^2 + 1)","A",0
321,1,116,0,0.473488," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x^2,x, algorithm=""fricas"")","-\frac{6 \, a c^{4} x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 18 \, a c^{4} x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + 16 \, a c^{4} x + {\left(2 \, a^{3} c^{4} x^{3} - 9 \, a^{2} c^{4} x^{2} + 16 \, a c^{4} x + 6 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, x}"," ",0,"-1/6*(6*a*c^4*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 18*a*c^4*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 16*a*c^4*x + (2*a^3*c^4*x^3 - 9*a^2*c^4*x^2 + 16*a*c^4*x + 6*c^4)*sqrt(-a^2*x^2 + 1))/x","A",0
322,1,125,0,0.485727," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x^3,x, algorithm=""fricas"")","-\frac{10 \, a^{2} c^{4} x^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 5 \, a^{2} c^{4} x^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 6 \, a^{2} c^{4} x^{2} + {\left(a^{3} c^{4} x^{3} - 6 \, a^{2} c^{4} x^{2} - 6 \, a c^{4} x + c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, x^{2}}"," ",0,"-1/2*(10*a^2*c^4*x^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 5*a^2*c^4*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 6*a^2*c^4*x^2 + (a^3*c^4*x^3 - 6*a^2*c^4*x^2 - 6*a*c^4*x + c^4)*sqrt(-a^2*x^2 + 1))/x^2","A",0
323,1,129,0,0.479222," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x^4,x, algorithm=""fricas"")","\frac{36 \, a^{3} c^{4} x^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 3 \, a^{3} c^{4} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 6 \, a^{3} c^{4} x^{3} - {\left(6 \, a^{3} c^{4} x^{3} + 16 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + 2 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, x^{3}}"," ",0,"1/6*(36*a^3*c^4*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*a^3*c^4*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 6*a^3*c^4*x^3 - (6*a^3*c^4*x^3 + 16*a^2*c^4*x^2 - 9*a*c^4*x + 2*c^4)*sqrt(-a^2*x^2 + 1))/x^3","A",0
324,1,106,0,0.512216," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x^5,x, algorithm=""fricas"")","-\frac{16 \, a^{4} c^{4} x^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 13 \, a^{4} c^{4} x^{4} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(11 \, a^{2} c^{4} x^{2} - 8 \, a c^{4} x + 2 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{8 \, x^{4}}"," ",0,"-1/8*(16*a^4*c^4*x^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 13*a^4*c^4*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (11*a^2*c^4*x^2 - 8*a*c^4*x + 2*c^4)*sqrt(-a^2*x^2 + 1))/x^4","A",0
325,1,95,0,0.487444," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x^6,x, algorithm=""fricas"")","\frac{105 \, a^{5} c^{4} x^{5} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(136 \, a^{4} c^{4} x^{4} + 15 \, a^{3} c^{4} x^{3} - 112 \, a^{2} c^{4} x^{2} + 90 \, a c^{4} x - 24 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, x^{5}}"," ",0,"1/120*(105*a^5*c^4*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (136*a^4*c^4*x^4 + 15*a^3*c^4*x^3 - 112*a^2*c^4*x^2 + 90*a*c^4*x - 24*c^4)*sqrt(-a^2*x^2 + 1))/x^5","A",0
326,1,106,0,0.536573," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x^7,x, algorithm=""fricas"")","-\frac{105 \, a^{6} c^{4} x^{6} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(176 \, a^{5} c^{4} x^{5} - 105 \, a^{4} c^{4} x^{4} - 32 \, a^{3} c^{4} x^{3} + 170 \, a^{2} c^{4} x^{2} - 144 \, a c^{4} x + 40 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{240 \, x^{6}}"," ",0,"-1/240*(105*a^6*c^4*x^6*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (176*a^5*c^4*x^5 - 105*a^4*c^4*x^4 - 32*a^3*c^4*x^3 + 170*a^2*c^4*x^2 - 144*a*c^4*x + 40*c^4)*sqrt(-a^2*x^2 + 1))/x^6","A",0
327,1,95,0,0.507754," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c),x, algorithm=""fricas"")","\frac{128 \, a x + 162 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(6 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 17 \, a^{2} x^{2} + 47 \, a x - 128\right)} \sqrt{-a^{2} x^{2} + 1} - 128}{24 \, {\left(a^{6} c x - a^{5} c\right)}}"," ",0,"1/24*(128*a*x + 162*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (6*a^4*x^4 + 10*a^3*x^3 + 17*a^2*x^2 + 47*a*x - 128)*sqrt(-a^2*x^2 + 1) - 128)/(a^6*c*x - a^5*c)","A",0
328,1,86,0,0.600644," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a*c*x+c),x, algorithm=""fricas"")","\frac{14 \, a x + 18 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(a^{3} x^{3} + 2 \, a^{2} x^{2} + 5 \, a x - 14\right)} \sqrt{-a^{2} x^{2} + 1} - 14}{3 \, {\left(a^{5} c x - a^{4} c\right)}}"," ",0,"1/3*(14*a*x + 18*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a^3*x^3 + 2*a^2*x^2 + 5*a*x - 14)*sqrt(-a^2*x^2 + 1) - 14)/(a^5*c*x - a^4*c)","A",0
329,1,78,0,0.498277," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a*c*x+c),x, algorithm=""fricas"")","\frac{8 \, a x + 10 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(a^{2} x^{2} + 3 \, a x - 8\right)} \sqrt{-a^{2} x^{2} + 1} - 8}{2 \, {\left(a^{4} c x - a^{3} c\right)}}"," ",0,"1/2*(8*a*x + 10*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a^2*x^2 + 3*a*x - 8)*sqrt(-a^2*x^2 + 1) - 8)/(a^4*c*x - a^3*c)","A",0
330,1,69,0,0.433533," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a*c*x+c),x, algorithm=""fricas"")","\frac{3 \, a x + 4 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a x - 3\right)} - 3}{a^{3} c x - a^{2} c}"," ",0,"(3*a*x + 4*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x - 3) - 3)/(a^3*c*x - a^2*c)","A",0
331,1,62,0,0.520925," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c),x, algorithm=""fricas"")","\frac{2 \, {\left(a x + {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - \sqrt{-a^{2} x^{2} + 1} - 1\right)}}{a^{2} c x - a c}"," ",0,"2*(a*x + (a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1) - 1)/(a^2*c*x - a*c)","A",0
332,1,56,0,0.494239," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c),x, algorithm=""fricas"")","\frac{2 \, a x + {\left(a x - 1\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} - 2}{a c x - c}"," ",0,"(2*a*x + (a*x - 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 2*sqrt(-a^2*x^2 + 1) - 2)/(a*c*x - c)","A",0
333,1,80,0,0.490945," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c),x, algorithm=""fricas"")","\frac{2 \, a^{2} x^{2} - 2 \, a x + 2 \, {\left(a^{2} x^{2} - a x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \sqrt{-a^{2} x^{2} + 1} {\left(3 \, a x - 1\right)}}{a c x^{2} - c x}"," ",0,"(2*a^2*x^2 - 2*a*x + 2*(a^2*x^2 - a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - sqrt(-a^2*x^2 + 1)*(3*a*x - 1))/(a*c*x^2 - c*x)","A",0
334,1,99,0,0.426638," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a*c*x+c),x, algorithm=""fricas"")","\frac{4 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + 5 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(8 \, a^{2} x^{2} - 3 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a c x^{3} - c x^{2}\right)}}"," ",0,"1/2*(4*a^3*x^3 - 4*a^2*x^2 + 5*(a^3*x^3 - a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (8*a^2*x^2 - 3*a*x - 1)*sqrt(-a^2*x^2 + 1))/(a*c*x^3 - c*x^2)","A",0
335,1,107,0,0.410522," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a*c*x+c),x, algorithm=""fricas"")","\frac{6 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 9 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(14 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 2 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1}}{3 \, {\left(a c x^{4} - c x^{3}\right)}}"," ",0,"1/3*(6*a^4*x^4 - 6*a^3*x^3 + 9*(a^4*x^4 - a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (14*a^3*x^3 - 5*a^2*x^2 - 2*a*x - 1)*sqrt(-a^2*x^2 + 1))/(a*c*x^4 - c*x^3)","A",0
336,1,125,0,0.450538," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{80 \, a^{2} x^{2} - 160 \, a x + 102 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{4} x^{4} + 5 \, a^{3} x^{3} + 18 \, a^{2} x^{2} - 109 \, a x + 80\right)} \sqrt{-a^{2} x^{2} + 1} + 80}{6 \, {\left(a^{7} c^{2} x^{2} - 2 \, a^{6} c^{2} x + a^{5} c^{2}\right)}}"," ",0,"-1/6*(80*a^2*x^2 - 160*a*x + 102*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^4*x^4 + 5*a^3*x^3 + 18*a^2*x^2 - 109*a*x + 80)*sqrt(-a^2*x^2 + 1) + 80)/(a^7*c^2*x^2 - 2*a^6*c^2*x + a^5*c^2)","A",0
337,1,117,0,0.429828," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{52 \, a^{2} x^{2} - 104 \, a x + 66 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 71 \, a x + 52\right)} \sqrt{-a^{2} x^{2} + 1} + 52}{6 \, {\left(a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right)}}"," ",0,"-1/6*(52*a^2*x^2 - 104*a*x + 66*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^3*x^3 + 12*a^2*x^2 - 71*a*x + 52)*sqrt(-a^2*x^2 + 1) + 52)/(a^6*c^2*x^2 - 2*a^5*c^2*x + a^4*c^2)","A",0
338,1,109,0,0.437307," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{2} x^{2} - 19 \, a x + 14\right)} \sqrt{-a^{2} x^{2} + 1} + 14}{3 \, {\left(a^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} c^{2}\right)}}"," ",0,"-1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^5*c^2*x^2 - 2*a^4*c^2*x + a^3*c^2)","A",0
339,1,102,0,0.433830," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a*c*x+c)^2,x, algorithm=""fricas"")","-\frac{5 \, a^{2} x^{2} - 10 \, a x + 6 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - \sqrt{-a^{2} x^{2} + 1} {\left(7 \, a x - 5\right)} + 5}{3 \, {\left(a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right)}}"," ",0,"-1/3*(5*a^2*x^2 - 10*a*x + 6*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1)*(7*a*x - 5) + 5)/(a^4*c^2*x^2 - 2*a^3*c^2*x + a^2*c^2)","A",0
340,1,60,0,0.475045," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{a^{2} x^{2} - 2 \, a x + \sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} + 1}{3 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"1/3*(a^2*x^2 - 2*a*x + sqrt(-a^2*x^2 + 1)*(a*x + 1) + 1)/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)","B",0
341,1,93,0,0.504278," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{7 \, a^{2} x^{2} - 14 \, a x + 3 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \sqrt{-a^{2} x^{2} + 1} {\left(5 \, a x - 7\right)} + 7}{3 \, {\left(a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right)}}"," ",0,"1/3*(7*a^2*x^2 - 14*a*x + 3*(a^2*x^2 - 2*a*x + 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - sqrt(-a^2*x^2 + 1)*(5*a*x - 7) + 7)/(a^2*c^2*x^2 - 2*a*c^2*x + c^2)","A",0
342,1,118,0,0.487056," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{13 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 13 \, a x + 9 \, {\left(a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(14 \, a^{2} x^{2} - 19 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1}}{3 \, {\left(a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right)}}"," ",0,"1/3*(13*a^3*x^3 - 26*a^2*x^2 + 13*a*x + 9*(a^3*x^3 - 2*a^2*x^2 + a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (14*a^2*x^2 - 19*a*x + 3)*sqrt(-a^2*x^2 + 1))/(a^2*c^2*x^3 - 2*a*c^2*x^2 + c^2*x)","A",0
343,1,136,0,0.464144," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{38 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 38 \, a^{2} x^{2} + 33 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(52 \, a^{3} x^{3} - 71 \, a^{2} x^{2} + 12 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{2} c^{2} x^{4} - 2 \, a c^{2} x^{3} + c^{2} x^{2}\right)}}"," ",0,"1/6*(38*a^4*x^4 - 76*a^3*x^3 + 38*a^2*x^2 + 33*(a^4*x^4 - 2*a^3*x^3 + a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (52*a^3*x^3 - 71*a^2*x^2 + 12*a*x + 3)*sqrt(-a^2*x^2 + 1))/(a^2*c^2*x^4 - 2*a*c^2*x^3 + c^2*x^2)","A",0
344,1,144,0,0.447802," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{50 \, a^{5} x^{5} - 100 \, a^{4} x^{4} + 50 \, a^{3} x^{3} + 51 \, {\left(a^{5} x^{5} - 2 \, a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(80 \, a^{4} x^{4} - 109 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 5 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{2} c^{2} x^{5} - 2 \, a c^{2} x^{4} + c^{2} x^{3}\right)}}"," ",0,"1/6*(50*a^5*x^5 - 100*a^4*x^4 + 50*a^3*x^3 + 51*(a^5*x^5 - 2*a^4*x^4 + a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (80*a^4*x^4 - 109*a^3*x^3 + 18*a^2*x^2 + 5*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*c^2*x^5 - 2*a*c^2*x^4 + c^2*x^3)","A",0
345,1,153,0,0.471184," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{448 \, a^{3} x^{3} - 1344 \, a^{2} x^{2} + 1344 \, a x + 570 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(15 \, a^{4} x^{4} + 75 \, a^{3} x^{3} - 713 \, a^{2} x^{2} + 1059 \, a x - 448\right)} \sqrt{-a^{2} x^{2} + 1} - 448}{30 \, {\left(a^{8} c^{3} x^{3} - 3 \, a^{7} c^{3} x^{2} + 3 \, a^{6} c^{3} x - a^{5} c^{3}\right)}}"," ",0,"1/30*(448*a^3*x^3 - 1344*a^2*x^2 + 1344*a*x + 570*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^4*x^4 + 75*a^3*x^3 - 713*a^2*x^2 + 1059*a*x - 448)*sqrt(-a^2*x^2 + 1) - 448)/(a^8*c^3*x^3 - 3*a^7*c^3*x^2 + 3*a^6*c^3*x - a^5*c^3)","A",0
346,1,145,0,0.487317," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{94 \, a^{3} x^{3} - 282 \, a^{2} x^{2} + 282 \, a x + 120 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(15 \, a^{3} x^{3} - 149 \, a^{2} x^{2} + 222 \, a x - 94\right)} \sqrt{-a^{2} x^{2} + 1} - 94}{15 \, {\left(a^{7} c^{3} x^{3} - 3 \, a^{6} c^{3} x^{2} + 3 \, a^{5} c^{3} x - a^{4} c^{3}\right)}}"," ",0,"1/15*(94*a^3*x^3 - 282*a^2*x^2 + 282*a*x + 120*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^3*x^3 - 149*a^2*x^2 + 222*a*x - 94)*sqrt(-a^2*x^2 + 1) - 94)/(a^7*c^3*x^3 - 3*a^6*c^3*x^2 + 3*a^5*c^3*x - a^4*c^3)","A",0
347,1,138,0,0.490536," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{8 \, a^{3} x^{3} - 24 \, a^{2} x^{2} + 24 \, a x + 10 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(13 \, a^{2} x^{2} - 19 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} - 8}{5 \, {\left(a^{6} c^{3} x^{3} - 3 \, a^{5} c^{3} x^{2} + 3 \, a^{4} c^{3} x - a^{3} c^{3}\right)}}"," ",0,"1/5*(8*a^3*x^3 - 24*a^2*x^2 + 24*a*x + 10*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (13*a^2*x^2 - 19*a*x + 8)*sqrt(-a^2*x^2 + 1) - 8)/(a^6*c^3*x^3 - 3*a^5*c^3*x^2 + 3*a^4*c^3*x - a^3*c^3)","A",0
348,1,91,0,0.616057," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a*c*x+c)^3,x, algorithm=""fricas"")","-\frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x + {\left(4 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1} - 1}{15 \, {\left(a^{5} c^{3} x^{3} - 3 \, a^{4} c^{3} x^{2} + 3 \, a^{3} c^{3} x - a^{2} c^{3}\right)}}"," ",0,"-1/15*(a^3*x^3 - 3*a^2*x^2 + 3*a*x + (4*a^2*x^2 + 3*a*x - 1)*sqrt(-a^2*x^2 + 1) - 1)/(a^5*c^3*x^3 - 3*a^4*c^3*x^2 + 3*a^3*c^3*x - a^2*c^3)","A",0
349,1,89,0,0.501075," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{4 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 12 \, a x + {\left(a^{2} x^{2} - 3 \, a x - 4\right)} \sqrt{-a^{2} x^{2} + 1} - 4}{15 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"1/15*(4*a^3*x^3 - 12*a^2*x^2 + 12*a*x + (a^2*x^2 - 3*a*x - 4)*sqrt(-a^2*x^2 + 1) - 4)/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)","A",0
350,1,130,0,0.592125," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{13 \, a^{3} x^{3} - 39 \, a^{2} x^{2} + 39 \, a x + 5 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(8 \, a^{2} x^{2} - 19 \, a x + 13\right)} \sqrt{-a^{2} x^{2} + 1} - 13}{5 \, {\left(a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} + 3 \, a c^{3} x - c^{3}\right)}}"," ",0,"1/5*(13*a^3*x^3 - 39*a^2*x^2 + 39*a*x + 5*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (8*a^2*x^2 - 19*a*x + 13)*sqrt(-a^2*x^2 + 1) - 13)/(a^3*c^3*x^3 - 3*a^2*c^3*x^2 + 3*a*c^3*x - c^3)","A",0
351,1,155,0,0.613181," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{104 \, a^{4} x^{4} - 312 \, a^{3} x^{3} + 312 \, a^{2} x^{2} - 104 \, a x + 60 \, {\left(a^{4} x^{4} - 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - a x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(94 \, a^{3} x^{3} - 222 \, a^{2} x^{2} + 149 \, a x - 15\right)} \sqrt{-a^{2} x^{2} + 1}}{15 \, {\left(a^{3} c^{3} x^{4} - 3 \, a^{2} c^{3} x^{3} + 3 \, a c^{3} x^{2} - c^{3} x\right)}}"," ",0,"1/15*(104*a^4*x^4 - 312*a^3*x^3 + 312*a^2*x^2 - 104*a*x + 60*(a^4*x^4 - 3*a^3*x^3 + 3*a^2*x^2 - a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (94*a^3*x^3 - 222*a^2*x^2 + 149*a*x - 15)*sqrt(-a^2*x^2 + 1))/(a^3*c^3*x^4 - 3*a^2*c^3*x^3 + 3*a*c^3*x^2 - c^3*x)","A",0
352,1,173,0,0.446526," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{398 \, a^{5} x^{5} - 1194 \, a^{4} x^{4} + 1194 \, a^{3} x^{3} - 398 \, a^{2} x^{2} + 285 \, {\left(a^{5} x^{5} - 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(448 \, a^{4} x^{4} - 1059 \, a^{3} x^{3} + 713 \, a^{2} x^{2} - 75 \, a x - 15\right)} \sqrt{-a^{2} x^{2} + 1}}{30 \, {\left(a^{3} c^{3} x^{5} - 3 \, a^{2} c^{3} x^{4} + 3 \, a c^{3} x^{3} - c^{3} x^{2}\right)}}"," ",0,"1/30*(398*a^5*x^5 - 1194*a^4*x^4 + 1194*a^3*x^3 - 398*a^2*x^2 + 285*(a^5*x^5 - 3*a^4*x^4 + 3*a^3*x^3 - a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (448*a^4*x^4 - 1059*a^3*x^3 + 713*a^2*x^2 - 75*a*x - 15)*sqrt(-a^2*x^2 + 1))/(a^3*c^3*x^5 - 3*a^2*c^3*x^4 + 3*a*c^3*x^3 - c^3*x^2)","A",0
353,1,181,0,0.468846," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a*c*x+c)^3,x, algorithm=""fricas"")","\frac{324 \, a^{6} x^{6} - 972 \, a^{5} x^{5} + 972 \, a^{4} x^{4} - 324 \, a^{3} x^{3} + 270 \, {\left(a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(424 \, a^{5} x^{5} - 1002 \, a^{4} x^{4} + 674 \, a^{3} x^{3} - 70 \, a^{2} x^{2} - 15 \, a x - 5\right)} \sqrt{-a^{2} x^{2} + 1}}{15 \, {\left(a^{3} c^{3} x^{6} - 3 \, a^{2} c^{3} x^{5} + 3 \, a c^{3} x^{4} - c^{3} x^{3}\right)}}"," ",0,"1/15*(324*a^6*x^6 - 972*a^5*x^5 + 972*a^4*x^4 - 324*a^3*x^3 + 270*(a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 - a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (424*a^5*x^5 - 1002*a^4*x^4 + 674*a^3*x^3 - 70*a^2*x^2 - 15*a*x - 5)*sqrt(-a^2*x^2 + 1))/(a^3*c^3*x^6 - 3*a^2*c^3*x^5 + 3*a*c^3*x^4 - c^3*x^3)","A",0
354,1,187,0,0.481420," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{4784 \, a^{4} x^{4} - 19136 \, a^{3} x^{3} + 28704 \, a^{2} x^{2} - 19136 \, a x + 6090 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(105 \, a^{5} x^{5} + 630 \, a^{4} x^{4} - 8404 \, a^{3} x^{3} + 18916 \, a^{2} x^{2} - 16091 \, a x + 4784\right)} \sqrt{-a^{2} x^{2} + 1} + 4784}{210 \, {\left(a^{10} c^{4} x^{4} - 4 \, a^{9} c^{4} x^{3} + 6 \, a^{8} c^{4} x^{2} - 4 \, a^{7} c^{4} x + a^{6} c^{4}\right)}}"," ",0,"-1/210*(4784*a^4*x^4 - 19136*a^3*x^3 + 28704*a^2*x^2 - 19136*a*x + 6090*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (105*a^5*x^5 + 630*a^4*x^4 - 8404*a^3*x^3 + 18916*a^2*x^2 - 16091*a*x + 4784)*sqrt(-a^2*x^2 + 1) + 4784)/(a^10*c^4*x^4 - 4*a^9*c^4*x^3 + 6*a^8*c^4*x^2 - 4*a^7*c^4*x + a^6*c^4)","A",0
355,1,179,0,0.483912," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{824 \, a^{4} x^{4} - 3296 \, a^{3} x^{3} + 4944 \, a^{2} x^{2} - 3296 \, a x + 1050 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(105 \, a^{4} x^{4} - 1444 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 2771 \, a x + 824\right)} \sqrt{-a^{2} x^{2} + 1} + 824}{105 \, {\left(a^{9} c^{4} x^{4} - 4 \, a^{8} c^{4} x^{3} + 6 \, a^{7} c^{4} x^{2} - 4 \, a^{6} c^{4} x + a^{5} c^{4}\right)}}"," ",0,"-1/105*(824*a^4*x^4 - 3296*a^3*x^3 + 4944*a^2*x^2 - 3296*a*x + 1050*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (105*a^4*x^4 - 1444*a^3*x^3 + 3256*a^2*x^2 - 2771*a*x + 824)*sqrt(-a^2*x^2 + 1) + 824)/(a^9*c^4*x^4 - 4*a^8*c^4*x^3 + 6*a^7*c^4*x^2 - 4*a^6*c^4*x + a^5*c^4)","A",0
356,1,172,0,0.602975," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{166 \, a^{4} x^{4} - 664 \, a^{3} x^{3} + 996 \, a^{2} x^{2} - 664 \, a x + 210 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(296 \, a^{3} x^{3} - 659 \, a^{2} x^{2} + 559 \, a x - 166\right)} \sqrt{-a^{2} x^{2} + 1} + 166}{105 \, {\left(a^{8} c^{4} x^{4} - 4 \, a^{7} c^{4} x^{3} + 6 \, a^{6} c^{4} x^{2} - 4 \, a^{5} c^{4} x + a^{4} c^{4}\right)}}"," ",0,"-1/105*(166*a^4*x^4 - 664*a^3*x^3 + 996*a^2*x^2 - 664*a*x + 210*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (296*a^3*x^3 - 659*a^2*x^2 + 559*a*x - 166)*sqrt(-a^2*x^2 + 1) + 166)/(a^8*c^4*x^4 - 4*a^7*c^4*x^3 + 6*a^6*c^4*x^2 - 4*a^5*c^4*x + a^4*c^4)","A",0
357,1,118,0,0.572003," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{2 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 8 \, a x + {\left(23 \, a^{3} x^{3} + 13 \, a^{2} x^{2} - 8 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} + 2}{105 \, {\left(a^{7} c^{4} x^{4} - 4 \, a^{6} c^{4} x^{3} + 6 \, a^{5} c^{4} x^{2} - 4 \, a^{4} c^{4} x + a^{3} c^{4}\right)}}"," ",0,"1/105*(2*a^4*x^4 - 8*a^3*x^3 + 12*a^2*x^2 - 8*a*x + (23*a^3*x^3 + 13*a^2*x^2 - 8*a*x + 2)*sqrt(-a^2*x^2 + 1) + 2)/(a^7*c^4*x^4 - 4*a^6*c^4*x^3 + 6*a^5*c^4*x^2 - 4*a^4*c^4*x + a^3*c^4)","A",0
358,1,116,0,0.671993," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a*c*x+c)^4,x, algorithm=""fricas"")","-\frac{a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + {\left(a^{3} x^{3} - 4 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{-a^{2} x^{2} + 1} + 1}{21 \, {\left(a^{6} c^{4} x^{4} - 4 \, a^{5} c^{4} x^{3} + 6 \, a^{4} c^{4} x^{2} - 4 \, a^{3} c^{4} x + a^{2} c^{4}\right)}}"," ",0,"-1/21*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + (a^3*x^3 - 4*a^2*x^2 - 4*a*x + 1)*sqrt(-a^2*x^2 + 1) + 1)/(a^6*c^4*x^4 - 4*a^5*c^4*x^3 + 6*a^4*c^4*x^2 - 4*a^3*c^4*x + a^2*c^4)","A",0
359,1,116,0,0.526685," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{23 \, a^{4} x^{4} - 92 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 92 \, a x + {\left(2 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 13 \, a x + 23\right)} \sqrt{-a^{2} x^{2} + 1} + 23}{105 \, {\left(a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"1/105*(23*a^4*x^4 - 92*a^3*x^3 + 138*a^2*x^2 - 92*a*x + (2*a^3*x^3 - 8*a^2*x^2 + 13*a*x + 23)*sqrt(-a^2*x^2 + 1) + 23)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)","A",0
360,1,163,0,0.462851," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{296 \, a^{4} x^{4} - 1184 \, a^{3} x^{3} + 1776 \, a^{2} x^{2} - 1184 \, a x + 105 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(166 \, a^{3} x^{3} - 559 \, a^{2} x^{2} + 659 \, a x - 296\right)} \sqrt{-a^{2} x^{2} + 1} + 296}{105 \, {\left(a^{4} c^{4} x^{4} - 4 \, a^{3} c^{4} x^{3} + 6 \, a^{2} c^{4} x^{2} - 4 \, a c^{4} x + c^{4}\right)}}"," ",0,"1/105*(296*a^4*x^4 - 1184*a^3*x^3 + 1776*a^2*x^2 - 1184*a*x + 105*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (166*a^3*x^3 - 559*a^2*x^2 + 659*a*x - 296)*sqrt(-a^2*x^2 + 1) + 296)/(a^4*c^4*x^4 - 4*a^3*c^4*x^3 + 6*a^2*c^4*x^2 - 4*a*c^4*x + c^4)","A",0
361,1,188,0,0.477465," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{1024 \, a^{5} x^{5} - 4096 \, a^{4} x^{4} + 6144 \, a^{3} x^{3} - 4096 \, a^{2} x^{2} + 1024 \, a x + 525 \, {\left(a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(824 \, a^{4} x^{4} - 2771 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 1444 \, a x + 105\right)} \sqrt{-a^{2} x^{2} + 1}}{105 \, {\left(a^{4} c^{4} x^{5} - 4 \, a^{3} c^{4} x^{4} + 6 \, a^{2} c^{4} x^{3} - 4 \, a c^{4} x^{2} + c^{4} x\right)}}"," ",0,"1/105*(1024*a^5*x^5 - 4096*a^4*x^4 + 6144*a^3*x^3 - 4096*a^2*x^2 + 1024*a*x + 525*(a^5*x^5 - 4*a^4*x^4 + 6*a^3*x^3 - 4*a^2*x^2 + a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (824*a^4*x^4 - 2771*a^3*x^3 + 3256*a^2*x^2 - 1444*a*x + 105)*sqrt(-a^2*x^2 + 1))/(a^4*c^4*x^5 - 4*a^3*c^4*x^4 + 6*a^2*c^4*x^3 - 4*a*c^4*x^2 + c^4*x)","A",0
362,1,206,0,0.470407," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{4834 \, a^{6} x^{6} - 19336 \, a^{5} x^{5} + 29004 \, a^{4} x^{4} - 19336 \, a^{3} x^{3} + 4834 \, a^{2} x^{2} + 3045 \, {\left(a^{6} x^{6} - 4 \, a^{5} x^{5} + 6 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(4784 \, a^{5} x^{5} - 16091 \, a^{4} x^{4} + 18916 \, a^{3} x^{3} - 8404 \, a^{2} x^{2} + 630 \, a x + 105\right)} \sqrt{-a^{2} x^{2} + 1}}{210 \, {\left(a^{4} c^{4} x^{6} - 4 \, a^{3} c^{4} x^{5} + 6 \, a^{2} c^{4} x^{4} - 4 \, a c^{4} x^{3} + c^{4} x^{2}\right)}}"," ",0,"1/210*(4834*a^6*x^6 - 19336*a^5*x^5 + 29004*a^4*x^4 - 19336*a^3*x^3 + 4834*a^2*x^2 + 3045*(a^6*x^6 - 4*a^5*x^5 + 6*a^4*x^4 - 4*a^3*x^3 + a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (4784*a^5*x^5 - 16091*a^4*x^4 + 18916*a^3*x^3 - 8404*a^2*x^2 + 630*a*x + 105)*sqrt(-a^2*x^2 + 1))/(a^4*c^4*x^6 - 4*a^3*c^4*x^5 + 6*a^2*c^4*x^4 - 4*a*c^4*x^3 + c^4*x^2)","A",0
363,1,38,0,0.600522," ","integrate((1+x)^2/(-x^2+1)^(1/2)*x,x, algorithm=""fricas"")","-\frac{1}{3} \, {\left(x^{2} + 3 \, x + 5\right)} \sqrt{-x^{2} + 1} - 2 \, \arctan\left(\frac{\sqrt{-x^{2} + 1} - 1}{x}\right)"," ",0,"-1/3*(x^2 + 3*x + 5)*sqrt(-x^2 + 1) - 2*arctan((sqrt(-x^2 + 1) - 1)/x)","A",0
364,1,33,0,0.443124," ","integrate((1+x)^2/(-x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{1}{2} \, \sqrt{-x^{2} + 1} {\left(x + 4\right)} - 3 \, \arctan\left(\frac{\sqrt{-x^{2} + 1} - 1}{x}\right)"," ",0,"-1/2*sqrt(-x^2 + 1)*(x + 4) - 3*arctan((sqrt(-x^2 + 1) - 1)/x)","A",0
365,1,45,0,0.469543," ","integrate((1+x)^3/(-x^2+1)^(1/2)*x,x, algorithm=""fricas"")","-\frac{1}{8} \, {\left(2 \, x^{3} + 8 \, x^{2} + 15 \, x + 24\right)} \sqrt{-x^{2} + 1} - \frac{15}{4} \, \arctan\left(\frac{\sqrt{-x^{2} + 1} - 1}{x}\right)"," ",0,"-1/8*(2*x^3 + 8*x^2 + 15*x + 24)*sqrt(-x^2 + 1) - 15/4*arctan((sqrt(-x^2 + 1) - 1)/x)","A",0
366,1,40,0,0.426268," ","integrate((1+x)^3/(-x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{1}{6} \, {\left(2 \, x^{2} + 9 \, x + 22\right)} \sqrt{-x^{2} + 1} - 5 \, \arctan\left(\frac{\sqrt{-x^{2} + 1} - 1}{x}\right)"," ",0,"-1/6*(2*x^2 + 9*x + 22)*sqrt(-x^2 + 1) - 5*arctan((sqrt(-x^2 + 1) - 1)/x)","A",0
367,1,11,0,0.494946," ","integrate(x/(-x^2+1)^(1/2),x, algorithm=""fricas"")","-\sqrt{-x^{2} + 1}"," ",0,"-sqrt(-x^2 + 1)","C",0
368,1,18,0,0.448560," ","integrate(1/(-x^2+1)^(1/2),x, algorithm=""fricas"")","-2 \, \arctan\left(\frac{\sqrt{-x^{2} + 1} - 1}{x}\right)"," ",0,"-2*arctan((sqrt(-x^2 + 1) - 1)/x)","B",0
369,1,44,0,0.474227," ","integrate(1/(1+x)/(-x^2+1)^(1/2)*x,x, algorithm=""fricas"")","-\frac{2 \, {\left(x + 1\right)} \arctan\left(\frac{\sqrt{-x^{2} + 1} - 1}{x}\right) - x - \sqrt{-x^{2} + 1} - 1}{x + 1}"," ",0,"-(2*(x + 1)*arctan((sqrt(-x^2 + 1) - 1)/x) - x - sqrt(-x^2 + 1) - 1)/(x + 1)","B",0
370,1,19,0,0.424713," ","integrate(1/(1+x)/(-x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{x + \sqrt{-x^{2} + 1} + 1}{x + 1}"," ",0,"-(x + sqrt(-x^2 + 1) + 1)/(x + 1)","A",0
371,1,31,0,0.622956," ","integrate((1+x)^(5/2)/(-x^2+1)^(1/2)*x,x, algorithm=""fricas"")","-\frac{2 \, {\left(3 \, x^{3} + 12 \, x^{2} + 23 \, x + 46\right)} \sqrt{-x^{2} + 1}}{21 \, \sqrt{x + 1}}"," ",0,"-2/21*(3*x^3 + 12*x^2 + 23*x + 46)*sqrt(-x^2 + 1)/sqrt(x + 1)","A",0
372,1,26,0,0.677166," ","integrate((1+x)^(5/2)/(-x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(3 \, x^{2} + 14 \, x + 43\right)} \sqrt{-x^{2} + 1}}{15 \, \sqrt{x + 1}}"," ",0,"-2/15*(3*x^2 + 14*x + 43)*sqrt(-x^2 + 1)/sqrt(x + 1)","A",0
373,1,38,0,0.517904," ","integrate((1+x)/(-x^2+1)^(1/2)*(1-x)^(3/2)*x,x, algorithm=""fricas"")","\frac{2 \, {\left(15 \, x^{3} - 18 \, x^{2} - 11 \, x + 22\right)} \sqrt{-x^{2} + 1} \sqrt{-x + 1}}{105 \, {\left(x - 1\right)}}"," ",0,"2/105*(15*x^3 - 18*x^2 - 11*x + 22)*sqrt(-x^2 + 1)*sqrt(-x + 1)/(x - 1)","A",0
374,1,33,0,0.423785," ","integrate((1+x)/(-x^2+1)^(1/2)*(1-x)^(3/2),x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, x^{2} - 4 \, x - 7\right)} \sqrt{-x^{2} + 1} \sqrt{-x + 1}}{15 \, {\left(x - 1\right)}}"," ",0,"2/15*(3*x^2 - 4*x - 7)*sqrt(-x^2 + 1)*sqrt(-x + 1)/(x - 1)","B",0
375,1,24,0,0.473252," ","integrate((1+x)^(3/2)/(-x^2+1)^(1/2)*x,x, algorithm=""fricas"")","-\frac{2 \, {\left(x^{2} + 3 \, x + 6\right)} \sqrt{-x^{2} + 1}}{5 \, \sqrt{x + 1}}"," ",0,"-2/5*(x^2 + 3*x + 6)*sqrt(-x^2 + 1)/sqrt(x + 1)","A",0
376,1,19,0,0.453754," ","integrate((1+x)^(3/2)/(-x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-x^{2} + 1} {\left(x + 5\right)}}{3 \, \sqrt{x + 1}}"," ",0,"-2/3*sqrt(-x^2 + 1)*(x + 5)/sqrt(x + 1)","A",0
377,1,31,0,0.470697," ","integrate((1+x)/(-x^2+1)^(1/2)*(1-x)^(1/2)*x,x, algorithm=""fricas"")","-\frac{2 \, {\left(3 \, x^{2} + x - 2\right)} \sqrt{-x^{2} + 1} \sqrt{-x + 1}}{15 \, {\left(x - 1\right)}}"," ",0,"-2/15*(3*x^2 + x - 2)*sqrt(-x^2 + 1)*sqrt(-x + 1)/(x - 1)","B",0
378,1,26,0,0.564091," ","integrate((1+x)/(-x^2+1)^(1/2)*(1-x)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-x^{2} + 1} {\left(x + 1\right)} \sqrt{-x + 1}}{3 \, {\left(x - 1\right)}}"," ",0,"-2/3*sqrt(-x^2 + 1)*(x + 1)*sqrt(-x + 1)/(x - 1)","B",0
379,1,19,0,0.460298," ","integrate((1+x)^(1/2)/(-x^2+1)^(1/2)*x,x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-x^{2} + 1} {\left(x + 2\right)}}{3 \, \sqrt{x + 1}}"," ",0,"-2/3*sqrt(-x^2 + 1)*(x + 2)/sqrt(x + 1)","A",0
380,1,16,0,0.413951," ","integrate((1+x)^(1/2)/(-x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-x^{2} + 1}}{\sqrt{x + 1}}"," ",0,"-2*sqrt(-x^2 + 1)/sqrt(x + 1)","C",0
381,1,79,0,0.475594," ","integrate((1+x)/(-x^2+1)^(1/2)*x/(1-x)^(1/2),x, algorithm=""fricas"")","\frac{3 \, \sqrt{2} {\left(x - 1\right)} \log\left(-\frac{x^{2} - 2 \, \sqrt{2} \sqrt{-x^{2} + 1} \sqrt{-x + 1} + 2 \, x - 3}{x^{2} - 2 \, x + 1}\right) + 2 \, \sqrt{-x^{2} + 1} {\left(x + 4\right)} \sqrt{-x + 1}}{3 \, {\left(x - 1\right)}}"," ",0,"1/3*(3*sqrt(2)*(x - 1)*log(-(x^2 - 2*sqrt(2)*sqrt(-x^2 + 1)*sqrt(-x + 1) + 2*x - 3)/(x^2 - 2*x + 1)) + 2*sqrt(-x^2 + 1)*(x + 4)*sqrt(-x + 1))/(x - 1)","B",0
382,1,74,0,0.476317," ","integrate((1+x)/(-x^2+1)^(1/2)/(1-x)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{2} {\left(x - 1\right)} \log\left(-\frac{x^{2} - 2 \, \sqrt{2} \sqrt{-x^{2} + 1} \sqrt{-x + 1} + 2 \, x - 3}{x^{2} - 2 \, x + 1}\right) + 2 \, \sqrt{-x^{2} + 1} \sqrt{-x + 1}}{x - 1}"," ",0,"(sqrt(2)*(x - 1)*log(-(x^2 - 2*sqrt(2)*sqrt(-x^2 + 1)*sqrt(-x + 1) + 2*x - 3)/(x^2 - 2*x + 1)) + 2*sqrt(-x^2 + 1)*sqrt(-x + 1))/(x - 1)","B",0
383,1,71,0,0.463395," ","integrate(1/(1+x)^(1/2)/(-x^2+1)^(1/2)*x,x, algorithm=""fricas"")","\frac{\sqrt{2} {\left(x + 1\right)} \log\left(-\frac{x^{2} - 2 \, \sqrt{2} \sqrt{-x^{2} + 1} \sqrt{x + 1} - 2 \, x - 3}{x^{2} + 2 \, x + 1}\right) - 4 \, \sqrt{-x^{2} + 1} \sqrt{x + 1}}{2 \, {\left(x + 1\right)}}"," ",0,"1/2*(sqrt(2)*(x + 1)*log(-(x^2 - 2*sqrt(2)*sqrt(-x^2 + 1)*sqrt(x + 1) - 2*x - 3)/(x^2 + 2*x + 1)) - 4*sqrt(-x^2 + 1)*sqrt(x + 1))/(x + 1)","B",0
384,1,45,0,0.600460," ","integrate(1/(1+x)^(1/2)/(-x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{1}{2} \, \sqrt{2} \log\left(-\frac{x^{2} + 2 \, \sqrt{2} \sqrt{-x^{2} + 1} \sqrt{x + 1} - 2 \, x - 3}{x^{2} + 2 \, x + 1}\right)"," ",0,"1/2*sqrt(2)*log(-(x^2 + 2*sqrt(2)*sqrt(-x^2 + 1)*sqrt(x + 1) - 2*x - 3)/(x^2 + 2*x + 1))","B",0
385,1,91,0,0.585446," ","integrate((1+x)/(-x^2+1)^(1/2)*x/(1-x)^(3/2),x, algorithm=""fricas"")","\frac{5 \, \sqrt{2} {\left(x^{2} - 2 \, x + 1\right)} \log\left(-\frac{x^{2} + 2 \, \sqrt{2} \sqrt{-x^{2} + 1} \sqrt{-x + 1} + 2 \, x - 3}{x^{2} - 2 \, x + 1}\right) - 4 \, \sqrt{-x^{2} + 1} {\left(2 \, x - 3\right)} \sqrt{-x + 1}}{4 \, {\left(x^{2} - 2 \, x + 1\right)}}"," ",0,"1/4*(5*sqrt(2)*(x^2 - 2*x + 1)*log(-(x^2 + 2*sqrt(2)*sqrt(-x^2 + 1)*sqrt(-x + 1) + 2*x - 3)/(x^2 - 2*x + 1)) - 4*sqrt(-x^2 + 1)*(2*x - 3)*sqrt(-x + 1))/(x^2 - 2*x + 1)","B",0
386,1,85,0,0.436946," ","integrate((1+x)/(-x^2+1)^(1/2)/(1-x)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{2} {\left(x^{2} - 2 \, x + 1\right)} \log\left(-\frac{x^{2} + 2 \, \sqrt{2} \sqrt{-x^{2} + 1} \sqrt{-x + 1} + 2 \, x - 3}{x^{2} - 2 \, x + 1}\right) + 4 \, \sqrt{-x^{2} + 1} \sqrt{-x + 1}}{4 \, {\left(x^{2} - 2 \, x + 1\right)}}"," ",0,"1/4*(sqrt(2)*(x^2 - 2*x + 1)*log(-(x^2 + 2*sqrt(2)*sqrt(-x^2 + 1)*sqrt(-x + 1) + 2*x - 3)/(x^2 - 2*x + 1)) + 4*sqrt(-x^2 + 1)*sqrt(-x + 1))/(x^2 - 2*x + 1)","B",0
387,0,0,0,0.572083," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} x^{m}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*x^m/(a*x - 1), x)","F",0
388,1,58,0,0.522579," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(15 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - 4 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{105 \, {\left(a^{4} x - a^{3}\right)}}"," ",0,"-2/105*(15*a^3*x^3 + 3*a^2*x^2 - 4*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^4*x - a^3)","A",0
389,1,49,0,0.572258," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(3 \, a^{2} x^{2} + a x - 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{15 \, {\left(a^{3} x - a^{2}\right)}}"," ",0,"-2/15*(3*a^2*x^2 + a*x - 2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^3*x - a^2)","A",0
390,1,39,0,0.567272," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 1\right)}}{3 \, {\left(a^{2} x - a\right)}}"," ",0,"-2/3*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 1)/(a^2*x - a)","A",0
391,1,183,0,0.493486," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left(a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a x - 1}, -\frac{2 \, {\left({\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right)}}{a x - 1}\right]"," ",0,"[((a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x - 1), -2*((a*x - 1)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x - 1)]","A",0
392,1,207,0,0.498694," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(1/2)/x^2,x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - a x\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{2 \, {\left(a x^{2} - x\right)}}, -\frac{{\left(a^{2} x^{2} - a x\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a x^{2} - x}\right]"," ",0,"[1/2*((a^2*x^2 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^2 - x), -((a^2*x^2 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^2 - x)]","A",0
393,1,44,0,0.649307," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(35 \, a^{4} x^{4} + 85 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 136 \, a x + 272\right)} \sqrt{-a c x + c}}{315 \, a^{4}}"," ",0,"-2/315*(35*a^4*x^4 + 85*a^3*x^3 + 102*a^2*x^2 + 136*a*x + 272)*sqrt(-a*c*x + c)/a^4","A",0
394,1,36,0,0.416968," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(15 \, a^{3} x^{3} + 39 \, a^{2} x^{2} + 52 \, a x + 104\right)} \sqrt{-a c x + c}}{105 \, a^{3}}"," ",0,"-2/105*(15*a^3*x^3 + 39*a^2*x^2 + 52*a*x + 104)*sqrt(-a*c*x + c)/a^3","A",0
395,1,27,0,0.650640," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a^{2} x^{2} + 3 \, a x + 6\right)} \sqrt{-a c x + c}}{5 \, a^{2}}"," ",0,"-2/5*(a^2*x^2 + 3*a*x + 6)*sqrt(-a*c*x + c)/a^2","A",0
396,1,19,0,0.678408," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-a c x + c} {\left(a x + 5\right)}}{3 \, a}"," ",0,"-2/3*sqrt(-a*c*x + c)*(a*x + 5)/a","A",0
397,1,82,0,0.709285," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^(1/2)/x,x, algorithm=""fricas"")","\left[\sqrt{c} \log\left(\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right) - 2 \, \sqrt{-a c x + c}, 2 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right) - 2 \, \sqrt{-a c x + c}\right]"," ",0,"[sqrt(c)*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) - 2*sqrt(-a*c*x + c), 2*sqrt(-c)*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - 2*sqrt(-a*c*x + c)]","A",0
398,1,96,0,0.503957," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^(1/2)/x^2,x, algorithm=""fricas"")","\left[\frac{3 \, a \sqrt{c} x \log\left(\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right) - 2 \, \sqrt{-a c x + c}}{2 \, x}, \frac{3 \, a \sqrt{-c} x \arctan\left(\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right) - \sqrt{-a c x + c}}{x}\right]"," ",0,"[1/2*(3*a*sqrt(c)*x*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) - 2*sqrt(-a*c*x + c))/x, (3*a*sqrt(-c)*x*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - sqrt(-a*c*x + c))/x]","A",0
399,1,117,0,0.557410," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^(1/2)/x^3,x, algorithm=""fricas"")","\left[\frac{7 \, a^{2} \sqrt{c} x^{2} \log\left(\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right) - 2 \, \sqrt{-a c x + c} {\left(7 \, a x + 2\right)}}{8 \, x^{2}}, \frac{7 \, a^{2} \sqrt{-c} x^{2} \arctan\left(\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right) - \sqrt{-a c x + c} {\left(7 \, a x + 2\right)}}{4 \, x^{2}}\right]"," ",0,"[1/8*(7*a^2*sqrt(c)*x^2*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) - 2*sqrt(-a*c*x + c)*(7*a*x + 2))/x^2, 1/4*(7*a^2*sqrt(-c)*x^2*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - sqrt(-a*c*x + c)*(7*a*x + 2))/x^2]","A",0
400,1,133,0,0.798331," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^(1/2)/x^4,x, algorithm=""fricas"")","\left[\frac{33 \, a^{3} \sqrt{c} x^{3} \log\left(\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right) - 2 \, {\left(33 \, a^{2} x^{2} + 22 \, a x + 8\right)} \sqrt{-a c x + c}}{48 \, x^{3}}, \frac{33 \, a^{3} \sqrt{-c} x^{3} \arctan\left(\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right) - {\left(33 \, a^{2} x^{2} + 22 \, a x + 8\right)} \sqrt{-a c x + c}}{24 \, x^{3}}\right]"," ",0,"[1/48*(33*a^3*sqrt(c)*x^3*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) - 2*(33*a^2*x^2 + 22*a*x + 8)*sqrt(-a*c*x + c))/x^3, 1/24*(33*a^3*sqrt(-c)*x^3*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - (33*a^2*x^2 + 22*a*x + 8)*sqrt(-a*c*x + c))/x^3]","A",0
401,1,149,0,0.748214," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^(1/2)/x^5,x, algorithm=""fricas"")","\left[\frac{75 \, a^{4} \sqrt{c} x^{4} \log\left(\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right) - 2 \, {\left(75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right)} \sqrt{-a c x + c}}{128 \, x^{4}}, \frac{75 \, a^{4} \sqrt{-c} x^{4} \arctan\left(\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right) - {\left(75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right)} \sqrt{-a c x + c}}{64 \, x^{4}}\right]"," ",0,"[1/128*(75*a^4*sqrt(c)*x^4*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) - 2*(75*a^3*x^3 + 50*a^2*x^2 + 40*a*x + 16)*sqrt(-a*c*x + c))/x^4, 1/64*(75*a^4*sqrt(-c)*x^4*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - (75*a^3*x^3 + 50*a^2*x^2 + 40*a*x + 16)*sqrt(-a*c*x + c))/x^4]","A",0
402,1,274,0,0.535795," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(315 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + {\left(35 \, a^{4} x^{4} + 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} + 236 \, a x + 788\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right)}}{315 \, {\left(a^{5} x - a^{4}\right)}}, \frac{2 \, {\left(630 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(35 \, a^{4} x^{4} + 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} + 236 \, a x + 788\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right)}}{315 \, {\left(a^{5} x - a^{4}\right)}}\right]"," ",0,"[2/315*(315*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + (35*a^4*x^4 + 95*a^3*x^3 + 138*a^2*x^2 + 236*a*x + 788)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^5*x - a^4), 2/315*(630*sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (35*a^4*x^4 + 95*a^3*x^3 + 138*a^2*x^2 + 236*a*x + 788)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^5*x - a^4)]","A",0
403,1,258,0,0.931482," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(21 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + {\left(3 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 16 \, a x + 52\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right)}}{21 \, {\left(a^{4} x - a^{3}\right)}}, \frac{2 \, {\left(42 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(3 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 16 \, a x + 52\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right)}}{21 \, {\left(a^{4} x - a^{3}\right)}}\right]"," ",0,"[2/21*(21*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + (3*a^3*x^3 + 9*a^2*x^2 + 16*a*x + 52)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^4*x - a^3), 2/21*(42*sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (3*a^3*x^3 + 9*a^2*x^2 + 16*a*x + 52)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^4*x - a^3)]","A",0
404,1,242,0,0.530642," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(15 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + {\left(3 \, a^{2} x^{2} + 11 \, a x + 38\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right)}}{15 \, {\left(a^{3} x - a^{2}\right)}}, \frac{2 \, {\left(30 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(3 \, a^{2} x^{2} + 11 \, a x + 38\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right)}}{15 \, {\left(a^{3} x - a^{2}\right)}}\right]"," ",0,"[2/15*(15*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + (3*a^2*x^2 + 11*a*x + 38)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*x - a^2), 2/15*(30*sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (3*a^2*x^2 + 11*a*x + 38)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^3*x - a^2)]","A",0
405,1,220,0,0.556297," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(3 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 7\right)}\right)}}{3 \, {\left(a^{2} x - a\right)}}, \frac{2 \, {\left(6 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 7\right)}\right)}}{3 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[2/3*(3*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 7))/(a^2*x - a), 2/3*(6*sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 7))/(a^2*x - a)]","A",0
406,1,321,0,0.589234," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + {\left(a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a x - 1}, \frac{2 \, {\left(2 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right)}}{a x - 1}\right]"," ",0,"[(2*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + (a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x - 1), 2*(2*sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - (a*x - 1)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x - 1)]","A",0
407,1,358,0,0.516086," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^2,x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{2} {\left(a^{2} x^{2} - a x\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 5 \, {\left(a^{2} x^{2} - a x\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{2 \, {\left(a x^{2} - x\right)}}, \frac{4 \, \sqrt{2} {\left(a^{2} x^{2} - a x\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - 5 \, {\left(a^{2} x^{2} - a x\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a x^{2} - x}\right]"," ",0,"[1/2*(4*sqrt(2)*(a^2*x^2 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 5*(a^2*x^2 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^2 - x), (4*sqrt(2)*(a^2*x^2 - a*x)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 5*(a^2*x^2 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^2 - x)]","A",0
408,1,391,0,0.488072," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^3,x, algorithm=""fricas"")","\left[\frac{16 \, \sqrt{2} {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 23 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(9 \, a x + 2\right)}}{8 \, {\left(a x^{3} - x^{2}\right)}}, \frac{16 \, \sqrt{2} {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - 23 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(9 \, a x + 2\right)}}{4 \, {\left(a x^{3} - x^{2}\right)}}\right]"," ",0,"[1/8*(16*sqrt(2)*(a^3*x^3 - a^2*x^2)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 23*(a^3*x^3 - a^2*x^2)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(9*a*x + 2))/(a*x^3 - x^2), 1/4*(16*sqrt(2)*(a^3*x^3 - a^2*x^2)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 23*(a^3*x^3 - a^2*x^2)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(9*a*x + 2))/(a*x^3 - x^2)]","A",0
409,1,407,0,0.553366," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^4,x, algorithm=""fricas"")","\left[\frac{96 \, \sqrt{2} {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 135 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, {\left(57 \, a^{2} x^{2} + 26 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{48 \, {\left(a x^{4} - x^{3}\right)}}, \frac{96 \, \sqrt{2} {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - 135 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(57 \, a^{2} x^{2} + 26 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{24 \, {\left(a x^{4} - x^{3}\right)}}\right]"," ",0,"[1/48*(96*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 135*(a^4*x^4 - a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*(57*a^2*x^2 + 26*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3), 1/24*(96*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 135*(a^4*x^4 - a^3*x^3)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (57*a^2*x^2 + 26*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3)]","A",0
410,1,423,0,0.536960," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^5,x, algorithm=""fricas"")","\left[\frac{768 \, \sqrt{2} {\left(a^{5} x^{5} - a^{4} x^{4}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right) + 1089 \, {\left(a^{5} x^{5} - a^{4} x^{4}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, {\left(447 \, a^{3} x^{3} + 214 \, a^{2} x^{2} + 136 \, a x + 48\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{384 \, {\left(a x^{5} - x^{4}\right)}}, \frac{768 \, \sqrt{2} {\left(a^{5} x^{5} - a^{4} x^{4}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - 1089 \, {\left(a^{5} x^{5} - a^{4} x^{4}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(447 \, a^{3} x^{3} + 214 \, a^{2} x^{2} + 136 \, a x + 48\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{192 \, {\left(a x^{5} - x^{4}\right)}}\right]"," ",0,"[1/384*(768*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 1089*(a^5*x^5 - a^4*x^4)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*(447*a^3*x^3 + 214*a^2*x^2 + 136*a*x + 48)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^5 - x^4), 1/192*(768*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 1089*(a^5*x^5 - a^4*x^4)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (447*a^3*x^3 + 214*a^2*x^2 + 136*a*x + 48)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^5 - x^4)]","A",0
411,0,0,0,0.573311," ","integrate(x^m*(-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} x^{m}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*x^m/(a*x + 1), x)","F",0
412,1,58,0,0.449158," ","integrate(x^2*(-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{2 \, {\left(15 \, a^{3} x^{3} - 39 \, a^{2} x^{2} + 52 \, a x - 104\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{105 \, {\left(a^{4} x - a^{3}\right)}}"," ",0,"2/105*(15*a^3*x^3 - 39*a^2*x^2 + 52*a*x - 104)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^4*x - a^3)","A",0
413,1,49,0,0.407367," ","integrate(x*(-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{2 \, {\left(a^{2} x^{2} - 3 \, a x + 6\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{5 \, {\left(a^{3} x - a^{2}\right)}}"," ",0,"2/5*(a^2*x^2 - 3*a*x + 6)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^3*x - a^2)","A",0
414,1,39,0,0.455735," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x - 5\right)}}{3 \, {\left(a^{2} x - a\right)}}"," ",0,"2/3*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x - 5)/(a^2*x - a)","A",0
415,1,184,0,0.462610," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left(a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a x - 1}, -\frac{2 \, {\left({\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right)}}{a x - 1}\right]"," ",0,"[((a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x - 1), -2*((a*x - 1)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x - 1)]","A",0
416,1,207,0,0.491396," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} x^{2} - a x\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{2 \, {\left(a x^{2} - x\right)}}, \frac{3 \, {\left(a^{2} x^{2} - a x\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a x^{2} - x}\right]"," ",0,"[1/2*(3*(a^2*x^2 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^2 - x), (3*(a^2*x^2 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^2 - x)]","A",0
417,1,232,0,0.487896," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^3,x, algorithm=""fricas"")","\left[\frac{7 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(7 \, a x - 2\right)}}{8 \, {\left(a x^{3} - x^{2}\right)}}, -\frac{7 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(7 \, a x - 2\right)}}{4 \, {\left(a x^{3} - x^{2}\right)}}\right]"," ",0,"[1/8*(7*(a^3*x^3 - a^2*x^2)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(7*a*x - 2))/(a*x^3 - x^2), -1/4*(7*(a^3*x^3 - a^2*x^2)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(7*a*x - 2))/(a*x^3 - x^2)]","A",0
418,1,248,0,0.602455," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^4,x, algorithm=""fricas"")","\left[\frac{33 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, {\left(33 \, a^{2} x^{2} - 22 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{48 \, {\left(a x^{4} - x^{3}\right)}}, \frac{33 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(33 \, a^{2} x^{2} - 22 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{24 \, {\left(a x^{4} - x^{3}\right)}}\right]"," ",0,"[1/48*(33*(a^4*x^4 - a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*(33*a^2*x^2 - 22*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3), 1/24*(33*(a^4*x^4 - a^3*x^3)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (33*a^2*x^2 - 22*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3)]","A",0
419,1,169,0,0.785234," ","integrate(x^3*(-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(315 \, \sqrt{2} \sqrt{c} \log\left(\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) - {\left(35 \, a^{4} x^{4} - 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 236 \, a x + 788\right)} \sqrt{-a c x + c}\right)}}{315 \, a^{4}}, -\frac{2 \, {\left(630 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) + {\left(35 \, a^{4} x^{4} - 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 236 \, a x + 788\right)} \sqrt{-a c x + c}\right)}}{315 \, a^{4}}\right]"," ",0,"[2/315*(315*sqrt(2)*sqrt(c)*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - (35*a^4*x^4 - 95*a^3*x^3 + 138*a^2*x^2 - 236*a*x + 788)*sqrt(-a*c*x + c))/a^4, -2/315*(630*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) + (35*a^4*x^4 - 95*a^3*x^3 + 138*a^2*x^2 - 236*a*x + 788)*sqrt(-a*c*x + c))/a^4]","A",0
420,1,154,0,0.553183," ","integrate(x^2*(-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(21 \, \sqrt{2} \sqrt{c} \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) - {\left(3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right)} \sqrt{-a c x + c}\right)}}{21 \, a^{3}}, \frac{2 \, {\left(42 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - {\left(3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right)} \sqrt{-a c x + c}\right)}}{21 \, a^{3}}\right]"," ",0,"[2/21*(21*sqrt(2)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - (3*a^3*x^3 - 9*a^2*x^2 + 16*a*x - 52)*sqrt(-a*c*x + c))/a^3, 2/21*(42*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - (3*a^3*x^3 - 9*a^2*x^2 + 16*a*x - 52)*sqrt(-a*c*x + c))/a^3]","A",0
421,1,137,0,0.457959," ","integrate(x*(-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(15 \, \sqrt{2} \sqrt{c} \log\left(\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) - {\left(3 \, a^{2} x^{2} - 11 \, a x + 38\right)} \sqrt{-a c x + c}\right)}}{15 \, a^{2}}, -\frac{2 \, {\left(30 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) + {\left(3 \, a^{2} x^{2} - 11 \, a x + 38\right)} \sqrt{-a c x + c}\right)}}{15 \, a^{2}}\right]"," ",0,"[2/15*(15*sqrt(2)*sqrt(c)*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - (3*a^2*x^2 - 11*a*x + 38)*sqrt(-a*c*x + c))/a^2, -2/15*(30*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) + (3*a^2*x^2 - 11*a*x + 38)*sqrt(-a*c*x + c))/a^2]","A",0
422,1,120,0,0.523981," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(3 \, \sqrt{2} \sqrt{c} \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) - \sqrt{-a c x + c} {\left(a x - 7\right)}\right)}}{3 \, a}, \frac{2 \, {\left(6 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - \sqrt{-a c x + c} {\left(a x - 7\right)}\right)}}{3 \, a}\right]"," ",0,"[2/3*(3*sqrt(2)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - sqrt(-a*c*x + c)*(a*x - 7))/a, 2/3*(6*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - sqrt(-a*c*x + c)*(a*x - 7))/a]","A",0
423,1,157,0,0.477148," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x,x, algorithm=""fricas"")","\left[2 \, \sqrt{2} \sqrt{c} \log\left(\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) + \sqrt{c} \log\left(\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right) - 2 \, \sqrt{-a c x + c}, -4 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) + 2 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right) - 2 \, \sqrt{-a c x + c}\right]"," ",0,"[2*sqrt(2)*sqrt(c)*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + sqrt(c)*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) - 2*sqrt(-a*c*x + c), -4*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) + 2*sqrt(-c)*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - 2*sqrt(-a*c*x + c)]","A",0
424,1,175,0,0.632277," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^2,x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{2} a \sqrt{c} x \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) + 5 \, a \sqrt{c} x \log\left(\frac{a c x - 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right) - 2 \, \sqrt{-a c x + c}}{2 \, x}, \frac{4 \, \sqrt{2} a \sqrt{-c} x \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - 5 \, a \sqrt{-c} x \arctan\left(\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right) - \sqrt{-a c x + c}}{x}\right]"," ",0,"[1/2*(4*sqrt(2)*a*sqrt(c)*x*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + 5*a*sqrt(c)*x*log((a*c*x - 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) - 2*sqrt(-a*c*x + c))/x, (4*sqrt(2)*a*sqrt(-c)*x*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 5*a*sqrt(-c)*x*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - sqrt(-a*c*x + c))/x]","A",0
425,1,204,0,0.700562," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^3,x, algorithm=""fricas"")","\left[\frac{16 \, \sqrt{2} a^{2} \sqrt{c} x^{2} \log\left(\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) + 23 \, a^{2} \sqrt{c} x^{2} \log\left(\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right) + 2 \, \sqrt{-a c x + c} {\left(9 \, a x - 2\right)}}{8 \, x^{2}}, -\frac{16 \, \sqrt{2} a^{2} \sqrt{-c} x^{2} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - 23 \, a^{2} \sqrt{-c} x^{2} \arctan\left(\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right) - \sqrt{-a c x + c} {\left(9 \, a x - 2\right)}}{4 \, x^{2}}\right]"," ",0,"[1/8*(16*sqrt(2)*a^2*sqrt(c)*x^2*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + 23*a^2*sqrt(c)*x^2*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) + 2*sqrt(-a*c*x + c)*(9*a*x - 2))/x^2, -1/4*(16*sqrt(2)*a^2*sqrt(-c)*x^2*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 23*a^2*sqrt(-c)*x^2*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - sqrt(-a*c*x + c)*(9*a*x - 2))/x^2]","A",0
426,1,220,0,0.500820," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm=""fricas"")","\left[\frac{96 \, \sqrt{2} a^{3} \sqrt{c} x^{3} \log\left(\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) + 135 \, a^{3} \sqrt{c} x^{3} \log\left(\frac{a c x - 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right) - 2 \, {\left(57 \, a^{2} x^{2} - 26 \, a x + 8\right)} \sqrt{-a c x + c}}{48 \, x^{3}}, \frac{96 \, \sqrt{2} a^{3} \sqrt{-c} x^{3} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - 135 \, a^{3} \sqrt{-c} x^{3} \arctan\left(\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right) - {\left(57 \, a^{2} x^{2} - 26 \, a x + 8\right)} \sqrt{-a c x + c}}{24 \, x^{3}}\right]"," ",0,"[1/48*(96*sqrt(2)*a^3*sqrt(c)*x^3*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + 135*a^3*sqrt(c)*x^3*log((a*c*x - 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) - 2*(57*a^2*x^2 - 26*a*x + 8)*sqrt(-a*c*x + c))/x^3, 1/24*(96*sqrt(2)*a^3*sqrt(-c)*x^3*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 135*a^3*sqrt(-c)*x^3*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - (57*a^2*x^2 - 26*a*x + 8)*sqrt(-a*c*x + c))/x^3]","A",0
427,1,236,0,0.562166," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x, algorithm=""fricas"")","\left[\frac{768 \, \sqrt{2} a^{4} \sqrt{c} x^{4} \log\left(\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right) + 1089 \, a^{4} \sqrt{c} x^{4} \log\left(\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right) + 2 \, {\left(447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right)} \sqrt{-a c x + c}}{384 \, x^{4}}, -\frac{768 \, \sqrt{2} a^{4} \sqrt{-c} x^{4} \arctan\left(\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right) - 1089 \, a^{4} \sqrt{-c} x^{4} \arctan\left(\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right) - {\left(447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right)} \sqrt{-a c x + c}}{192 \, x^{4}}\right]"," ",0,"[1/384*(768*sqrt(2)*a^4*sqrt(c)*x^4*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + 1089*a^4*sqrt(c)*x^4*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) + 2*(447*a^3*x^3 - 214*a^2*x^2 + 136*a*x - 48)*sqrt(-a*c*x + c))/x^4, -1/192*(768*sqrt(2)*a^4*sqrt(-c)*x^4*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 1089*a^4*sqrt(-c)*x^4*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - (447*a^3*x^3 - 214*a^2*x^2 + 136*a*x - 48)*sqrt(-a*c*x + c))/x^4]","A",0
428,1,76,0,0.479771," ","integrate(x^3*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(5 \, a^{5} x^{5} - 20 \, a^{4} x^{4} + 41 \, a^{3} x^{3} - 82 \, a^{2} x^{2} + 328 \, a x + 656\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{45 \, {\left(a^{6} x^{2} - a^{4}\right)}}"," ",0,"-2/45*(5*a^5*x^5 - 20*a^4*x^4 + 41*a^3*x^3 - 82*a^2*x^2 + 328*a*x + 656)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^6*x^2 - a^4)","A",0
429,1,68,0,0.496437," ","integrate(x^2*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(15 \, a^{4} x^{4} - 66 \, a^{3} x^{3} + 167 \, a^{2} x^{2} - 668 \, a x - 1336\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{105 \, {\left(a^{5} x^{2} - a^{3}\right)}}"," ",0,"-2/105*(15*a^4*x^4 - 66*a^3*x^3 + 167*a^2*x^2 - 668*a*x - 1336)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^5*x^2 - a^3)","A",0
430,1,60,0,0.448874," ","integrate(x*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 79 \, a x + 158\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{15 \, {\left(a^{4} x^{2} - a^{2}\right)}}"," ",0,"-2/15*(3*a^3*x^3 - 16*a^2*x^2 + 79*a*x + 158)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^4*x^2 - a^2)","A",0
431,1,49,0,0.465842," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a^{2} x^{2} - 10 \, a x - 23\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{3 \, {\left(a^{3} x^{2} - a\right)}}"," ",0,"-2/3*(a^2*x^2 - 10*a*x - 23)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^3*x^2 - a)","A",0
432,1,209,0,0.470900," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 5\right)}}{a^{2} x^{2} - 1}, -\frac{2 \, {\left({\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(a x + 5\right)}\right)}}{a^{2} x^{2} - 1}\right]"," ",0,"[((a^2*x^2 - 1)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 5))/(a^2*x^2 - 1), -2*((a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 5))/(a^2*x^2 - 1)]","A",0
433,1,223,0,0.738697," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm=""fricas"")","\left[\frac{7 \, {\left(a^{3} x^{3} - a x\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(9 \, a x + 1\right)}}{2 \, {\left(a^{2} x^{3} - x\right)}}, \frac{7 \, {\left(a^{3} x^{3} - a x\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} {\left(9 \, a x + 1\right)}}{a^{2} x^{3} - x}\right]"," ",0,"[1/2*(7*(a^3*x^3 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(9*a*x + 1))/(a^2*x^3 - x), (7*(a^3*x^3 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(9*a*x + 1))/(a^2*x^3 - x)]","A",0
434,1,252,0,0.465780," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^3,x, algorithm=""fricas"")","\left[\frac{47 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) - 2 \, {\left(47 \, a^{2} x^{2} + 13 \, a x - 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{8 \, {\left(a^{2} x^{4} - x^{2}\right)}}, -\frac{47 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(47 \, a^{2} x^{2} + 13 \, a x - 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{4 \, {\left(a^{2} x^{4} - x^{2}\right)}}\right]"," ",0,"[1/8*(47*(a^4*x^4 - a^2*x^2)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) - 2*(47*a^2*x^2 + 13*a*x - 2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*x^4 - x^2), -1/4*(47*(a^4*x^4 - a^2*x^2)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (47*a^2*x^2 + 13*a*x - 2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*x^4 - x^2)]","A",0
435,1,268,0,0.498108," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^4,x, algorithm=""fricas"")","\left[\frac{357 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) + 2 \, {\left(357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{48 \, {\left(a^{2} x^{5} - x^{3}\right)}}, \frac{357 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{24 \, {\left(a^{2} x^{5} - x^{3}\right)}}\right]"," ",0,"[1/48*(357*(a^5*x^5 - a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*(357*a^3*x^3 + 119*a^2*x^2 - 38*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*x^5 - x^3), 1/24*(357*(a^5*x^5 - a^3*x^3)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (357*a^3*x^3 + 119*a^2*x^2 - 38*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*x^5 - x^3)]","A",0
436,1,284,0,0.474690," ","integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm=""fricas"")","\left[\frac{3345 \, {\left(a^{6} x^{6} - a^{4} x^{4}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right) - 2 \, {\left(3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{384 \, {\left(a^{2} x^{6} - x^{4}\right)}}, -\frac{3345 \, {\left(a^{6} x^{6} - a^{4} x^{4}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{192 \, {\left(a^{2} x^{6} - x^{4}\right)}}\right]"," ",0,"[1/384*(3345*(a^6*x^6 - a^4*x^4)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) - 2*(3345*a^4*x^4 + 1115*a^3*x^3 - 446*a^2*x^2 + 200*a*x - 48)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*x^6 - x^4), -1/192*(3345*(a^6*x^6 - a^4*x^4)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (3345*a^4*x^4 + 1115*a^3*x^3 - 446*a^2*x^2 + 200*a*x - 48)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*x^6 - x^4)]","A",0
437,0,0,0,0.443660," ","integrate((-a*c*x+c)^p/exp(2*p*arctanh(a*x)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(-a c x + c\right)}^{p}}{\left(\frac{a x + 1}{a x - 1}\right)^{p}}, x\right)"," ",0,"integral((-a*c*x + c)^p/((a*x + 1)/(a*x - 1))^p, x)","F",0
438,1,37,0,0.555893," ","integrate(exp(2*p*arctanh(a*x))*(-a*c*x+c)^p,x, algorithm=""fricas"")","\frac{{\left(a x + 1\right)} {\left(-a c x + c\right)}^{p} \left(\frac{a x + 1}{a x - 1}\right)^{p}}{a p + a}"," ",0,"(a*x + 1)*(-a*c*x + c)^p*((a*x + 1)/(a*x - 1))^p/(a*p + a)","A",0
439,0,0,0,0.474485," ","integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-a c x + c\right)}^{p} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral((-a*c*x + c)^p*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
440,0,0,0,0.477233," ","integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^3,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} + 3 \, a c^{3} x - c^{3}\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(-(a^3*c^3*x^3 - 3*a^2*c^3*x^2 + 3*a*c^3*x - c^3)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
441,0,0,0,0.440930," ","integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral((a^2*c^2*x^2 - 2*a*c^2*x + c^2)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
442,0,0,0,0.442579," ","integrate(exp(n*arctanh(a*x))*(-a*c*x+c),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a c x - c\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(-(a*c*x - c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
443,0,0,0,0.545008," ","integrate(exp(n*arctanh(a*x))/(-a*c*x+c),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c x - c}, x\right)"," ",0,"integral(-((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c), x)","F",0
444,1,58,0,0.601141," ","integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^2,x, algorithm=""fricas"")","\frac{{\left(a x + 1\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c^{2} n + 2 \, a c^{2} - {\left(a^{2} c^{2} n + 2 \, a^{2} c^{2}\right)} x}"," ",0,"(a*x + 1)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^2*n + 2*a*c^2 - (a^2*c^2*n + 2*a^2*c^2)*x)","A",0
445,1,128,0,0.437024," ","integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^3,x, algorithm=""fricas"")","-\frac{{\left(a^{2} x^{2} - {\left(a n + 2 \, a\right)} x - n - 3\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c^{3} n^{2} + 6 \, a c^{3} n + 8 \, a c^{3} + {\left(a^{3} c^{3} n^{2} + 6 \, a^{3} c^{3} n + 8 \, a^{3} c^{3}\right)} x^{2} - 2 \, {\left(a^{2} c^{3} n^{2} + 6 \, a^{2} c^{3} n + 8 \, a^{2} c^{3}\right)} x}"," ",0,"-(a^2*x^2 - (a*n + 2*a)*x - n - 3)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^3*n^2 + 6*a*c^3*n + 8*a*c^3 + (a^3*c^3*n^2 + 6*a^3*c^3*n + 8*a^3*c^3)*x^2 - 2*(a^2*c^3*n^2 + 6*a^2*c^3*n + 8*a^2*c^3)*x)","A",0
446,1,228,0,0.729673," ","integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^4,x, algorithm=""fricas"")","\frac{{\left(2 \, a^{3} x^{3} - 2 \, {\left(a^{2} n + 3 \, a^{2}\right)} x^{2} + n^{2} + {\left(a n^{2} + 6 \, a n + 6 \, a\right)} x + 8 \, n + 14\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c^{4} n^{3} + 12 \, a c^{4} n^{2} + 44 \, a c^{4} n + 48 \, a c^{4} - {\left(a^{4} c^{4} n^{3} + 12 \, a^{4} c^{4} n^{2} + 44 \, a^{4} c^{4} n + 48 \, a^{4} c^{4}\right)} x^{3} + 3 \, {\left(a^{3} c^{4} n^{3} + 12 \, a^{3} c^{4} n^{2} + 44 \, a^{3} c^{4} n + 48 \, a^{3} c^{4}\right)} x^{2} - 3 \, {\left(a^{2} c^{4} n^{3} + 12 \, a^{2} c^{4} n^{2} + 44 \, a^{2} c^{4} n + 48 \, a^{2} c^{4}\right)} x}"," ",0,"(2*a^3*x^3 - 2*(a^2*n + 3*a^2)*x^2 + n^2 + (a*n^2 + 6*a*n + 6*a)*x + 8*n + 14)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^4*n^3 + 12*a*c^4*n^2 + 44*a*c^4*n + 48*a*c^4 - (a^4*c^4*n^3 + 12*a^4*c^4*n^2 + 44*a^4*c^4*n + 48*a^4*c^4)*x^3 + 3*(a^3*c^4*n^3 + 12*a^3*c^4*n^2 + 44*a^3*c^4*n + 48*a^3*c^4)*x^2 - 3*(a^2*c^4*n^3 + 12*a^2*c^4*n^2 + 44*a^2*c^4*n + 48*a^2*c^4)*x)","A",0
447,0,0,0,0.552233," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} \left(\frac{a c x - c}{a x}\right)^{p}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*((a*c*x - c)/(a*x))^p/(a*x - 1), x)","F",0
448,1,132,0,0.521011," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^4,x, algorithm=""fricas"")","\frac{36 \, a^{3} c^{4} x^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 3 \, a^{3} c^{4} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 6 \, a^{3} c^{4} x^{3} - {\left(6 \, a^{3} c^{4} x^{3} + 16 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + 2 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}}"," ",0,"1/6*(36*a^3*c^4*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*a^3*c^4*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 6*a^3*c^4*x^3 - (6*a^3*c^4*x^3 + 16*a^2*c^4*x^2 - 9*a*c^4*x + 2*c^4)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)","A",0
449,1,121,0,0.602950," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^3,x, algorithm=""fricas"")","\frac{8 \, a^{2} c^{3} x^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - a^{2} c^{3} x^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 2 \, a^{2} c^{3} x^{2} - {\left(2 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, a^{3} x^{2}}"," ",0,"1/2*(8*a^2*c^3*x^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - a^2*c^3*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 2*a^2*c^3*x^2 - (2*a^2*c^3*x^2 + 4*a*c^3*x - c^3)*sqrt(-a^2*x^2 + 1))/(a^3*x^2)","A",0
450,1,94,0,0.458518," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^2,x, algorithm=""fricas"")","\frac{2 \, a c^{2} x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - a c^{2} x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - a c^{2} x - {\left(a c^{2} x + c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{a^{2} x}"," ",0,"(2*a*c^2*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - a*c^2*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) - a*c^2*x - (a*c^2*x + c^2)*sqrt(-a^2*x^2 + 1))/(a^2*x)","A",0
451,1,41,0,0.425435," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x),x, algorithm=""fricas"")","-\frac{c \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + \sqrt{-a^{2} x^{2} + 1} c}{a}"," ",0,"-(c*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*c)/a","A",0
452,1,68,0,0.516151," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x),x, algorithm=""fricas"")","-\frac{3 \, a x + 4 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a x - 3\right)} - 3}{a^{2} c x - a c}"," ",0,"-(3*a*x + 4*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x - 3) - 3)/(a^2*c*x - a*c)","A",0
453,1,107,0,0.486476," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^2,x, algorithm=""fricas"")","-\frac{14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{2} x^{2} - 19 \, a x + 14\right)} \sqrt{-a^{2} x^{2} + 1} + 14}{3 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"-1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)","A",0
454,1,143,0,0.472140," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^3,x, algorithm=""fricas"")","-\frac{94 \, a^{3} x^{3} - 282 \, a^{2} x^{2} + 282 \, a x + 120 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(15 \, a^{3} x^{3} - 149 \, a^{2} x^{2} + 222 \, a x - 94\right)} \sqrt{-a^{2} x^{2} + 1} - 94}{15 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/15*(94*a^3*x^3 - 282*a^2*x^2 + 282*a*x + 120*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^3*x^3 - 149*a^2*x^2 + 222*a*x - 94)*sqrt(-a^2*x^2 + 1) - 94)/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)","A",0
455,1,177,0,0.476341," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^4,x, algorithm=""fricas"")","-\frac{824 \, a^{4} x^{4} - 3296 \, a^{3} x^{3} + 4944 \, a^{2} x^{2} - 3296 \, a x + 1050 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(105 \, a^{4} x^{4} - 1444 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 2771 \, a x + 824\right)} \sqrt{-a^{2} x^{2} + 1} + 824}{105 \, {\left(a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/105*(824*a^4*x^4 - 3296*a^3*x^3 + 4944*a^2*x^2 - 3296*a*x + 1050*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (105*a^4*x^4 - 1444*a^3*x^3 + 3256*a^2*x^2 - 2771*a*x + 824)*sqrt(-a^2*x^2 + 1) + 824)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)","A",0
456,0,0,0,0.502063," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x + 1\right)} \left(\frac{a c x - c}{a x}\right)^{p}}{a x - 1}, x\right)"," ",0,"integral(-(a*x + 1)*((a*c*x - c)/(a*x))^p/(a*x - 1), x)","F",0
457,1,67,0,0.548089," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^5,x, algorithm=""fricas"")","-\frac{4 \, a^{5} c^{5} x^{5} - 12 \, a^{4} c^{5} x^{4} \log\left(x\right) - 8 \, a^{3} c^{5} x^{3} - 4 \, a^{2} c^{5} x^{2} + 4 \, a c^{5} x - c^{5}}{4 \, a^{5} x^{4}}"," ",0,"-1/4*(4*a^5*c^5*x^5 - 12*a^4*c^5*x^4*log(x) - 8*a^3*c^5*x^3 - 4*a^2*c^5*x^2 + 4*a*c^5*x - c^5)/(a^5*x^4)","A",0
458,1,43,0,0.569692," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^4,x, algorithm=""fricas"")","-\frac{3 \, a^{4} c^{4} x^{4} - 6 \, a^{3} c^{4} x^{3} \log\left(x\right) - 3 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}}"," ",0,"-1/3*(3*a^4*c^4*x^4 - 6*a^3*c^4*x^3*log(x) - 3*a*c^4*x + c^4)/(a^4*x^3)","A",0
459,1,45,0,0.411337," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^3,x, algorithm=""fricas"")","-\frac{2 \, a^{3} c^{3} x^{3} - 2 \, a^{2} c^{3} x^{2} \log\left(x\right) + 2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}}"," ",0,"-1/2*(2*a^3*c^3*x^3 - 2*a^2*c^3*x^2*log(x) + 2*a*c^3*x - c^3)/(a^3*x^2)","A",0
460,1,22,0,0.560448," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^2,x, algorithm=""fricas"")","-\frac{a^{2} c^{2} x^{2} + c^{2}}{a^{2} x}"," ",0,"-(a^2*c^2*x^2 + c^2)/(a^2*x)","A",0
461,1,14,0,0.512434," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x),x, algorithm=""fricas"")","-\frac{a c x + c \log\left(x\right)}{a}"," ",0,"-(a*c*x + c*log(x))/a","A",0
462,1,41,0,0.455948," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x),x, algorithm=""fricas"")","-\frac{a^{2} x^{2} - a x + 3 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 2}{a^{2} c x - a c}"," ",0,"-(a^2*x^2 - a*x + 3*(a*x - 1)*log(a*x - 1) - 2)/(a^2*c*x - a*c)","A",0
463,1,71,0,0.420733," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^2,x, algorithm=""fricas"")","-\frac{a^{3} x^{3} - 2 \, a^{2} x^{2} - 4 \, a x + 4 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) + 4}{a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}}"," ",0,"-(a^3*x^3 - 2*a^2*x^2 - 4*a*x + 4*(a^2*x^2 - 2*a*x + 1)*log(a*x - 1) + 4)/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)","A",0
464,1,100,0,0.490217," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^3,x, algorithm=""fricas"")","-\frac{6 \, a^{4} x^{4} - 18 \, a^{3} x^{3} - 36 \, a^{2} x^{2} + 81 \, a x + 30 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \log\left(a x - 1\right) - 37}{6 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/6*(6*a^4*x^4 - 18*a^3*x^3 - 36*a^2*x^2 + 81*a*x + 30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(a*x - 1) - 37)/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)","A",0
465,1,126,0,0.424037," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^4,x, algorithm=""fricas"")","-\frac{2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 60 \, a^{2} x^{2} - 56 \, a x + 12 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \log\left(a x - 1\right) + 17}{2 \, {\left(a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/2*(2*a^5*x^5 - 8*a^4*x^4 - 16*a^3*x^3 + 60*a^2*x^2 - 56*a*x + 12*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(a*x - 1) + 17)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)","A",0
466,1,132,0,0.653633," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^4,x, algorithm=""fricas"")","-\frac{12 \, a^{3} c^{4} x^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 9 \, a^{3} c^{4} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 6 \, a^{3} c^{4} x^{3} - {\left(6 \, a^{3} c^{4} x^{3} + 8 \, a^{2} c^{4} x^{2} + 3 \, a c^{4} x - 2 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}}"," ",0,"-1/6*(12*a^3*c^4*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 9*a^3*c^4*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 6*a^3*c^4*x^3 - (6*a^3*c^4*x^3 + 8*a^2*c^4*x^2 + 3*a*c^4*x - 2*c^4)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)","A",0
467,1,78,0,0.491593," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^3,x, algorithm=""fricas"")","\frac{3 \, a^{2} c^{3} x^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + 2 \, a^{2} c^{3} x^{2} + {\left(2 \, a^{2} c^{3} x^{2} + c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, a^{3} x^{2}}"," ",0,"1/2*(3*a^2*c^3*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 2*a^2*c^3*x^2 + (2*a^2*c^3*x^2 + c^3)*sqrt(-a^2*x^2 + 1))/(a^3*x^2)","A",0
468,1,93,0,0.578337," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^2,x, algorithm=""fricas"")","\frac{2 \, a c^{2} x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + a c^{2} x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + a c^{2} x + {\left(a c^{2} x - c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{a^{2} x}"," ",0,"(2*a*c^2*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + a*c^2*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) + a*c^2*x + (a*c^2*x - c^2)*sqrt(-a^2*x^2 + 1))/(a^2*x)","A",0
469,1,66,0,0.428167," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x),x, algorithm=""fricas"")","\frac{4 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - c \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + \sqrt{-a^{2} x^{2} + 1} c}{a}"," ",0,"(4*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - c*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*c)/a","A",0
470,1,101,0,0.470037," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x),x, algorithm=""fricas"")","\frac{19 \, a^{2} x^{2} - 38 \, a x + 24 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{2} x^{2} - 26 \, a x + 19\right)} \sqrt{-a^{2} x^{2} + 1} + 19}{3 \, {\left(a^{3} c x^{2} - 2 \, a^{2} c x + a c\right)}}"," ",0,"1/3*(19*a^2*x^2 - 38*a*x + 24*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 26*a*x + 19)*sqrt(-a^2*x^2 + 1) + 19)/(a^3*c*x^2 - 2*a^2*c*x + a*c)","A",0
471,1,143,0,0.492860," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^2,x, algorithm=""fricas"")","\frac{118 \, a^{3} x^{3} - 354 \, a^{2} x^{2} + 354 \, a x + 150 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(15 \, a^{3} x^{3} - 188 \, a^{2} x^{2} + 279 \, a x - 118\right)} \sqrt{-a^{2} x^{2} + 1} - 118}{15 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}"," ",0,"1/15*(118*a^3*x^3 - 354*a^2*x^2 + 354*a*x + 150*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^3*x^3 - 188*a^2*x^2 + 279*a*x - 118)*sqrt(-a^2*x^2 + 1) - 118)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)","A",0
472,1,177,0,0.507043," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^3,x, algorithm=""fricas"")","\frac{66 \, a^{4} x^{4} - 264 \, a^{3} x^{3} + 396 \, a^{2} x^{2} - 264 \, a x + 84 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(7 \, a^{4} x^{4} - 116 \, a^{3} x^{3} + 261 \, a^{2} x^{2} - 222 \, a x + 66\right)} \sqrt{-a^{2} x^{2} + 1} + 66}{7 \, {\left(a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"1/7*(66*a^4*x^4 - 264*a^3*x^3 + 396*a^2*x^2 - 264*a*x + 84*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (7*a^4*x^4 - 116*a^3*x^3 + 261*a^2*x^2 - 222*a*x + 66)*sqrt(-a^2*x^2 + 1) + 66)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)","A",0
473,1,213,0,0.792321," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^4,x, algorithm=""fricas"")","\frac{3464 \, a^{5} x^{5} - 17320 \, a^{4} x^{4} + 34640 \, a^{3} x^{3} - 34640 \, a^{2} x^{2} + 17320 \, a x + 4410 \, {\left(a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(315 \, a^{5} x^{5} - 6539 \, a^{4} x^{4} + 19780 \, a^{3} x^{3} - 25347 \, a^{2} x^{2} + 15115 \, a x - 3464\right)} \sqrt{-a^{2} x^{2} + 1} - 3464}{315 \, {\left(a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"1/315*(3464*a^5*x^5 - 17320*a^4*x^4 + 34640*a^3*x^3 - 34640*a^2*x^2 + 17320*a*x + 4410*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (315*a^5*x^5 - 6539*a^4*x^4 + 19780*a^3*x^3 - 25347*a^2*x^2 + 15115*a*x - 3464)*sqrt(-a^2*x^2 + 1) - 3464)/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4)","A",0
474,0,0,0,0.506684," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^p,x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} x^{2} + 2 \, a x + 1\right)} \left(\frac{a c x - c}{a x}\right)^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral((a^2*x^2 + 2*a*x + 1)*((a*c*x - c)/(a*x))^p/(a^2*x^2 - 2*a*x + 1), x)","F",0
475,1,67,0,0.440779," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^5,x, algorithm=""fricas"")","\frac{12 \, a^{5} c^{5} x^{5} - 12 \, a^{4} c^{5} x^{4} \log\left(x\right) + 24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}}"," ",0,"1/12*(12*a^5*c^5*x^5 - 12*a^4*c^5*x^4*log(x) + 24*a^3*c^5*x^3 - 12*a^2*c^5*x^2 - 4*a*c^5*x + 3*c^5)/(a^5*x^4)","A",0
476,1,36,0,0.491064," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^4,x, algorithm=""fricas"")","\frac{3 \, a^{4} c^{4} x^{4} + 6 \, a^{2} c^{4} x^{2} - c^{4}}{3 \, a^{4} x^{3}}"," ",0,"1/3*(3*a^4*c^4*x^4 + 6*a^2*c^4*x^2 - c^4)/(a^4*x^3)","A",0
477,1,43,0,0.552478," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^3,x, algorithm=""fricas"")","\frac{2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} \log\left(x\right) + 2 \, a c^{3} x + c^{3}}{2 \, a^{3} x^{2}}"," ",0,"1/2*(2*a^3*c^3*x^3 + 2*a^2*c^3*x^2*log(x) + 2*a*c^3*x + c^3)/(a^3*x^2)","A",0
478,1,32,0,0.407952," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^2,x, algorithm=""fricas"")","\frac{a^{2} c^{2} x^{2} + 2 \, a c^{2} x \log\left(x\right) - c^{2}}{a^{2} x}"," ",0,"(a^2*c^2*x^2 + 2*a*c^2*x*log(x) - c^2)/(a^2*x)","A",0
479,1,23,0,0.518854," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x),x, algorithm=""fricas"")","\frac{a c x + 4 \, c \log\left(a x - 1\right) - c \log\left(x\right)}{a}"," ",0,"(a*c*x + 4*c*log(a*x - 1) - c*log(x))/a","A",0
480,1,64,0,0.504111," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(c-c/a/x),x, algorithm=""fricas"")","\frac{a^{3} x^{3} - 2 \, a^{2} x^{2} - 7 \, a x + 5 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) + 6}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}"," ",0,"(a^3*x^3 - 2*a^2*x^2 - 7*a*x + 5*(a^2*x^2 - 2*a*x + 1)*log(a*x - 1) + 6)/(a^3*c*x^2 - 2*a^2*c*x + a*c)","A",0
481,1,100,0,0.450803," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(c-c/a/x)^2,x, algorithm=""fricas"")","\frac{3 \, a^{4} x^{4} - 9 \, a^{3} x^{3} - 30 \, a^{2} x^{2} + 57 \, a x + 18 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \log\left(a x - 1\right) - 25}{3 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}"," ",0,"1/3*(3*a^4*x^4 - 9*a^3*x^3 - 30*a^2*x^2 + 57*a*x + 18*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(a*x - 1) - 25)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)","A",0
482,1,126,0,0.513655," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(c-c/a/x)^3,x, algorithm=""fricas"")","\frac{6 \, a^{5} x^{5} - 24 \, a^{4} x^{4} - 78 \, a^{3} x^{3} + 243 \, a^{2} x^{2} - 218 \, a x + 42 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \log\left(a x - 1\right) + 65}{6 \, {\left(a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"1/6*(6*a^5*x^5 - 24*a^4*x^4 - 78*a^3*x^3 + 243*a^2*x^2 - 218*a*x + 42*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(a*x - 1) + 65)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)","A",0
483,1,154,0,0.425306," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(c-c/a/x)^4,x, algorithm=""fricas"")","\frac{15 \, a^{6} x^{6} - 75 \, a^{5} x^{5} - 240 \, a^{4} x^{4} + 1080 \, a^{3} x^{3} - 1480 \, a^{2} x^{2} + 890 \, a x + 120 \, {\left(a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right)} \log\left(a x - 1\right) - 202}{15 \, {\left(a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"1/15*(15*a^6*x^6 - 75*a^5*x^5 - 240*a^4*x^4 + 1080*a^3*x^3 - 1480*a^2*x^2 + 890*a*x + 120*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*log(a*x - 1) - 202)/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4)","A",0
484,0,0,0,0.529548," ","integrate((c-c/a/x)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} \left(\frac{a c x - c}{a x}\right)^{p}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*((a*c*x - c)/(a*x))^p/(a*x + 1), x)","F",0
485,1,132,0,0.433311," ","integrate((c-c/a/x)^4/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{60 \, a^{3} c^{4} x^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 75 \, a^{3} c^{4} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 6 \, a^{3} c^{4} x^{3} - {\left(6 \, a^{3} c^{4} x^{3} - 64 \, a^{2} c^{4} x^{2} + 15 \, a c^{4} x - 2 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}}"," ",0,"-1/6*(60*a^3*c^4*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 75*a^3*c^4*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 6*a^3*c^4*x^3 - (6*a^3*c^4*x^3 - 64*a^2*c^4*x^2 + 15*a*c^4*x - 2*c^4)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)","A",0
486,1,119,0,0.426906," ","integrate((c-c/a/x)^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{16 \, a^{2} c^{3} x^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 13 \, a^{2} c^{3} x^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 2 \, a^{2} c^{3} x^{2} - {\left(2 \, a^{2} c^{3} x^{2} - 8 \, a c^{3} x + c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, a^{3} x^{2}}"," ",0,"-1/2*(16*a^2*c^3*x^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 13*a^2*c^3*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 2*a^2*c^3*x^2 - (2*a^2*c^3*x^2 - 8*a*c^3*x + c^3)*sqrt(-a^2*x^2 + 1))/(a^3*x^2)","A",0
487,1,97,0,0.483922," ","integrate((c-c/a/x)^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{6 \, a c^{2} x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 3 \, a c^{2} x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - a c^{2} x - {\left(a c^{2} x - c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{a^{2} x}"," ",0,"-(6*a*c^2*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*a*c^2*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) - a*c^2*x - (a*c^2*x - c^2)*sqrt(-a^2*x^2 + 1))/(a^2*x)","A",0
488,1,67,0,0.519104," ","integrate((c-c/a/x)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{4 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + c \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \sqrt{-a^{2} x^{2} + 1} c}{a}"," ",0,"-(4*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + c*log((sqrt(-a^2*x^2 + 1) - 1)/x) - sqrt(-a^2*x^2 + 1)*c)/a","A",0
489,1,19,0,0.552901," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} x^{2} + 1}}{a c}"," ",0,"sqrt(-a^2*x^2 + 1)/(a*c)","A",0
490,1,71,0,0.472611," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^2,x, algorithm=""fricas"")","\frac{2 \, a x + 2 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a x - 2\right)} - 2}{a^{2} c^{2} x - a c^{2}}"," ",0,"(2*a*x + 2*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x - 2) - 2)/(a^2*c^2*x - a*c^2)","A",0
491,1,107,0,0.643180," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^3,x, algorithm=""fricas"")","\frac{10 \, a^{2} x^{2} - 20 \, a x + 12 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{2} x^{2} - 14 \, a x + 10\right)} \sqrt{-a^{2} x^{2} + 1} + 10}{3 \, {\left(a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"1/3*(10*a^2*x^2 - 20*a*x + 12*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 14*a*x + 10)*sqrt(-a^2*x^2 + 1) + 10)/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3)","A",0
492,1,143,0,0.527704," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^4,x, algorithm=""fricas"")","\frac{24 \, a^{3} x^{3} - 72 \, a^{2} x^{2} + 72 \, a x + 30 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(5 \, a^{3} x^{3} - 39 \, a^{2} x^{2} + 57 \, a x - 24\right)} \sqrt{-a^{2} x^{2} + 1} - 24}{5 \, {\left(a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"1/5*(24*a^3*x^3 - 72*a^2*x^2 + 72*a*x + 30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (5*a^3*x^3 - 39*a^2*x^2 + 57*a*x - 24)*sqrt(-a^2*x^2 + 1) - 24)/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)","A",0
493,0,0,0,0.563271," ","integrate((c-c/a/x)^p/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x - 1\right)} \left(\frac{a c x - c}{a x}\right)^{p}}{a x + 1}, x\right)"," ",0,"integral(-(a*x - 1)*((a*c*x - c)/(a*x))^p/(a*x + 1), x)","F",0
494,1,71,0,0.450719," ","integrate((c-c/a/x)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{3 \, a^{4} c^{4} x^{4} - 96 \, a^{3} c^{4} x^{3} \log\left(a x + 1\right) + 78 \, a^{3} c^{4} x^{3} \log\left(x\right) + 48 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}}"," ",0,"-1/3*(3*a^4*c^4*x^4 - 96*a^3*c^4*x^3*log(a*x + 1) + 78*a^3*c^4*x^3*log(x) + 48*a^2*c^4*x^2 - 9*a*c^4*x + c^4)/(a^4*x^3)","A",0
495,1,62,0,0.415427," ","integrate((c-c/a/x)^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{2 \, a^{3} c^{3} x^{3} - 32 \, a^{2} c^{3} x^{2} \log\left(a x + 1\right) + 22 \, a^{2} c^{3} x^{2} \log\left(x\right) + 10 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}}"," ",0,"-1/2*(2*a^3*c^3*x^3 - 32*a^2*c^3*x^2*log(a*x + 1) + 22*a^2*c^3*x^2*log(x) + 10*a*c^3*x - c^3)/(a^3*x^2)","A",0
496,1,44,0,0.434110," ","integrate((c-c/a/x)^2/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{a^{2} c^{2} x^{2} - 8 \, a c^{2} x \log\left(a x + 1\right) + 4 \, a c^{2} x \log\left(x\right) + c^{2}}{a^{2} x}"," ",0,"-(a^2*c^2*x^2 - 8*a*c^2*x*log(a*x + 1) + 4*a*c^2*x*log(x) + c^2)/(a^2*x)","A",0
497,1,23,0,0.506968," ","integrate((c-c/a/x)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{a c x - 4 \, c \log\left(a x + 1\right) + c \log\left(x\right)}{a}"," ",0,"-(a*c*x - 4*c*log(a*x + 1) + c*log(x))/a","A",0
498,1,20,0,0.553302," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x),x, algorithm=""fricas"")","-\frac{a x - \log\left(a x + 1\right)}{a c}"," ",0,"-(a*x - log(a*x + 1))/(a*c)","A",0
499,1,27,0,0.522163," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^2,x, algorithm=""fricas"")","-\frac{2 \, a x - \log\left(a x + 1\right) + \log\left(a x - 1\right)}{2 \, a c^{2}}"," ",0,"-1/2*(2*a*x - log(a*x + 1) + log(a*x - 1))/(a*c^2)","A",0
500,1,59,0,0.398519," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^3,x, algorithm=""fricas"")","-\frac{4 \, a^{2} x^{2} - 4 \, a x - {\left(a x - 1\right)} \log\left(a x + 1\right) + 5 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 2}{4 \, {\left(a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/4*(4*a^2*x^2 - 4*a*x - (a*x - 1)*log(a*x + 1) + 5*(a*x - 1)*log(a*x - 1) - 2)/(a^2*c^3*x - a*c^3)","A",0
501,1,93,0,0.406904," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^4,x, algorithm=""fricas"")","-\frac{8 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 6 \, a x - {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) + 17 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) + 12}{8 \, {\left(a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/8*(8*a^3*x^3 - 16*a^2*x^2 - 6*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 17*(a^2*x^2 - 2*a*x + 1)*log(a*x - 1) + 12)/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)","A",0
502,1,177,0,0.473161," ","integrate((c-c/a/x)^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{66 \, a^{3} c^{3} x^{3} + 66 \, a^{2} c^{3} x^{2} - 24 \, {\left(a^{3} c^{3} x^{3} + a^{2} c^{3} x^{2}\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 33 \, {\left(a^{3} c^{3} x^{3} + a^{2} c^{3} x^{2}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(2 \, a^{3} c^{3} x^{3} + 78 \, a^{2} c^{3} x^{2} + 11 \, a c^{3} x - c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a^{4} x^{3} + a^{3} x^{2}\right)}}"," ",0,"-1/2*(66*a^3*c^3*x^3 + 66*a^2*c^3*x^2 - 24*(a^3*c^3*x^3 + a^2*c^3*x^2)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 33*(a^3*c^3*x^3 + a^2*c^3*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (2*a^3*c^3*x^3 + 78*a^2*c^3*x^2 + 11*a*c^3*x - c^3)*sqrt(-a^2*x^2 + 1))/(a^4*x^3 + a^3*x^2)","A",0
503,1,149,0,0.604839," ","integrate((c-c/a/x)^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{17 \, a^{2} c^{2} x^{2} + 17 \, a c^{2} x - 10 \, {\left(a^{2} c^{2} x^{2} + a c^{2} x\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 5 \, {\left(a^{2} c^{2} x^{2} + a c^{2} x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + {\left(a^{2} c^{2} x^{2} + 18 \, a c^{2} x + c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{a^{3} x^{2} + a^{2} x}"," ",0,"-(17*a^2*c^2*x^2 + 17*a*c^2*x - 10*(a^2*c^2*x^2 + a*c^2*x)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 5*(a^2*c^2*x^2 + a*c^2*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (a^2*c^2*x^2 + 18*a*c^2*x + c^2)*sqrt(-a^2*x^2 + 1))/(a^3*x^2 + a^2*x)","A",0
504,1,97,0,0.433832," ","integrate((c-c/a/x)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{9 \, a c x - 8 \, {\left(a c x + c\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(a c x + c\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a c x + 9 \, c\right)} + 9 \, c}{a^{2} x + a}"," ",0,"-(9*a*c*x - 8*(a*c*x + c)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a*c*x + c)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*(a*c*x + 9*c) + 9*c)/(a^2*x + a)","A",0
505,1,67,0,0.469997," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x),x, algorithm=""fricas"")","-\frac{3 \, a x - 4 \, {\left(a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a x + 3\right)} + 3}{a^{2} c x + a c}"," ",0,"-(3*a*x - 4*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x + 3) + 3)/(a^2*c*x + a*c)","A",0
506,1,71,0,0.544924," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^2,x, algorithm=""fricas"")","-\frac{2 \, a x - 2 \, {\left(a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a x + 2\right)} + 2}{a^{2} c^{2} x + a c^{2}}"," ",0,"-(2*a*x - 2*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x + 2) + 2)/(a^2*c^2*x + a*c^2)","A",0
507,1,53,0,1.365042," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^3,x, algorithm=""fricas"")","-\frac{2 \, a^{2} x^{2} + {\left(a^{2} x^{2} - 2\right)} \sqrt{-a^{2} x^{2} + 1} - 2}{a^{3} c^{3} x^{2} - a c^{3}}"," ",0,"-(2*a^2*x^2 + (a^2*x^2 - 2)*sqrt(-a^2*x^2 + 1) - 2)/(a^3*c^3*x^2 - a*c^3)","A",0
508,1,142,0,0.854418," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^4,x, algorithm=""fricas"")","-\frac{8 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 8 \, a x + 6 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} + 8}{3 \, {\left(a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/3*(8*a^3*x^3 - 8*a^2*x^2 - 8*a*x + 6*(a^3*x^3 - a^2*x^2 - a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^3*x^3 - 7*a^2*x^2 - 5*a*x + 8)*sqrt(-a^2*x^2 + 1) + 8)/(a^4*c^4*x^3 - a^3*c^4*x^2 - a^2*c^4*x + a*c^4)","A",0
509,1,151,0,0.578152," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^5,x, algorithm=""fricas"")","-\frac{56 \, a^{4} x^{4} - 112 \, a^{3} x^{3} + 112 \, a x + 60 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(15 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 82 \, a x - 56\right)} \sqrt{-a^{2} x^{2} + 1} - 56}{15 \, {\left(a^{5} c^{5} x^{4} - 2 \, a^{4} c^{5} x^{3} + 2 \, a^{2} c^{5} x - a c^{5}\right)}}"," ",0,"-1/15*(56*a^4*x^4 - 112*a^3*x^3 + 112*a*x + 60*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^4*x^4 - 76*a^3*x^3 + 32*a^2*x^2 + 82*a*x - 56)*sqrt(-a^2*x^2 + 1) - 56)/(a^5*c^5*x^4 - 2*a^4*c^5*x^3 + 2*a^2*c^5*x - a*c^5)","A",0
510,1,386,0,0.552151," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(9/2),x, algorithm=""fricas"")","\left[\frac{735 \, {\left(a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(105 \, a^{4} c^{4} x^{4} + 292 \, a^{3} c^{4} x^{3} - 356 \, a^{2} c^{4} x^{2} + 162 \, a c^{4} x - 30 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{420 \, {\left(a^{5} x^{4} - a^{4} x^{3}\right)}}, \frac{735 \, {\left(a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(105 \, a^{4} c^{4} x^{4} + 292 \, a^{3} c^{4} x^{3} - 356 \, a^{2} c^{4} x^{2} + 162 \, a c^{4} x - 30 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{210 \, {\left(a^{5} x^{4} - a^{4} x^{3}\right)}}\right]"," ",0,"[1/420*(735*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(105*a^4*c^4*x^4 + 292*a^3*c^4*x^3 - 356*a^2*c^4*x^2 + 162*a*c^4*x - 30*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x^3), 1/210*(735*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(105*a^4*c^4*x^4 + 292*a^3*c^4*x^3 - 356*a^2*c^4*x^2 + 162*a*c^4*x - 30*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x^3)]","A",0
511,1,364,0,0.701630," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(7/2),x, algorithm=""fricas"")","\left[\frac{75 \, {\left(a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(15 \, a^{3} c^{3} x^{3} + 56 \, a^{2} c^{3} x^{2} - 28 \, a c^{3} x + 6 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{60 \, {\left(a^{4} x^{3} - a^{3} x^{2}\right)}}, \frac{75 \, {\left(a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(15 \, a^{3} c^{3} x^{3} + 56 \, a^{2} c^{3} x^{2} - 28 \, a c^{3} x + 6 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{30 \, {\left(a^{4} x^{3} - a^{3} x^{2}\right)}}\right]"," ",0,"[1/60*(75*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(15*a^3*c^3*x^3 + 56*a^2*c^3*x^2 - 28*a*c^3*x + 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2), 1/30*(75*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(15*a^3*c^3*x^3 + 56*a^2*c^3*x^2 - 28*a*c^3*x + 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2)]","A",0
512,1,330,0,1.677871," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(5/2),x, algorithm=""fricas"")","\left[\frac{9 \, {\left(a^{2} c^{2} x^{2} - a c^{2} x\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(3 \, a^{2} c^{2} x^{2} + 10 \, a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{12 \, {\left(a^{3} x^{2} - a^{2} x\right)}}, \frac{9 \, {\left(a^{2} c^{2} x^{2} - a c^{2} x\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(3 \, a^{2} c^{2} x^{2} + 10 \, a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{6 \, {\left(a^{3} x^{2} - a^{2} x\right)}}\right]"," ",0,"[1/12*(9*(a^2*c^2*x^2 - a*c^2*x)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(3*a^2*c^2*x^2 + 10*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x), 1/6*(9*(a^2*c^2*x^2 - a*c^2*x)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(3*a^2*c^2*x^2 + 10*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x)]","A",0
513,1,266,0,0.596224," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a c x - c\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, \sqrt{-a^{2} x^{2} + 1} {\left(a c x + 2 \, c\right)} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{2} x - a\right)}}, \frac{{\left(a c x - c\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} {\left(a c x + 2 \, c\right)} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[1/4*((a*c*x - c)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*sqrt(-a^2*x^2 + 1)*(a*c*x + 2*c)*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a), 1/2*((a*c*x - c)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*sqrt(-a^2*x^2 + 1)*(a*c*x + 2*c)*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a)]","A",0
514,1,249,0,0.586005," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[-\frac{4 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} - {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right)}{4 \, {\left(a^{2} x - a\right)}}, -\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} + {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right)}{2 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[-1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - (a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)))/(a^2*x - a), -1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + (a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)))/(a^2*x - a)]","A",0
515,1,447,0,0.570192," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[-\frac{4 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} - 2 \, \sqrt{2} {\left(a c x - c\right)} \sqrt{-\frac{1}{c}} \log\left(-\frac{17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{1}{c}} \sqrt{\frac{a c x - c}{a x}} - 13 \, a x - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 3 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right)}{4 \, {\left(a^{2} c x - a c\right)}}, -\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} + 3 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - \frac{2 \, \sqrt{2} {\left(a c x - c\right)} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}}}{{\left(3 \, a^{2} x^{2} - 2 \, a x - 1\right)} \sqrt{c}}\right)}{\sqrt{c}}}{2 \, {\left(a^{2} c x - a c\right)}}\right]"," ",0,"[-1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - 2*sqrt(2)*(a*c*x - c)*sqrt(-1/c)*log(-(17*a^3*x^3 - 3*a^2*x^2 - 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-1/c)*sqrt((a*c*x - c)/(a*x)) - 13*a*x - 1)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 3*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)))/(a^2*c*x - a*c), -1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + 3*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*sqrt(2)*(a*c*x - c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x))/((3*a^2*x^2 - 2*a*x - 1)*sqrt(c)))/sqrt(c))/(a^2*c*x - a*c)]","A",0
516,1,526,0,0.568405," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(3/2),x, algorithm=""fricas"")","\left[-\frac{7 \, \sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 10 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 8 \, {\left(a^{2} x^{2} - 2 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}, \frac{7 \, \sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - 10 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 4 \, {\left(a^{2} x^{2} - 2 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}\right]"," ",0,"[-1/8*(7*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 10*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(a^2*x^2 - 2*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), 1/4*(7*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 10*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 4*(a^2*x^2 - 2*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]","A",0
517,1,600,0,0.641470," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x, algorithm=""fricas"")","\left[-\frac{79 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 112 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 8 \, {\left(8 \, a^{3} x^{3} - 35 \, a^{2} x^{2} + 23 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{64 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}, \frac{79 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - 112 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 4 \, {\left(8 \, a^{3} x^{3} - 35 \, a^{2} x^{2} + 23 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{32 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}\right]"," ",0,"[-1/64*(79*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 112*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(8*a^3*x^3 - 35*a^2*x^2 + 23*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3), 1/32*(79*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 112*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 4*(8*a^3*x^3 - 35*a^2*x^2 + 23*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)]","A",0
518,1,234,0,0.457338," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(9/2),x, algorithm=""fricas"")","\left[\frac{105 \, a^{3} c^{\frac{9}{2}} x^{3} \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(21 \, a^{4} c^{4} x^{4} + 92 \, a^{3} c^{4} x^{3} + 4 \, a^{2} c^{4} x^{2} - 18 \, a c^{4} x + 6 \, c^{4}\right)} \sqrt{\frac{a c x - c}{a x}}}{42 \, a^{4} x^{3}}, -\frac{105 \, a^{3} \sqrt{-c} c^{4} x^{3} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) + {\left(21 \, a^{4} c^{4} x^{4} + 92 \, a^{3} c^{4} x^{3} + 4 \, a^{2} c^{4} x^{2} - 18 \, a c^{4} x + 6 \, c^{4}\right)} \sqrt{\frac{a c x - c}{a x}}}{21 \, a^{4} x^{3}}\right]"," ",0,"[1/42*(105*a^3*c^(9/2)*x^3*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(21*a^4*c^4*x^4 + 92*a^3*c^4*x^3 + 4*a^2*c^4*x^2 - 18*a*c^4*x + 6*c^4)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3), -1/21*(105*a^3*sqrt(-c)*c^4*x^3*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (21*a^4*c^4*x^4 + 92*a^3*c^4*x^3 + 4*a^2*c^4*x^2 - 18*a*c^4*x + 6*c^4)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3)]","A",0
519,1,212,0,0.412128," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(7/2),x, algorithm=""fricas"")","\left[\frac{15 \, a^{2} c^{\frac{7}{2}} x^{2} \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(5 \, a^{3} c^{3} x^{3} + 8 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - 2 \, c^{3}\right)} \sqrt{\frac{a c x - c}{a x}}}{10 \, a^{3} x^{2}}, -\frac{15 \, a^{2} \sqrt{-c} c^{3} x^{2} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) + {\left(5 \, a^{3} c^{3} x^{3} + 8 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - 2 \, c^{3}\right)} \sqrt{\frac{a c x - c}{a x}}}{5 \, a^{3} x^{2}}\right]"," ",0,"[1/10*(15*a^2*c^(7/2)*x^2*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(5*a^3*c^3*x^3 + 8*a^2*c^3*x^2 + 4*a*c^3*x - 2*c^3)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2), -1/5*(15*a^2*sqrt(-c)*c^3*x^2*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (5*a^3*c^3*x^3 + 8*a^2*c^3*x^2 + 4*a*c^3*x - 2*c^3)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2)]","A",0
520,1,182,0,0.537458," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, a c^{\frac{5}{2}} x \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(3 \, a^{2} c^{2} x^{2} - 2 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{\frac{a c x - c}{a x}}}{6 \, a^{2} x}, -\frac{3 \, a \sqrt{-c} c^{2} x \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) + {\left(3 \, a^{2} c^{2} x^{2} - 2 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} x}\right]"," ",0,"[1/6*(3*a*c^(5/2)*x*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(3*a^2*c^2*x^2 - 2*a*c^2*x + 2*c^2)*sqrt((a*c*x - c)/(a*x)))/(a^2*x), -1/3*(3*a*sqrt(-c)*c^2*x*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (3*a^2*c^2*x^2 - 2*a*c^2*x + 2*c^2)*sqrt((a*c*x - c)/(a*x)))/(a^2*x)]","A",0
521,1,136,0,0.441317," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(3/2),x, algorithm=""fricas"")","\left[\frac{c^{\frac{3}{2}} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(a c x - 2 \, c\right)} \sqrt{\frac{a c x - c}{a x}}}{2 \, a}, \frac{\sqrt{-c} c \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - {\left(a c x - 2 \, c\right)} \sqrt{\frac{a c x - c}{a x}}}{a}\right]"," ",0,"[1/2*(c^(3/2)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/a, (sqrt(-c)*c*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/a]","A",0
522,1,125,0,0.446501," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, a x \sqrt{\frac{a c x - c}{a x}} - 3 \, \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{2 \, a}, -\frac{a x \sqrt{\frac{a c x - c}{a x}} - 3 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{a}\right]"," ",0,"[-1/2*(2*a*x*sqrt((a*c*x - c)/(a*x)) - 3*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a, -(a*x*sqrt((a*c*x - c)/(a*x)) - 3*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a]","A",0
523,1,175,0,0.441943," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[\frac{5 \, {\left(a x - 1\right)} \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(a^{2} x^{2} - 5 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{2} c x - a c\right)}}, \frac{5 \, {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - {\left(a^{2} x^{2} - 5 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{a^{2} c x - a c}\right]"," ",0,"[1/2*(5*(a*x - 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(a^2*x^2 - 5*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^2*c*x - a*c), (5*(a*x - 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (a^2*x^2 - 5*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^2*c*x - a*c)]","A",0
524,1,238,0,0.655004," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^(3/2),x, algorithm=""fricas"")","\left[\frac{21 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{6 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}, \frac{21 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - {\left(3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{3 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}\right]"," ",0,"[1/6*(21*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(3*a^3*x^3 - 28*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), 1/3*(21*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (3*a^3*x^3 - 28*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]","A",0
525,1,294,0,0.437534," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^(5/2),x, algorithm=""fricas"")","\left[\frac{45 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(5 \, a^{4} x^{4} - 69 \, a^{3} x^{3} + 105 \, a^{2} x^{2} - 45 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{10 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}, \frac{45 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - {\left(5 \, a^{4} x^{4} - 69 \, a^{3} x^{3} + 105 \, a^{2} x^{2} - 45 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{5 \, {\left(a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right)}}\right]"," ",0,"[1/10*(45*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(5*a^4*x^4 - 69*a^3*x^3 + 105*a^2*x^2 - 45*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3), 1/5*(45*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (5*a^4*x^4 - 69*a^3*x^3 + 105*a^2*x^2 - 45*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)]","A",0
526,1,346,0,0.442141," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^(7/2),x, algorithm=""fricas"")","\left[\frac{1155 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{210 \, {\left(a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right)}}, \frac{1155 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - {\left(105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{105 \, {\left(a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right)}}\right]"," ",0,"[1/210*(1155*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(105*a^5*x^5 - 1936*a^4*x^4 + 4466*a^3*x^3 - 3850*a^2*x^2 + 1155*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4), 1/105*(1155*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (105*a^5*x^5 - 1936*a^4*x^4 + 4466*a^3*x^3 - 3850*a^2*x^2 + 1155*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)]","A",0
527,1,386,0,0.601262," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(9/2),x, algorithm=""fricas"")","\left[\frac{105 \, {\left(a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(35 \, a^{4} c^{4} x^{4} + 164 \, a^{3} c^{4} x^{3} - 12 \, a^{2} c^{4} x^{2} - 26 \, a c^{4} x + 10 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{140 \, {\left(a^{5} x^{4} - a^{4} x^{3}\right)}}, -\frac{105 \, {\left(a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(35 \, a^{4} c^{4} x^{4} + 164 \, a^{3} c^{4} x^{3} - 12 \, a^{2} c^{4} x^{2} - 26 \, a c^{4} x + 10 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{70 \, {\left(a^{5} x^{4} - a^{4} x^{3}\right)}}\right]"," ",0,"[1/140*(105*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(35*a^4*c^4*x^4 + 164*a^3*c^4*x^3 - 12*a^2*c^4*x^2 - 26*a*c^4*x + 10*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x^3), -1/70*(105*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(35*a^4*c^4*x^4 + 164*a^3*c^4*x^3 - 12*a^2*c^4*x^2 - 26*a*c^4*x + 10*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x^3)]","A",0
528,1,364,0,0.466337," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(7/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(15 \, a^{3} c^{3} x^{3} + 44 \, a^{2} c^{3} x^{2} + 8 \, a c^{3} x - 6 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{60 \, {\left(a^{4} x^{3} - a^{3} x^{2}\right)}}, -\frac{15 \, {\left(a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(15 \, a^{3} c^{3} x^{3} + 44 \, a^{2} c^{3} x^{2} + 8 \, a c^{3} x - 6 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{30 \, {\left(a^{4} x^{3} - a^{3} x^{2}\right)}}\right]"," ",0,"[1/60*(15*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(15*a^3*c^3*x^3 + 44*a^2*c^3*x^2 + 8*a*c^3*x - 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2), -1/30*(15*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(15*a^3*c^3*x^3 + 44*a^2*c^3*x^2 + 8*a*c^3*x - 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2)]","A",0
529,1,330,0,0.533566," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} c^{2} x^{2} - a c^{2} x\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(3 \, a^{2} c^{2} x^{2} + 2 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{12 \, {\left(a^{3} x^{2} - a^{2} x\right)}}, \frac{3 \, {\left(a^{2} c^{2} x^{2} - a c^{2} x\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) + 2 \, {\left(3 \, a^{2} c^{2} x^{2} + 2 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{6 \, {\left(a^{3} x^{2} - a^{2} x\right)}}\right]"," ",0,"[1/12*(3*(a^2*c^2*x^2 - a*c^2*x)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(3*a^2*c^2*x^2 + 2*a*c^2*x + 2*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x), 1/6*(3*(a^2*c^2*x^2 - a*c^2*x)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(3*a^2*c^2*x^2 + 2*a*c^2*x + 2*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x)]","A",0
530,1,268,0,0.537669," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a c x - c\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} {\left(a c x - 2 \, c\right)} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{2} x - a\right)}}, \frac{3 \, {\left(a c x - c\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} {\left(a c x - 2 \, c\right)} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[1/4*(3*(a*c*x - c)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*(a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a), 1/2*(3*(a*c*x - c)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*sqrt(-a^2*x^2 + 1)*(a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a)]","A",0
531,1,442,0,0.534877," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} + 4 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 5 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right)}{4 \, {\left(a^{2} x - a\right)}}, \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} - 4 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) + 5 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right)}{2 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + 4*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 5*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)))/(a^2*x - a), 1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - 4*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) + 5*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)))/(a^2*x - a)]","A",0
532,1,513,0,0.834882," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[\frac{5 \, \sqrt{2} {\left(a^{2} c x^{2} - 2 \, a c x + c\right)} \sqrt{-\frac{1}{c}} \log\left(-\frac{17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{1}{c}} \sqrt{\frac{a c x - c}{a x}} - 13 \, a x - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) - 7 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(a^{2} x^{2} - 3 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{3} c x^{2} - 2 \, a^{2} c x + a c\right)}}, \frac{7 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) + 2 \, {\left(a^{2} x^{2} - 3 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}} - \frac{5 \, \sqrt{2} {\left(a^{2} c x^{2} - 2 \, a c x + c\right)} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}}}{{\left(3 \, a^{2} x^{2} - 2 \, a x - 1\right)} \sqrt{c}}\right)}{\sqrt{c}}}{2 \, {\left(a^{3} c x^{2} - 2 \, a^{2} c x + a c\right)}}\right]"," ",0,"[1/4*(5*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)*sqrt(-1/c)*log(-(17*a^3*x^3 - 3*a^2*x^2 + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-1/c)*sqrt((a*c*x - c)/(a*x)) - 13*a*x - 1)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) - 7*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(a^2*x^2 - 3*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c*x^2 - 2*a^2*c*x + a*c), 1/2*(7*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(a^2*x^2 - 3*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)) - 5*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x))/((3*a^2*x^2 - 2*a*x - 1)*sqrt(c)))/sqrt(c))/(a^3*c*x^2 - 2*a^2*c*x + a*c)]","A",0
533,1,600,0,0.603714," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(3/2),x, algorithm=""fricas"")","\left[-\frac{51 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 72 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 8 \, {\left(4 \, a^{3} x^{3} - 23 \, a^{2} x^{2} + 15 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{32 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}, -\frac{51 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - 72 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 4 \, {\left(4 \, a^{3} x^{3} - 23 \, a^{2} x^{2} + 15 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{16 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}\right]"," ",0,"[-1/32*(51*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 72*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 8*(4*a^3*x^3 - 23*a^2*x^2 + 15*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2), -1/16*(51*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 72*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 4*(4*a^3*x^3 - 23*a^2*x^2 + 15*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)]","A",0
534,1,668,0,0.530198," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x, algorithm=""fricas"")","\left[-\frac{747 \, \sqrt{2} {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 1056 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 8 \, {\left(48 \, a^{4} x^{4} - 415 \, a^{3} x^{3} + 554 \, a^{2} x^{2} - 219 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{384 \, {\left(a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right)}}, -\frac{747 \, \sqrt{2} {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - 1056 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 4 \, {\left(48 \, a^{4} x^{4} - 415 \, a^{3} x^{3} + 554 \, a^{2} x^{2} - 219 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{192 \, {\left(a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right)}}\right]"," ",0,"[-1/384*(747*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 1056*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 8*(48*a^4*x^4 - 415*a^3*x^3 + 554*a^2*x^2 - 219*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3), -1/192*(747*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 1056*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 4*(48*a^4*x^4 - 415*a^3*x^3 + 554*a^2*x^2 - 219*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)]","A",0
535,1,386,0,0.456805," ","integrate((c-c/a/x)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{1155 \, {\left(a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(105 \, a^{4} c^{4} x^{4} - 4156 \, a^{3} c^{4} x^{3} + 1028 \, a^{2} c^{4} x^{2} - 246 \, a c^{4} x + 30 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{420 \, {\left(a^{5} x^{4} - a^{4} x^{3}\right)}}, -\frac{1155 \, {\left(a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(105 \, a^{4} c^{4} x^{4} - 4156 \, a^{3} c^{4} x^{3} + 1028 \, a^{2} c^{4} x^{2} - 246 \, a c^{4} x + 30 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{210 \, {\left(a^{5} x^{4} - a^{4} x^{3}\right)}}\right]"," ",0,"[1/420*(1155*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(105*a^4*c^4*x^4 - 4156*a^3*c^4*x^3 + 1028*a^2*c^4*x^2 - 246*a*c^4*x + 30*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x^3), -1/210*(1155*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(105*a^4*c^4*x^4 - 4156*a^3*c^4*x^3 + 1028*a^2*c^4*x^2 - 246*a*c^4*x + 30*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x^3)]","A",0
536,1,364,0,0.699432," ","integrate((c-c/a/x)^(7/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{45 \, {\left(a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(5 \, a^{3} c^{3} x^{3} - 92 \, a^{2} c^{3} x^{2} + 16 \, a c^{3} x - 2 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{20 \, {\left(a^{4} x^{3} - a^{3} x^{2}\right)}}, -\frac{45 \, {\left(a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(5 \, a^{3} c^{3} x^{3} - 92 \, a^{2} c^{3} x^{2} + 16 \, a c^{3} x - 2 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{10 \, {\left(a^{4} x^{3} - a^{3} x^{2}\right)}}\right]"," ",0,"[1/20*(45*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(5*a^3*c^3*x^3 - 92*a^2*c^3*x^2 + 16*a*c^3*x - 2*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2), -1/10*(45*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(5*a^3*c^3*x^3 - 92*a^2*c^3*x^2 + 16*a*c^3*x - 2*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2)]","A",0
537,1,330,0,0.669871," ","integrate((c-c/a/x)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{21 \, {\left(a^{2} c^{2} x^{2} - a c^{2} x\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(3 \, a^{2} c^{2} x^{2} - 22 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{12 \, {\left(a^{3} x^{2} - a^{2} x\right)}}, -\frac{21 \, {\left(a^{2} c^{2} x^{2} - a c^{2} x\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(3 \, a^{2} c^{2} x^{2} - 22 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{6 \, {\left(a^{3} x^{2} - a^{2} x\right)}}\right]"," ",0,"[1/12*(21*(a^2*c^2*x^2 - a*c^2*x)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(3*a^2*c^2*x^2 - 22*a*c^2*x + 2*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x), -1/6*(21*(a^2*c^2*x^2 - a*c^2*x)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(3*a^2*c^2*x^2 - 22*a*c^2*x + 2*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x)]","A",0
538,1,268,0,0.593087," ","integrate((c-c/a/x)^(3/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{5 \, {\left(a c x - c\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} {\left(a c x - 2 \, c\right)} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{2} x - a\right)}}, -\frac{5 \, {\left(a c x - c\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} {\left(a c x - 2 \, c\right)} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[1/4*(5*(a*c*x - c)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*(a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a), -1/2*(5*(a*c*x - c)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*sqrt(-a^2*x^2 + 1)*(a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a)]","A",0
539,1,250,0,0.448623," ","integrate((c-c/a/x)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} + 3 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right)}{4 \, {\left(a^{2} x - a\right)}}, \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} - 3 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right)}{2 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + 3*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)))/(a^2*x - a), 1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - 3*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)))/(a^2*x - a)]","A",0
540,1,254,0,0.498310," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} - {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right)}{4 \, {\left(a^{2} c x - a c\right)}}, \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} - {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right)}{2 \, {\left(a^{2} c x - a c\right)}}\right]"," ",0,"[1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - (a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)))/(a^2*c*x - a*c), 1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - (a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)))/(a^2*c*x - a*c)]","A",0
541,1,453,0,0.560040," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} + \sqrt{2} {\left(a c x - c\right)} \sqrt{-\frac{1}{c}} \log\left(-\frac{17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{1}{c}} \sqrt{\frac{a c x - c}{a x}} - 13 \, a x - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) - {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right)}{4 \, {\left(a^{2} c^{2} x - a c^{2}\right)}}, \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} + {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - \frac{\sqrt{2} {\left(a c x - c\right)} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}}}{{\left(3 \, a^{2} x^{2} - 2 \, a x - 1\right)} \sqrt{c}}\right)}{\sqrt{c}}}{2 \, {\left(a^{2} c^{2} x - a c^{2}\right)}}\right]"," ",0,"[1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + sqrt(2)*(a*c*x - c)*sqrt(-1/c)*log(-(17*a^3*x^3 - 3*a^2*x^2 + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-1/c)*sqrt((a*c*x - c)/(a*x)) - 13*a*x - 1)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) - (a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)))/(a^2*c^2*x - a*c^2), 1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + (a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - sqrt(2)*(a*c*x - c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x))/((3*a^2*x^2 - 2*a*x - 1)*sqrt(c)))/sqrt(c))/(a^2*c^2*x - a*c^2)]","A",0
542,1,528,0,0.570745," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x, algorithm=""fricas"")","\left[-\frac{9 \, \sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 12 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 8 \, {\left(2 \, a^{2} x^{2} - 3 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{16 \, {\left(a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right)}}, -\frac{9 \, \sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - 12 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 4 \, {\left(2 \, a^{2} x^{2} - 3 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \, {\left(a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right)}}\right]"," ",0,"[-1/16*(9*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 12*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 8*(2*a^2*x^2 - 3*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3), -1/8*(9*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 12*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 4*(2*a^2*x^2 - 3*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3)]","A",0
543,1,600,0,0.599159," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(7/2),x, algorithm=""fricas"")","\left[-\frac{115 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 160 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 8 \, {\left(16 \, a^{3} x^{3} - 55 \, a^{2} x^{2} + 35 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{128 \, {\left(a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right)}}, -\frac{115 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - 160 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 4 \, {\left(16 \, a^{3} x^{3} - 55 \, a^{2} x^{2} + 35 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{64 \, {\left(a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right)}}\right]"," ",0,"[-1/128*(115*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 160*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 8*(16*a^3*x^3 - 55*a^2*x^2 + 35*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4), -1/64*(115*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 160*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 4*(16*a^3*x^3 - 55*a^2*x^2 + 35*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)]","A",0
544,1,342,0,0.591439," ","integrate((c-c/a/x)^(9/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{6720 \, \sqrt{2} a^{3} c^{\frac{9}{2}} x^{3} \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) + 1365 \, a^{3} c^{\frac{9}{2}} x^{3} \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(105 \, a^{4} c^{4} x^{4} - 6428 \, a^{3} c^{4} x^{3} + 1196 \, a^{2} c^{4} x^{2} - 258 \, a c^{4} x + 30 \, c^{4}\right)} \sqrt{\frac{a c x - c}{a x}}}{210 \, a^{4} x^{3}}, \frac{6720 \, \sqrt{2} a^{3} \sqrt{-c} c^{4} x^{3} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) - 1365 \, a^{3} \sqrt{-c} c^{4} x^{3} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - {\left(105 \, a^{4} c^{4} x^{4} - 6428 \, a^{3} c^{4} x^{3} + 1196 \, a^{2} c^{4} x^{2} - 258 \, a c^{4} x + 30 \, c^{4}\right)} \sqrt{\frac{a c x - c}{a x}}}{105 \, a^{4} x^{3}}\right]"," ",0,"[1/210*(6720*sqrt(2)*a^3*c^(9/2)*x^3*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + 1365*a^3*c^(9/2)*x^3*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(105*a^4*c^4*x^4 - 6428*a^3*c^4*x^3 + 1196*a^2*c^4*x^2 - 258*a*c^4*x + 30*c^4)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3), 1/105*(6720*sqrt(2)*a^3*sqrt(-c)*c^4*x^3*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 1365*a^3*sqrt(-c)*c^4*x^3*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (105*a^4*c^4*x^4 - 6428*a^3*c^4*x^3 + 1196*a^2*c^4*x^2 - 258*a*c^4*x + 30*c^4)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3)]","A",0
545,1,320,0,0.499885," ","integrate((c-c/a/x)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{480 \, \sqrt{2} a^{2} c^{\frac{7}{2}} x^{2} \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) + 165 \, a^{2} c^{\frac{7}{2}} x^{2} \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(15 \, a^{3} c^{3} x^{3} - 376 \, a^{2} c^{3} x^{2} + 52 \, a c^{3} x - 6 \, c^{3}\right)} \sqrt{\frac{a c x - c}{a x}}}{30 \, a^{3} x^{2}}, \frac{480 \, \sqrt{2} a^{2} \sqrt{-c} c^{3} x^{2} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) - 165 \, a^{2} \sqrt{-c} c^{3} x^{2} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - {\left(15 \, a^{3} c^{3} x^{3} - 376 \, a^{2} c^{3} x^{2} + 52 \, a c^{3} x - 6 \, c^{3}\right)} \sqrt{\frac{a c x - c}{a x}}}{15 \, a^{3} x^{2}}\right]"," ",0,"[1/30*(480*sqrt(2)*a^2*c^(7/2)*x^2*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + 165*a^2*c^(7/2)*x^2*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(15*a^3*c^3*x^3 - 376*a^2*c^3*x^2 + 52*a*c^3*x - 6*c^3)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2), 1/15*(480*sqrt(2)*a^2*sqrt(-c)*c^3*x^2*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 165*a^2*sqrt(-c)*c^3*x^2*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (15*a^3*c^3*x^3 - 376*a^2*c^3*x^2 + 52*a*c^3*x - 6*c^3)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2)]","A",0
546,1,282,0,0.481593," ","integrate((c-c/a/x)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{48 \, \sqrt{2} a c^{\frac{5}{2}} x \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) + 27 \, a c^{\frac{5}{2}} x \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(3 \, a^{2} c^{2} x^{2} - 26 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{\frac{a c x - c}{a x}}}{6 \, a^{2} x}, \frac{48 \, \sqrt{2} a \sqrt{-c} c^{2} x \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) - 27 \, a \sqrt{-c} c^{2} x \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - {\left(3 \, a^{2} c^{2} x^{2} - 26 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} x}\right]"," ",0,"[1/6*(48*sqrt(2)*a*c^(5/2)*x*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + 27*a*c^(5/2)*x*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(3*a^2*c^2*x^2 - 26*a*c^2*x + 2*c^2)*sqrt((a*c*x - c)/(a*x)))/(a^2*x), 1/3*(48*sqrt(2)*a*sqrt(-c)*c^2*x*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 27*a*sqrt(-c)*c^2*x*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (3*a^2*c^2*x^2 - 26*a*c^2*x + 2*c^2)*sqrt((a*c*x - c)/(a*x)))/(a^2*x)]","A",0
547,1,231,0,0.475862," ","integrate((c-c/a/x)^(3/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{8 \, \sqrt{2} c^{\frac{3}{2}} \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) + 7 \, c^{\frac{3}{2}} \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, {\left(a c x - 2 \, c\right)} \sqrt{\frac{a c x - c}{a x}}}{2 \, a}, \frac{8 \, \sqrt{2} \sqrt{-c} c \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) - 7 \, \sqrt{-c} c \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - {\left(a c x - 2 \, c\right)} \sqrt{\frac{a c x - c}{a x}}}{a}\right]"," ",0,"[1/2*(8*sqrt(2)*c^(3/2)*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + 7*c^(3/2)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/a, (8*sqrt(2)*sqrt(-c)*c*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 7*sqrt(-c)*c*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/a]","A",0
548,1,217,0,0.449833," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, a x \sqrt{\frac{a c x - c}{a x}} - 4 \, \sqrt{2} \sqrt{c} \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) - 5 \, \sqrt{c} \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{2 \, a}, -\frac{a x \sqrt{\frac{a c x - c}{a x}} - 4 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + 5 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{a}\right]"," ",0,"[-1/2*(2*a*x*sqrt((a*c*x - c)/(a*x)) - 4*sqrt(2)*sqrt(c)*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) - 5*sqrt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a, -(a*x*sqrt((a*c*x - c)/(a*x)) - 4*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + 5*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a]","A",0
549,1,232,0,0.438441," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, a x \sqrt{\frac{a c x - c}{a x}} - 2 \, \sqrt{2} \sqrt{c} \log\left(\frac{\frac{2 \, \sqrt{2} a x \sqrt{\frac{a c x - c}{a x}}}{\sqrt{c}} - 3 \, a x + 1}{a x + 1}\right) - 3 \, \sqrt{c} \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{2 \, a c}, \frac{2 \, \sqrt{2} c \sqrt{-\frac{1}{c}} \arctan\left(\frac{\sqrt{2} a x \sqrt{-\frac{1}{c}} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}\right) - a x \sqrt{\frac{a c x - c}{a x}} - 3 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{a c}\right]"," ",0,"[-1/2*(2*a*x*sqrt((a*c*x - c)/(a*x)) - 2*sqrt(2)*sqrt(c)*log((2*sqrt(2)*a*x*sqrt((a*c*x - c)/(a*x))/sqrt(c) - 3*a*x + 1)/(a*x + 1)) - 3*sqrt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/(a*c), (2*sqrt(2)*c*sqrt(-1/c)*arctan(sqrt(2)*a*x*sqrt(-1/c)*sqrt((a*c*x - c)/(a*x))/(a*x - 1)) - a*x*sqrt((a*c*x - c)/(a*x)) - 3*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/(a*c)]","A",0
550,1,231,0,0.425296," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(3/2),x, algorithm=""fricas"")","\left[-\frac{2 \, a x \sqrt{\frac{a c x - c}{a x}} - \sqrt{2} \sqrt{c} \log\left(\frac{\frac{2 \, \sqrt{2} a x \sqrt{\frac{a c x - c}{a x}}}{\sqrt{c}} - 3 \, a x + 1}{a x + 1}\right) - \sqrt{c} \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{2 \, a c^{2}}, \frac{\sqrt{2} c \sqrt{-\frac{1}{c}} \arctan\left(\frac{\sqrt{2} a x \sqrt{-\frac{1}{c}} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}\right) - a x \sqrt{\frac{a c x - c}{a x}} - \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{a c^{2}}\right]"," ",0,"[-1/2*(2*a*x*sqrt((a*c*x - c)/(a*x)) - sqrt(2)*sqrt(c)*log((2*sqrt(2)*a*x*sqrt((a*c*x - c)/(a*x))/sqrt(c) - 3*a*x + 1)/(a*x + 1)) - sqrt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/(a*c^2), (sqrt(2)*c*sqrt(-1/c)*arctan(sqrt(2)*a*x*sqrt(-1/c)*sqrt((a*c*x - c)/(a*x))/(a*x - 1)) - a*x*sqrt((a*c*x - c)/(a*x)) - sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/(a*c^2)]","A",0
551,1,284,0,0.436536," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(5/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} {\left(a x - 1\right)} \sqrt{c} \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) + 2 \, {\left(a x - 1\right)} \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 4 \, {\left(a^{2} x^{2} - 2 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{2} c^{3} x - a c^{3}\right)}}, \frac{\sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + 2 \, {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - 2 \, {\left(a^{2} x^{2} - 2 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{2} c^{3} x - a c^{3}\right)}}\right]"," ",0,"[1/4*(sqrt(2)*(a*x - 1)*sqrt(c)*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + 2*(a*x - 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 4*(a^2*x^2 - 2*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^2*c^3*x - a*c^3), 1/2*(sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + 2*(a*x - 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 2*(a^2*x^2 - 2*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^2*c^3*x - a*c^3)]","A",0
552,1,356,0,0.437111," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(7/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) + 36 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 4 \, {\left(6 \, a^{3} x^{3} - 29 \, a^{2} x^{2} + 21 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{24 \, {\left(a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}, \frac{3 \, \sqrt{2} {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + 36 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - 2 \, {\left(6 \, a^{3} x^{3} - 29 \, a^{2} x^{2} + 21 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{12 \, {\left(a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}\right]"," ",0,"[1/24*(3*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + 36*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 4*(6*a^3*x^3 - 29*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4), 1/12*(3*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + 36*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 2*(6*a^3*x^3 - 29*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)]","A",0
553,1,428,0,0.415417," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(9/2),x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) + 600 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 4 \, {\left(60 \, a^{4} x^{4} - 497 \, a^{3} x^{3} + 740 \, a^{2} x^{2} - 315 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{240 \, {\left(a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right)}}, \frac{15 \, \sqrt{2} {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + 600 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - 2 \, {\left(60 \, a^{4} x^{4} - 497 \, a^{3} x^{3} + 740 \, a^{2} x^{2} - 315 \, a x\right)} \sqrt{\frac{a c x - c}{a x}}}{120 \, {\left(a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right)}}\right]"," ",0,"[1/240*(15*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + 600*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 4*(60*a^4*x^4 - 497*a^3*x^3 + 740*a^2*x^2 - 315*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5), 1/120*(15*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + 600*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 2*(60*a^4*x^4 - 497*a^3*x^3 + 740*a^2*x^2 - 315*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5)]","A",0
554,1,408,0,0.465300," ","integrate((c-c/a/x)^(9/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{105 \, {\left(a^{5} c^{4} x^{5} - a^{3} c^{4} x^{3}\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{28 \, {\left(a^{6} x^{5} - a^{4} x^{3}\right)}}, \frac{105 \, {\left(a^{5} c^{4} x^{5} - a^{3} c^{4} x^{3}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{14 \, {\left(a^{6} x^{5} - a^{4} x^{3}\right)}}\right]"," ",0,"[1/28*(105*(a^5*c^4*x^5 - a^3*c^4*x^3)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(7*a^5*c^4*x^5 + 1755*a^4*c^4*x^4 + 720*a^3*c^4*x^3 - 110*a^2*c^4*x^2 + 20*a*c^4*x - 2*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^6*x^5 - a^4*x^3), 1/14*(105*(a^5*c^4*x^5 - a^3*c^4*x^3)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(7*a^5*c^4*x^5 + 1755*a^4*c^4*x^4 + 720*a^3*c^4*x^3 - 110*a^2*c^4*x^2 + 20*a*c^4*x - 2*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^6*x^5 - a^4*x^3)]","A",0
555,1,386,0,0.552686," ","integrate((c-c/a/x)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{195 \, {\left(a^{4} c^{3} x^{4} - a^{2} c^{3} x^{2}\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(15 \, a^{4} c^{3} x^{4} + 1591 \, a^{3} c^{3} x^{3} + 548 \, a^{2} c^{3} x^{2} - 62 \, a c^{3} x + 6 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{60 \, {\left(a^{5} x^{4} - a^{3} x^{2}\right)}}, \frac{195 \, {\left(a^{4} c^{3} x^{4} - a^{2} c^{3} x^{2}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(15 \, a^{4} c^{3} x^{4} + 1591 \, a^{3} c^{3} x^{3} + 548 \, a^{2} c^{3} x^{2} - 62 \, a c^{3} x + 6 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{30 \, {\left(a^{5} x^{4} - a^{3} x^{2}\right)}}\right]"," ",0,"[1/60*(195*(a^4*c^3*x^4 - a^2*c^3*x^2)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(15*a^4*c^3*x^4 + 1591*a^3*c^3*x^3 + 548*a^2*c^3*x^2 - 62*a*c^3*x + 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^3*x^2), 1/30*(195*(a^4*c^3*x^4 - a^2*c^3*x^2)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(15*a^4*c^3*x^4 + 1591*a^3*c^3*x^3 + 548*a^2*c^3*x^2 - 62*a*c^3*x + 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^3*x^2)]","A",0
556,1,352,0,0.547040," ","integrate((c-c/a/x)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{33 \, {\left(a^{3} c^{2} x^{3} - a c^{2} x\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(3 \, a^{3} c^{2} x^{3} + 133 \, a^{2} c^{2} x^{2} + 32 \, a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{12 \, {\left(a^{4} x^{3} - a^{2} x\right)}}, \frac{33 \, {\left(a^{3} c^{2} x^{3} - a c^{2} x\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(3 \, a^{3} c^{2} x^{3} + 133 \, a^{2} c^{2} x^{2} + 32 \, a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{6 \, {\left(a^{4} x^{3} - a^{2} x\right)}}\right]"," ",0,"[1/12*(33*(a^3*c^2*x^3 - a*c^2*x)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(3*a^3*c^2*x^3 + 133*a^2*c^2*x^2 + 32*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^2*x), 1/6*(33*(a^3*c^2*x^3 - a*c^2*x)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(3*a^3*c^2*x^3 + 133*a^2*c^2*x^2 + 32*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^2*x)]","A",0
557,1,298,0,0.556271," ","integrate((c-c/a/x)^(3/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{9 \, {\left(a^{2} c x^{2} - c\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(a^{2} c x^{2} + 19 \, a c x + 2 \, c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{3} x^{2} - a\right)}}, \frac{9 \, {\left(a^{2} c x^{2} - c\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(a^{2} c x^{2} + 19 \, a c x + 2 \, c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{3} x^{2} - a\right)}}\right]"," ",0,"[1/4*(9*(a^2*c*x^2 - c)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(a^2*c*x^2 + 19*a*c*x + 2*c)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a), 1/2*(9*(a^2*c*x^2 - c)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(a^2*c*x^2 + 19*a*c*x + 2*c)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a)]","A",0
558,1,282,0,0.505497," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{7 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(a^{2} x^{2} + 9 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{3} x^{2} - a\right)}}, \frac{7 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(a^{2} x^{2} + 9 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{3} x^{2} - a\right)}}\right]"," ",0,"[1/4*(7*(a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(a^2*x^2 + 9*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a), 1/2*(7*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(a^2*x^2 + 9*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a)]","A",0
559,1,286,0,0.534953," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[-\frac{5 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(a^{2} x^{2} + 5 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{3} c x^{2} - a c\right)}}, \frac{5 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(a^{2} x^{2} + 5 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{3} c x^{2} - a c\right)}}\right]"," ",0,"[-1/4*(5*(a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(a^2*x^2 + 5*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c*x^2 - a*c), 1/2*(5*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(a^2*x^2 + 5*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c*x^2 - a*c)]","A",0
560,1,294,0,0.447919," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(a^{2} x^{2} + 3 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{3} c^{2} x^{2} - a c^{2}\right)}}, \frac{3 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(a^{2} x^{2} + 3 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{3} c^{2} x^{2} - a c^{2}\right)}}\right]"," ",0,"[-1/4*(3*(a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(a^2*x^2 + 3*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - a*c^2), 1/2*(3*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(a^2*x^2 + 3*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - a*c^2)]","A",0
561,1,492,0,0.680063," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{2} {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 2 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 8 \, {\left(a^{2} x^{2} + 2 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \, {\left(a^{3} c^{3} x^{2} - a c^{3}\right)}}, \frac{\sqrt{2} {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) + 2 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 4 \, {\left(a^{2} x^{2} + 2 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{3} c^{3} x^{2} - a c^{3}\right)}}\right]"," ",0,"[-1/8*(sqrt(2)*(a^2*x^2 - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*(a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(a^2*x^2 + 2*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^3*x^2 - a*c^3), 1/4*(sqrt(2)*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) + 2*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 4*(a^2*x^2 + 2*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^3*x^2 - a*c^3)]","A",0
562,1,596,0,0.592686," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(7/2),x, algorithm=""fricas"")","\left[-\frac{11 \, \sqrt{2} {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 8 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 8 \, {\left(4 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{32 \, {\left(a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right)}}, \frac{11 \, \sqrt{2} {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - 8 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 4 \, {\left(4 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{16 \, {\left(a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right)}}\right]"," ",0,"[-1/32*(11*sqrt(2)*(a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 8*(a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(4*a^3*x^3 + a^2*x^2 - 7*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - a^3*c^4*x^2 - a^2*c^4*x + a*c^4), 1/16*(11*sqrt(2)*(a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 8*(a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 4*(4*a^3*x^3 + a^2*x^2 - 7*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - a^3*c^4*x^2 - a^2*c^4*x + a*c^4)]","A",0
563,1,120,0,0.431695," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)/x^3,x, algorithm=""fricas"")","-\frac{20 \, a^{4} x^{3} - 40 \, a^{3} x^{2} + 20 \, a^{2} x + 12 \, {\left(a^{4} x^{3} - 2 \, a^{3} x^{2} + a^{2} x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(19 \, a^{3} x^{2} - 26 \, a^{2} x + 3 \, a\right)} \sqrt{-a^{2} x^{2} + 1}}{3 \, {\left(a^{2} c x^{3} - 2 \, a c x^{2} + c x\right)}}"," ",0,"-1/3*(20*a^4*x^3 - 40*a^3*x^2 + 20*a^2*x + 12*(a^4*x^3 - 2*a^3*x^2 + a^2*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (19*a^3*x^2 - 26*a^2*x + 3*a)*sqrt(-a^2*x^2 + 1))/(a^2*c*x^3 - 2*a*c*x^2 + c*x)","A",0
564,0,0,0,0.432490," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} x^{m} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*x^m*sqrt((a*c*x - c)/(a*x))/(a*x - 1), x)","F",0
565,1,292,0,0.637765," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(8 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{96 \, {\left(a^{4} x - a^{3}\right)}}, -\frac{3 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) + 2 \, {\left(8 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{48 \, {\left(a^{4} x - a^{3}\right)}}\right]"," ",0,"[1/96*(3*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(8*a^3*x^3 + 2*a^2*x^2 - 3*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3), -1/48*(3*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(8*a^3*x^3 + 2*a^2*x^2 - 3*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3)]","A",0
566,1,272,0,0.514165," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{16 \, {\left(a^{3} x - a^{2}\right)}}, \frac{{\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \, {\left(a^{3} x - a^{2}\right)}}\right]"," ",0,"[1/16*((a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2), 1/8*((a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2)]","A",0
567,1,249,0,0.480021," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[-\frac{4 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} - {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right)}{4 \, {\left(a^{2} x - a\right)}}, -\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} + {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right)}{2 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[-1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - (a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)))/(a^2*x - a), -1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + (a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)))/(a^2*x - a)]","A",0
568,1,236,0,0.933050," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a x - 1\right)}}, -\frac{{\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}\right]"," ",0,"[1/2*((a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x - 1), -((a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x - 1)]","A",0
569,1,47,0,0.496854," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(1/2)/x^2,x, algorithm=""fricas"")","\frac{2 \, \sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} \sqrt{\frac{a c x - c}{a x}}}{3 \, {\left(a x^{2} - x\right)}}"," ",0,"2/3*sqrt(-a^2*x^2 + 1)*(a*x + 1)*sqrt((a*c*x - c)/(a*x))/(a*x^2 - x)","A",0
570,1,58,0,0.755454," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(1/2)/x^3,x, algorithm=""fricas"")","-\frac{2 \, {\left(2 \, a^{2} x^{2} - a x - 3\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{15 \, {\left(a x^{3} - x^{2}\right)}}"," ",0,"-2/15*(2*a^2*x^2 - a*x - 3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))/(a*x^3 - x^2)","A",0
571,1,66,0,1.121994," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(1/2)/x^4,x, algorithm=""fricas"")","\frac{2 \, {\left(8 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + 3 \, a x + 15\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{105 \, {\left(a x^{4} - x^{3}\right)}}"," ",0,"2/105*(8*a^3*x^3 - 4*a^2*x^2 + 3*a*x + 15)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))/(a*x^4 - x^3)","A",0
572,1,74,0,0.401492," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(1/2)/x^5,x, algorithm=""fricas"")","-\frac{2 \, {\left(16 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 5 \, a x - 35\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{315 \, {\left(a x^{5} - x^{4}\right)}}"," ",0,"-2/315*(16*a^4*x^4 - 8*a^3*x^3 + 6*a^2*x^2 - 5*a*x - 35)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))/(a*x^5 - x^4)","A",0
573,1,179,0,0.449138," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(16 \, a^{4} x^{4} + 40 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 75 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} - 75 \, \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{128 \, a^{4}}, -\frac{{\left(16 \, a^{4} x^{4} + 40 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 75 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} - 75 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{64 \, a^{4}}\right]"," ",0,"[-1/128*(2*(16*a^4*x^4 + 40*a^3*x^3 + 50*a^2*x^2 + 75*a*x)*sqrt((a*c*x - c)/(a*x)) - 75*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a^4, -1/64*((16*a^4*x^4 + 40*a^3*x^3 + 50*a^2*x^2 + 75*a*x)*sqrt((a*c*x - c)/(a*x)) - 75*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a^4]","A",0
574,1,163,0,0.460142," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(8 \, a^{3} x^{3} + 22 \, a^{2} x^{2} + 33 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} - 33 \, \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{48 \, a^{3}}, -\frac{{\left(8 \, a^{3} x^{3} + 22 \, a^{2} x^{2} + 33 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} - 33 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{24 \, a^{3}}\right]"," ",0,"[-1/48*(2*(8*a^3*x^3 + 22*a^2*x^2 + 33*a*x)*sqrt((a*c*x - c)/(a*x)) - 33*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a^3, -1/24*((8*a^3*x^3 + 22*a^2*x^2 + 33*a*x)*sqrt((a*c*x - c)/(a*x)) - 33*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a^3]","A",0
575,1,147,0,0.425827," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(2 \, a^{2} x^{2} + 7 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} - 7 \, \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{8 \, a^{2}}, -\frac{{\left(2 \, a^{2} x^{2} + 7 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} - 7 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{4 \, a^{2}}\right]"," ",0,"[-1/8*(2*(2*a^2*x^2 + 7*a*x)*sqrt((a*c*x - c)/(a*x)) - 7*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a^2, -1/4*((2*a^2*x^2 + 7*a*x)*sqrt((a*c*x - c)/(a*x)) - 7*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a^2]","A",0
576,1,125,0,0.534202," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, a x \sqrt{\frac{a c x - c}{a x}} - 3 \, \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{2 \, a}, -\frac{a x \sqrt{\frac{a c x - c}{a x}} - 3 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{a}\right]"," ",0,"[-1/2*(2*a*x*sqrt((a*c*x - c)/(a*x)) - 3*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a, -(a*x*sqrt((a*c*x - c)/(a*x)) - 3*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a]","A",0
577,1,111,0,0.513789," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x,x, algorithm=""fricas"")","\left[\sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, \sqrt{\frac{a c x - c}{a x}}, 2 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - 2 \, \sqrt{\frac{a c x - c}{a x}}\right]"," ",0,"[sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*sqrt((a*c*x - c)/(a*x)), 2*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 2*sqrt((a*c*x - c)/(a*x))]","A",0
578,1,28,0,0.414212," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^2,x, algorithm=""fricas"")","-\frac{2 \, {\left(5 \, a x + 1\right)} \sqrt{\frac{a c x - c}{a x}}}{3 \, x}"," ",0,"-2/3*(5*a*x + 1)*sqrt((a*c*x - c)/(a*x))/x","A",0
579,1,36,0,0.436258," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^3,x, algorithm=""fricas"")","-\frac{2 \, {\left(6 \, a^{2} x^{2} + 3 \, a x + 1\right)} \sqrt{\frac{a c x - c}{a x}}}{5 \, x^{2}}"," ",0,"-2/5*(6*a^2*x^2 + 3*a*x + 1)*sqrt((a*c*x - c)/(a*x))/x^2","A",0
580,1,44,0,0.504164," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^4,x, algorithm=""fricas"")","-\frac{2 \, {\left(104 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 39 \, a x + 15\right)} \sqrt{\frac{a c x - c}{a x}}}{105 \, x^{3}}"," ",0,"-2/105*(104*a^3*x^3 + 52*a^2*x^2 + 39*a*x + 15)*sqrt((a*c*x - c)/(a*x))/x^3","A",0
581,1,52,0,0.617382," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^5,x, algorithm=""fricas"")","-\frac{2 \, {\left(272 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 85 \, a x + 35\right)} \sqrt{\frac{a c x - c}{a x}}}{315 \, x^{4}}"," ",0,"-2/315*(272*a^4*x^4 + 136*a^3*x^3 + 102*a^2*x^2 + 85*a*x + 35)*sqrt((a*c*x - c)/(a*x))/x^4","A",0
582,1,500,0,0.545601," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[\frac{768 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 1089 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(48 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} + 447 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{768 \, {\left(a^{5} x - a^{4}\right)}}, -\frac{768 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - 1089 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(48 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} + 447 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{384 \, {\left(a^{5} x - a^{4}\right)}}\right]"," ",0,"[1/768*(768*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 1089*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(48*a^4*x^4 + 136*a^3*x^3 + 214*a^2*x^2 + 447*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x - a^4), -1/384*(768*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 1089*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(48*a^4*x^4 + 136*a^3*x^3 + 214*a^2*x^2 + 447*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x - a^4)]","A",0
583,1,484,0,0.652421," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[\frac{96 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 135 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(8 \, a^{3} x^{3} + 26 \, a^{2} x^{2} + 57 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{96 \, {\left(a^{4} x - a^{3}\right)}}, -\frac{96 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - 135 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(8 \, a^{3} x^{3} + 26 \, a^{2} x^{2} + 57 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{48 \, {\left(a^{4} x - a^{3}\right)}}\right]"," ",0,"[1/96*(96*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 135*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(8*a^3*x^3 + 26*a^2*x^2 + 57*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3), -1/48*(96*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 135*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(8*a^3*x^3 + 26*a^2*x^2 + 57*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3)]","A",0
584,1,468,0,0.606872," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[\frac{16 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 23 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(2 \, a^{2} x^{2} + 9 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{16 \, {\left(a^{3} x - a^{2}\right)}}, -\frac{16 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - 23 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(2 \, a^{2} x^{2} + 9 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \, {\left(a^{3} x - a^{2}\right)}}\right]"," ",0,"[1/16*(16*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 23*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(2*a^2*x^2 + 9*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2), -1/8*(16*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 23*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(2*a^2*x^2 + 9*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2)]","A",0
585,1,442,0,0.555440," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} + 4 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 5 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right)}{4 \, {\left(a^{2} x - a\right)}}, \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} - 4 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) + 5 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right)}{2 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + 4*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 5*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)))/(a^2*x - a), 1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - 4*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) + 5*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)))/(a^2*x - a)]","A",0
586,1,429,0,0.914107," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a x - 1\right)}}, -\frac{2 \, \sqrt{2} {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}\right]"," ",0,"[1/2*(2*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + (a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x - 1), -(2*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - (a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x - 1)]","A",0
587,1,309,0,0.554394," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x^2,x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{2} {\left(a^{2} x^{2} - a x\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} {\left(7 \, a x + 1\right)} \sqrt{\frac{a c x - c}{a x}}}{3 \, {\left(a x^{2} - x\right)}}, -\frac{2 \, {\left(3 \, \sqrt{2} {\left(a^{2} x^{2} - a x\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - \sqrt{-a^{2} x^{2} + 1} {\left(7 \, a x + 1\right)} \sqrt{\frac{a c x - c}{a x}}\right)}}{3 \, {\left(a x^{2} - x\right)}}\right]"," ",0,"[1/3*(3*sqrt(2)*(a^2*x^2 - a*x)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*sqrt(-a^2*x^2 + 1)*(7*a*x + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^2 - x), -2/3*(3*sqrt(2)*(a^2*x^2 - a*x)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - sqrt(-a^2*x^2 + 1)*(7*a*x + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^2 - x)]","A",0
588,1,337,0,0.529095," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x^3,x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{2} {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 2 \, {\left(38 \, a^{2} x^{2} + 11 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{15 \, {\left(a x^{3} - x^{2}\right)}}, -\frac{2 \, {\left(15 \, \sqrt{2} {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - {\left(38 \, a^{2} x^{2} + 11 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}\right)}}{15 \, {\left(a x^{3} - x^{2}\right)}}\right]"," ",0,"[1/15*(15*sqrt(2)*(a^3*x^3 - a^2*x^2)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*(38*a^2*x^2 + 11*a*x + 3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^3 - x^2), -2/15*(15*sqrt(2)*(a^3*x^3 - a^2*x^2)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - (38*a^2*x^2 + 11*a*x + 3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^3 - x^2)]","A",0
589,1,353,0,0.584826," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x^4,x, algorithm=""fricas"")","\left[\frac{21 \, \sqrt{2} {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 2 \, {\left(52 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 9 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{21 \, {\left(a x^{4} - x^{3}\right)}}, -\frac{2 \, {\left(21 \, \sqrt{2} {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - {\left(52 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 9 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}\right)}}{21 \, {\left(a x^{4} - x^{3}\right)}}\right]"," ",0,"[1/21*(21*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*(52*a^3*x^3 + 16*a^2*x^2 + 9*a*x + 3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^4 - x^3), -2/21*(21*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - (52*a^3*x^3 + 16*a^2*x^2 + 9*a*x + 3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^4 - x^3)]","A",0
590,1,369,0,0.475747," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x^5,x, algorithm=""fricas"")","\left[\frac{315 \, \sqrt{2} {\left(a^{5} x^{5} - a^{4} x^{4}\right)} \sqrt{-c} \log\left(-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2} {\left(3 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right) + 2 \, {\left(788 \, a^{4} x^{4} + 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} + 95 \, a x + 35\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{315 \, {\left(a x^{5} - x^{4}\right)}}, -\frac{2 \, {\left(315 \, \sqrt{2} {\left(a^{5} x^{5} - a^{4} x^{4}\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right) - {\left(788 \, a^{4} x^{4} + 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} + 95 \, a x + 35\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}\right)}}{315 \, {\left(a x^{5} - x^{4}\right)}}\right]"," ",0,"[1/315*(315*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*(788*a^4*x^4 + 236*a^3*x^3 + 138*a^2*x^2 + 95*a*x + 35)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^5 - x^4), -2/315*(315*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - (788*a^4*x^4 + 236*a^3*x^3 + 138*a^2*x^2 + 95*a*x + 35)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^5 - x^4)]","A",0
591,0,0,0,0.497861," ","integrate(x^m*(c-c/a/x)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} x^{m} \sqrt{\frac{a c x - c}{a x}}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*x^m*sqrt((a*c*x - c)/(a*x))/(a*x + 1), x)","F",0
592,1,292,0,0.491342," ","integrate(x^2*(c-c/a/x)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{33 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(8 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 33 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{96 \, {\left(a^{4} x - a^{3}\right)}}, -\frac{33 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(8 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 33 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{48 \, {\left(a^{4} x - a^{3}\right)}}\right]"," ",0,"[1/96*(33*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(8*a^3*x^3 - 22*a^2*x^2 + 33*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3), -1/48*(33*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(8*a^3*x^3 - 22*a^2*x^2 + 33*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3)]","A",0
593,1,276,0,0.539238," ","integrate(x*(c-c/a/x)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{7 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, {\left(2 \, a^{2} x^{2} - 7 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{16 \, {\left(a^{3} x - a^{2}\right)}}, \frac{7 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) + 2 \, {\left(2 \, a^{2} x^{2} - 7 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \, {\left(a^{3} x - a^{2}\right)}}\right]"," ",0,"[1/16*(7*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(2*a^2*x^2 - 7*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2), 1/8*(7*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(2*a^2*x^2 - 7*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2)]","A",0
594,1,250,0,0.554957," ","integrate((c-c/a/x)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} + 3 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right)}{4 \, {\left(a^{2} x - a\right)}}, \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} - 3 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right)}{2 \, {\left(a^{2} x - a\right)}}\right]"," ",0,"[1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + 3*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)))/(a^2*x - a), 1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - 3*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)))/(a^2*x - a)]","A",0
595,1,235,0,0.626995," ","integrate((c-c/a/x)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left(a x - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a x - 1\right)}}, \frac{{\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}\right]"," ",0,"[1/2*((a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x - 1), ((a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x - 1)]","A",0
596,1,48,0,0.456920," ","integrate((c-c/a/x)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-a^{2} x^{2} + 1} {\left(5 \, a x - 1\right)} \sqrt{\frac{a c x - c}{a x}}}{3 \, {\left(a x^{2} - x\right)}}"," ",0,"-2/3*sqrt(-a^2*x^2 + 1)*(5*a*x - 1)*sqrt((a*c*x - c)/(a*x))/(a*x^2 - x)","A",0
597,1,58,0,0.428273," ","integrate((c-c/a/x)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^3,x, algorithm=""fricas"")","\frac{2 \, {\left(6 \, a^{2} x^{2} - 3 \, a x + 1\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{5 \, {\left(a x^{3} - x^{2}\right)}}"," ",0,"2/5*(6*a^2*x^2 - 3*a*x + 1)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))/(a*x^3 - x^2)","A",0
598,1,66,0,0.543573," ","integrate((c-c/a/x)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^4,x, algorithm=""fricas"")","-\frac{2 \, {\left(104 \, a^{3} x^{3} - 52 \, a^{2} x^{2} + 39 \, a x - 15\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{105 \, {\left(a x^{4} - x^{3}\right)}}"," ",0,"-2/105*(104*a^3*x^3 - 52*a^2*x^2 + 39*a*x - 15)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))/(a*x^4 - x^3)","A",0
599,1,274,0,0.634593," ","integrate(x^3*(c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{768 \, \sqrt{2} \sqrt{c} \log\left(-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right) - 2 \, {\left(48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} - 447 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} + 1089 \, \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{384 \, a^{4}}, -\frac{768 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + {\left(48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} - 447 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} - 1089 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{192 \, a^{4}}\right]"," ",0,"[1/384*(768*sqrt(2)*sqrt(c)*log(-(2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + 3*a*c*x - c)/(a*x + 1)) - 2*(48*a^4*x^4 - 136*a^3*x^3 + 214*a^2*x^2 - 447*a*x)*sqrt((a*c*x - c)/(a*x)) + 1089*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a^4, -1/192*(768*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (48*a^4*x^4 - 136*a^3*x^3 + 214*a^2*x^2 - 447*a*x)*sqrt((a*c*x - c)/(a*x)) - 1089*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a^4]","A",0
600,1,256,0,0.580576," ","integrate(x^2*(c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{96 \, \sqrt{2} \sqrt{c} \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) - 2 \, {\left(8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 57 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} + 135 \, \sqrt{c} \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{48 \, a^{3}}, \frac{96 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) - {\left(8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 57 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} - 135 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{24 \, a^{3}}\right]"," ",0,"[1/48*(96*sqrt(2)*sqrt(c)*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) - 2*(8*a^3*x^3 - 26*a^2*x^2 + 57*a*x)*sqrt((a*c*x - c)/(a*x)) + 135*sqrt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a^3, 1/24*(96*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (8*a^3*x^3 - 26*a^2*x^2 + 57*a*x)*sqrt((a*c*x - c)/(a*x)) - 135*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a^3]","A",0
601,1,242,0,0.692584," ","integrate(x*(c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{16 \, \sqrt{2} \sqrt{c} \log\left(-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right) - 2 \, {\left(2 \, a^{2} x^{2} - 9 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} + 23 \, \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{8 \, a^{2}}, -\frac{16 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + {\left(2 \, a^{2} x^{2} - 9 \, a x\right)} \sqrt{\frac{a c x - c}{a x}} - 23 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{4 \, a^{2}}\right]"," ",0,"[1/8*(16*sqrt(2)*sqrt(c)*log(-(2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + 3*a*c*x - c)/(a*x + 1)) - 2*(2*a^2*x^2 - 9*a*x)*sqrt((a*c*x - c)/(a*x)) + 23*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a^2, -1/4*(16*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (2*a^2*x^2 - 9*a*x)*sqrt((a*c*x - c)/(a*x)) - 23*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a^2]","A",0
602,1,217,0,0.469432," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, a x \sqrt{\frac{a c x - c}{a x}} - 4 \, \sqrt{2} \sqrt{c} \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) - 5 \, \sqrt{c} \log\left(-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right)}{2 \, a}, -\frac{a x \sqrt{\frac{a c x - c}{a x}} - 4 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + 5 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right)}{a}\right]"," ",0,"[-1/2*(2*a*x*sqrt((a*c*x - c)/(a*x)) - 4*sqrt(2)*sqrt(c)*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) - 5*sqrt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a, -(a*x*sqrt((a*c*x - c)/(a*x)) - 4*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + 5*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a]","A",0
603,1,206,0,0.573933," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x,x, algorithm=""fricas"")","\left[2 \, \sqrt{2} \sqrt{c} \log\left(-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right) + \sqrt{c} \log\left(-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right) - 2 \, \sqrt{\frac{a c x - c}{a x}}, -4 \, \sqrt{2} \sqrt{-c} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + 2 \, \sqrt{-c} \arctan\left(\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right) - 2 \, \sqrt{\frac{a c x - c}{a x}}\right]"," ",0,"[2*sqrt(2)*sqrt(c)*log(-(2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + 3*a*c*x - c)/(a*x + 1)) + sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*sqrt((a*c*x - c)/(a*x)), -4*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + 2*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 2*sqrt((a*c*x - c)/(a*x))]","A",0
604,1,157,0,0.625211," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(3 \, \sqrt{2} a \sqrt{c} x \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) + {\left(7 \, a x - 1\right)} \sqrt{\frac{a c x - c}{a x}}\right)}}{3 \, x}, \frac{2 \, {\left(6 \, \sqrt{2} a \sqrt{-c} x \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + {\left(7 \, a x - 1\right)} \sqrt{\frac{a c x - c}{a x}}\right)}}{3 \, x}\right]"," ",0,"[2/3*(3*sqrt(2)*a*sqrt(c)*x*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + (7*a*x - 1)*sqrt((a*c*x - c)/(a*x)))/x, 2/3*(6*sqrt(2)*a*sqrt(-c)*x*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (7*a*x - 1)*sqrt((a*c*x - c)/(a*x)))/x]","A",0
605,1,185,0,0.570234," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^3,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(15 \, \sqrt{2} a^{2} \sqrt{c} x^{2} \log\left(-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right) - {\left(38 \, a^{2} x^{2} - 11 \, a x + 3\right)} \sqrt{\frac{a c x - c}{a x}}\right)}}{15 \, x^{2}}, -\frac{2 \, {\left(30 \, \sqrt{2} a^{2} \sqrt{-c} x^{2} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + {\left(38 \, a^{2} x^{2} - 11 \, a x + 3\right)} \sqrt{\frac{a c x - c}{a x}}\right)}}{15 \, x^{2}}\right]"," ",0,"[2/15*(15*sqrt(2)*a^2*sqrt(c)*x^2*log(-(2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + 3*a*c*x - c)/(a*x + 1)) - (38*a^2*x^2 - 11*a*x + 3)*sqrt((a*c*x - c)/(a*x)))/x^2, -2/15*(30*sqrt(2)*a^2*sqrt(-c)*x^2*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (38*a^2*x^2 - 11*a*x + 3)*sqrt((a*c*x - c)/(a*x)))/x^2]","A",0
606,1,197,0,0.639343," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(21 \, \sqrt{2} a^{3} \sqrt{c} x^{3} \log\left(\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right) + {\left(52 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 9 \, a x - 3\right)} \sqrt{\frac{a c x - c}{a x}}\right)}}{21 \, x^{3}}, \frac{2 \, {\left(42 \, \sqrt{2} a^{3} \sqrt{-c} x^{3} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + {\left(52 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 9 \, a x - 3\right)} \sqrt{\frac{a c x - c}{a x}}\right)}}{21 \, x^{3}}\right]"," ",0,"[2/21*(21*sqrt(2)*a^3*sqrt(c)*x^3*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + (52*a^3*x^3 - 16*a^2*x^2 + 9*a*x - 3)*sqrt((a*c*x - c)/(a*x)))/x^3, 2/21*(42*sqrt(2)*a^3*sqrt(-c)*x^3*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (52*a^3*x^3 - 16*a^2*x^2 + 9*a*x - 3)*sqrt((a*c*x - c)/(a*x)))/x^3]","A",0
607,1,217,0,0.512558," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(315 \, \sqrt{2} a^{4} \sqrt{c} x^{4} \log\left(-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right) - {\left(788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right)} \sqrt{\frac{a c x - c}{a x}}\right)}}{315 \, x^{4}}, -\frac{2 \, {\left(630 \, \sqrt{2} a^{4} \sqrt{-c} x^{4} \arctan\left(\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right) + {\left(788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right)} \sqrt{\frac{a c x - c}{a x}}\right)}}{315 \, x^{4}}\right]"," ",0,"[2/315*(315*sqrt(2)*a^4*sqrt(c)*x^4*log(-(2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + 3*a*c*x - c)/(a*x + 1)) - (788*a^4*x^4 - 236*a^3*x^3 + 138*a^2*x^2 - 95*a*x + 35)*sqrt((a*c*x - c)/(a*x)))/x^4, -2/315*(630*sqrt(2)*a^4*sqrt(-c)*x^4*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (788*a^4*x^4 - 236*a^3*x^3 + 138*a^2*x^2 - 95*a*x + 35)*sqrt((a*c*x - c)/(a*x)))/x^4]","A",0
608,1,336,0,0.690916," ","integrate(x^3*(c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{3345 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(48 \, a^{5} x^{5} - 200 \, a^{4} x^{4} + 446 \, a^{3} x^{3} - 1115 \, a^{2} x^{2} - 3345 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{768 \, {\left(a^{6} x^{2} - a^{4}\right)}}, -\frac{3345 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) + 2 \, {\left(48 \, a^{5} x^{5} - 200 \, a^{4} x^{4} + 446 \, a^{3} x^{3} - 1115 \, a^{2} x^{2} - 3345 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{384 \, {\left(a^{6} x^{2} - a^{4}\right)}}\right]"," ",0,"[1/768*(3345*(a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(48*a^5*x^5 - 200*a^4*x^4 + 446*a^3*x^3 - 1115*a^2*x^2 - 3345*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^6*x^2 - a^4), -1/384*(3345*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(48*a^5*x^5 - 200*a^4*x^4 + 446*a^3*x^3 - 1115*a^2*x^2 - 3345*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^6*x^2 - a^4)]","A",0
609,1,320,0,0.664869," ","integrate(x^2*(c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{357 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{96 \, {\left(a^{5} x^{2} - a^{3}\right)}}, \frac{357 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{48 \, {\left(a^{5} x^{2} - a^{3}\right)}}\right]"," ",0,"[1/96*(357*(a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(8*a^4*x^4 - 38*a^3*x^3 + 119*a^2*x^2 + 357*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^2 - a^3), 1/48*(357*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(8*a^4*x^4 - 38*a^3*x^3 + 119*a^2*x^2 + 357*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^2 - a^3)]","A",0
610,1,304,0,0.628941," ","integrate(x*(c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{47 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(2 \, a^{3} x^{3} - 13 \, a^{2} x^{2} - 47 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{16 \, {\left(a^{4} x^{2} - a^{2}\right)}}, -\frac{47 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) + 2 \, {\left(2 \, a^{3} x^{3} - 13 \, a^{2} x^{2} - 47 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \, {\left(a^{4} x^{2} - a^{2}\right)}}\right]"," ",0,"[1/16*(47*(a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(2*a^3*x^3 - 13*a^2*x^2 - 47*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^2 - a^2), -1/8*(47*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(2*a^3*x^3 - 13*a^2*x^2 - 47*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^2 - a^2)]","A",0
611,1,282,0,0.704201," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{7 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) - 4 \, {\left(a^{2} x^{2} + 9 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \, {\left(a^{3} x^{2} - a\right)}}, \frac{7 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, {\left(a^{2} x^{2} + 9 \, a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{3} x^{2} - a\right)}}\right]"," ",0,"[1/4*(7*(a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(a^2*x^2 + 9*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a), 1/2*(7*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(a^2*x^2 + 9*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a)]","A",0
612,1,264,0,0.578774," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left(2 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right) + 4 \, \sqrt{-a^{2} x^{2} + 1} {\left(5 \, a x + 1\right)} \sqrt{\frac{a c x - c}{a x}}}{2 \, {\left(a^{2} x^{2} - 1\right)}}, -\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} {\left(5 \, a x + 1\right)} \sqrt{\frac{a c x - c}{a x}}}{a^{2} x^{2} - 1}\right]"," ",0,"[1/2*((a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*(5*a*x + 1)*sqrt((a*c*x - c)/(a*x)))/(a^2*x^2 - 1), -((a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*sqrt(-a^2*x^2 + 1)*(5*a*x + 1)*sqrt((a*c*x - c)/(a*x)))/(a^2*x^2 - 1)]","A",0
613,1,58,0,0.612714," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm=""fricas"")","-\frac{2 \, {\left(23 \, a^{2} x^{2} + 10 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{3 \, {\left(a^{2} x^{3} - x\right)}}"," ",0,"-2/3*(23*a^2*x^2 + 10*a*x - 1)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))/(a^2*x^3 - x)","A",0
614,1,68,0,0.548553," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^3,x, algorithm=""fricas"")","\frac{2 \, {\left(158 \, a^{3} x^{3} + 79 \, a^{2} x^{2} - 16 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{15 \, {\left(a^{2} x^{4} - x^{2}\right)}}"," ",0,"2/15*(158*a^3*x^3 + 79*a^2*x^2 - 16*a*x + 3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))/(a^2*x^4 - x^2)","A",0
615,1,76,0,0.650339," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^4,x, algorithm=""fricas"")","-\frac{2 \, {\left(1336 \, a^{4} x^{4} + 668 \, a^{3} x^{3} - 167 \, a^{2} x^{2} + 66 \, a x - 15\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{105 \, {\left(a^{2} x^{5} - x^{3}\right)}}"," ",0,"-2/105*(1336*a^4*x^4 + 668*a^3*x^3 - 167*a^2*x^2 + 66*a*x - 15)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))/(a^2*x^5 - x^3)","A",0
616,1,84,0,0.542036," ","integrate((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm=""fricas"")","\frac{2 \, {\left(656 \, a^{5} x^{5} + 328 \, a^{4} x^{4} - 82 \, a^{3} x^{3} + 41 \, a^{2} x^{2} - 20 \, a x + 5\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{45 \, {\left(a^{2} x^{6} - x^{4}\right)}}"," ",0,"2/45*(656*a^5*x^5 + 328*a^4*x^4 - 82*a^3*x^3 + 41*a^2*x^2 - 20*a*x + 5)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))/(a^2*x^6 - x^4)","A",0
617,0,0,0,0.587789," ","integrate(exp(n*arctanh(a*x))*(c-c/a/x)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \left(\frac{a c x - c}{a x}\right)^{p}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n)*((a*c*x - c)/(a*x))^p, x)","F",0
618,0,0,0,0.486210," ","integrate((c-c/a/x)^p/exp(2*p*arctanh(a*x)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{a c x - c}{a x}\right)^{p}}{\left(\frac{a x + 1}{a x - 1}\right)^{p}}, x\right)"," ",0,"integral(((a*c*x - c)/(a*x))^p/((a*x + 1)/(a*x - 1))^p, x)","F",0
619,0,0,0,0.700343," ","integrate(exp(2*p*arctanh(a*x))*(c-c/a/x)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\frac{a x + 1}{a x - 1}\right)^{p} \left(\frac{a c x - c}{a x}\right)^{p}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^p*((a*c*x - c)/(a*x))^p, x)","F",0
620,0,0,0,0.659819," ","integrate(exp(n*arctanh(a*x))*(c-c/a/x)^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} x^{2}}, x\right)"," ",0,"integral((a^2*c^2*x^2 - 2*a*c^2*x + c^2)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*x^2), x)","F",0
621,0,0,0,0.595109," ","integrate(exp(n*arctanh(a*x))*(c-c/a/x),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a c x - c\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a x}, x\right)"," ",0,"integral((a*c*x - c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*x), x)","F",0
622,0,0,0,0.662578," ","integrate(exp(n*arctanh(a*x))/(c-c/a/x),x, algorithm=""fricas"")","{\rm integral}\left(\frac{a x \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c x - c}, x\right)"," ",0,"integral(a*x*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c), x)","F",0
623,0,0,0,0.462016," ","integrate(exp(n*arctanh(a*x))/(c-c/a/x)^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{a^{2} x^{2} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, x\right)"," ",0,"integral(a^2*x^2*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c^2*x^2 - 2*a*c^2*x + c^2), x)","F",0
624,0,0,0,0.522939," ","integrate(exp(n*arctanh(a*x))*(c-c/a/x)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a c x - c\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \sqrt{\frac{a c x - c}{a x}}}{a x}, x\right)"," ",0,"integral((a*c*x - c)*((a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a*c*x - c)/(a*x))/(a*x), x)","F",0
625,0,0,0,0.540809," ","integrate(exp(n*arctanh(a*x))*(c-c/a/x)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \sqrt{\frac{a c x - c}{a x}}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a*c*x - c)/(a*x)), x)","F",0
626,0,0,0,0.617854," ","integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{a x \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \sqrt{\frac{a c x - c}{a x}}}{a c x - c}, x\right)"," ",0,"integral(a*x*((a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a*c*x - c)/(a*x))/(a*c*x - c), x)","F",0
627,0,0,0,0.558168," ","integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{a^{2} x^{2} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \sqrt{\frac{a c x - c}{a x}}}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, x\right)"," ",0,"integral(a^2*x^2*((a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a*c*x - c)/(a*x))/(a^2*c^2*x^2 - 2*a*c^2*x + c^2), x)","F",0
628,1,175,0,0.637994," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","-\frac{3360 \, a^{7} c^{4} x^{7} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 3675 \, a^{7} c^{4} x^{7} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + 1680 \, a^{7} c^{4} x^{7} + {\left(1680 \, a^{7} c^{4} x^{7} - 2816 \, a^{6} c^{4} x^{6} + 3045 \, a^{5} c^{4} x^{5} + 1952 \, a^{4} c^{4} x^{4} - 1330 \, a^{3} c^{4} x^{3} - 1056 \, a^{2} c^{4} x^{2} + 280 \, a c^{4} x + 240 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{1680 \, a^{8} x^{7}}"," ",0,"-1/1680*(3360*a^7*c^4*x^7*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3675*a^7*c^4*x^7*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 1680*a^7*c^4*x^7 + (1680*a^7*c^4*x^7 - 2816*a^6*c^4*x^6 + 3045*a^5*c^4*x^5 + 1952*a^4*c^4*x^4 - 1330*a^3*c^4*x^3 - 1056*a^2*c^4*x^2 + 280*a*c^4*x + 240*c^4)*sqrt(-a^2*x^2 + 1))/(a^8*x^7)","A",0
629,1,153,0,0.624605," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","-\frac{240 \, a^{5} c^{3} x^{5} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 225 \, a^{5} c^{3} x^{5} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + 120 \, a^{5} c^{3} x^{5} + {\left(120 \, a^{5} c^{3} x^{5} - 184 \, a^{4} c^{3} x^{4} + 135 \, a^{3} c^{3} x^{3} + 88 \, a^{2} c^{3} x^{2} - 30 \, a c^{3} x - 24 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, a^{6} x^{5}}"," ",0,"-1/120*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 225*a^5*c^3*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 120*a^5*c^3*x^5 + (120*a^5*c^3*x^5 - 184*a^4*c^3*x^4 + 135*a^3*c^3*x^3 + 88*a^2*c^3*x^2 - 30*a*c^3*x - 24*c^3)*sqrt(-a^2*x^2 + 1))/(a^6*x^5)","A",0
630,1,131,0,0.396821," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","-\frac{12 \, a^{3} c^{2} x^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 9 \, a^{3} c^{2} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + 6 \, a^{3} c^{2} x^{3} + {\left(6 \, a^{3} c^{2} x^{3} - 8 \, a^{2} c^{2} x^{2} + 3 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}}"," ",0,"-1/6*(12*a^3*c^2*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 9*a^3*c^2*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 6*a^3*c^2*x^3 + (6*a^3*c^2*x^3 - 8*a^2*c^2*x^2 + 3*a*c^2*x + 2*c^2)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)","A",0
631,1,84,0,0.421183," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2),x, algorithm=""fricas"")","-\frac{2 \, a c x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + a c x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + a c x + \sqrt{-a^{2} x^{2} + 1} {\left(a c x - c\right)}}{a^{2} x}"," ",0,"-(2*a*c*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + a*c*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) + a*c*x + sqrt(-a^2*x^2 + 1)*(a*c*x - c))/(a^2*x)","A",0
632,1,68,0,0.771851," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2),x, algorithm=""fricas"")","-\frac{2 \, a x + 2 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a x - 2\right)} - 2}{a^{2} c x - a c}"," ",0,"-(2*a*x + 2*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x - 2) - 2)/(a^2*c*x - a*c)","A",0
633,1,142,0,0.422561," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","-\frac{8 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 8 \, a x + 6 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} + 8}{3 \, {\left(a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} - a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"-1/3*(8*a^3*x^3 - 8*a^2*x^2 - 8*a*x + 6*(a^3*x^3 - a^2*x^2 - a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^3*x^3 - 7*a^2*x^2 - 5*a*x + 8)*sqrt(-a^2*x^2 + 1) + 8)/(a^4*c^2*x^3 - a^3*c^2*x^2 - a^2*c^2*x + a*c^2)","A",0
634,1,211,0,0.533215," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","-\frac{48 \, a^{5} x^{5} - 48 \, a^{4} x^{4} - 96 \, a^{3} x^{3} + 96 \, a^{2} x^{2} + 48 \, a x + 30 \, {\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(15 \, a^{5} x^{5} - 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} + 87 \, a^{2} x^{2} + 33 \, a x - 48\right)} \sqrt{-a^{2} x^{2} + 1} - 48}{15 \, {\left(a^{6} c^{3} x^{5} - a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/15*(48*a^5*x^5 - 48*a^4*x^4 - 96*a^3*x^3 + 96*a^2*x^2 + 48*a*x + 30*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^5*x^5 - 38*a^4*x^4 - 52*a^3*x^3 + 87*a^2*x^2 + 33*a*x - 48)*sqrt(-a^2*x^2 + 1) - 48)/(a^6*c^3*x^5 - a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 + a^2*c^3*x - a*c^3)","A",0
635,1,282,0,0.633537," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","-\frac{384 \, a^{7} x^{7} - 384 \, a^{6} x^{6} - 1152 \, a^{5} x^{5} + 1152 \, a^{4} x^{4} + 1152 \, a^{3} x^{3} - 1152 \, a^{2} x^{2} - 384 \, a x + 210 \, {\left(a^{7} x^{7} - a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(105 \, a^{7} x^{7} - 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} + 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} - 1065 \, a^{2} x^{2} - 279 \, a x + 384\right)} \sqrt{-a^{2} x^{2} + 1} + 384}{105 \, {\left(a^{8} c^{4} x^{7} - a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/105*(384*a^7*x^7 - 384*a^6*x^6 - 1152*a^5*x^5 + 1152*a^4*x^4 + 1152*a^3*x^3 - 1152*a^2*x^2 - 384*a*x + 210*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2*x^2 - a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (105*a^7*x^7 - 281*a^6*x^6 - 559*a^5*x^5 + 965*a^4*x^4 + 715*a^3*x^3 - 1065*a^2*x^2 - 279*a*x + 384)*sqrt(-a^2*x^2 + 1) + 384)/(a^8*c^4*x^7 - a^7*c^4*x^6 - 3*a^6*c^4*x^5 + 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 - 3*a^3*c^4*x^2 - a^2*c^4*x + a*c^4)","A",0
636,1,122,0,0.409930," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^5,x, algorithm=""fricas"")","-\frac{1260 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log\left(x\right) + 3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}}"," ",0,"-1/1260*(1260*a^10*c^5*x^10 + 2520*a^9*c^5*x^9*log(x) + 3780*a^8*c^5*x^8 + 5040*a^7*c^5*x^7 - 840*a^6*c^5*x^6 - 3780*a^5*c^5*x^5 - 504*a^4*c^5*x^4 + 1680*a^3*c^5*x^3 + 540*a^2*c^5*x^2 - 315*a*c^5*x - 140*c^5)/(a^10*x^9)","A",0
637,1,89,0,0.612930," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","-\frac{210 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log\left(x\right) + 420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}}"," ",0,"-1/210*(210*a^8*c^4*x^8 + 420*a^7*c^4*x^7*log(x) + 420*a^6*c^4*x^6 + 630*a^5*c^4*x^5 - 315*a^3*c^4*x^3 - 84*a^2*c^4*x^2 + 70*a*c^4*x + 30*c^4)/(a^8*x^7)","A",0
638,1,78,0,0.730525," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","-\frac{30 \, a^{6} c^{3} x^{6} + 60 \, a^{5} c^{3} x^{5} \log\left(x\right) + 30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}}"," ",0,"-1/30*(30*a^6*c^3*x^6 + 60*a^5*c^3*x^5*log(x) + 30*a^4*c^3*x^4 + 60*a^3*c^3*x^3 + 10*a^2*c^3*x^2 - 15*a*c^3*x - 6*c^3)/(a^6*x^5)","A",0
639,1,43,0,0.679902," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","-\frac{3 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} \log\left(x\right) + 3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}}"," ",0,"-1/3*(3*a^4*c^2*x^4 + 6*a^3*c^2*x^3*log(x) + 3*a*c^2*x + c^2)/(a^4*x^3)","A",0
640,1,27,0,0.466110," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2),x, algorithm=""fricas"")","-\frac{a^{2} c x^{2} + 2 \, a c x \log\left(x\right) - c}{a^{2} x}"," ",0,"-(a^2*c*x^2 + 2*a*c*x*log(x) - c)/(a^2*x)","A",0
641,1,41,0,0.484767," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2),x, algorithm=""fricas"")","-\frac{a^{2} x^{2} - a x + 2 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 1}{a^{2} c x - a c}"," ",0,"-(a^2*x^2 - a*x + 2*(a*x - 1)*log(a*x - 1) - 1)/(a^2*c*x - a*c)","A",0
642,1,93,0,0.487186," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","-\frac{8 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 6 \, a x - {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) + 17 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) + 12}{8 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"-1/8*(8*a^3*x^3 - 16*a^2*x^2 - 6*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 17*(a^2*x^2 - 2*a*x + 1)*log(a*x - 1) + 12)/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)","A",0
643,1,137,0,0.573625," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","-\frac{12 \, a^{5} x^{5} - 24 \, a^{4} x^{4} - 30 \, a^{3} x^{3} + 48 \, a^{2} x^{2} + 14 \, a x - 3 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x + 1\right) + 27 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x - 1\right) - 22}{12 \, {\left(a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/12*(12*a^5*x^5 - 24*a^4*x^4 - 30*a^3*x^3 + 48*a^2*x^2 + 14*a*x - 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x + 1) + 27*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x - 1) - 22)/(a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^2*c^3*x - a*c^3)","A",0
644,1,233,0,0.568155," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","-\frac{384 \, a^{7} x^{7} - 768 \, a^{6} x^{6} - 1638 \, a^{5} x^{5} + 2508 \, a^{4} x^{4} + 1732 \, a^{3} x^{3} - 2516 \, a^{2} x^{2} - 550 \, a x - 141 \, {\left(a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) + 909 \, {\left(a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) + 800}{384 \, {\left(a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/384*(384*a^7*x^7 - 768*a^6*x^6 - 1638*a^5*x^5 + 2508*a^4*x^4 + 1732*a^3*x^3 - 2516*a^2*x^2 - 550*a*x - 141*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 909*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(a*x - 1) + 800)/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)","A",0
645,1,175,0,1.311668," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","\frac{3360 \, a^{7} c^{4} x^{7} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 525 \, a^{7} c^{4} x^{7} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + 560 \, a^{7} c^{4} x^{7} + {\left(560 \, a^{7} c^{4} x^{7} - 2496 \, a^{6} c^{4} x^{6} - 525 \, a^{5} c^{4} x^{5} + 992 \, a^{4} c^{4} x^{4} + 770 \, a^{3} c^{4} x^{3} - 96 \, a^{2} c^{4} x^{2} - 280 \, a c^{4} x - 80 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{560 \, a^{8} x^{7}}"," ",0,"1/560*(3360*a^7*c^4*x^7*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 525*a^7*c^4*x^7*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 560*a^7*c^4*x^7 + (560*a^7*c^4*x^7 - 2496*a^6*c^4*x^6 - 525*a^5*c^4*x^5 + 992*a^4*c^4*x^4 + 770*a^3*c^4*x^3 - 96*a^2*c^4*x^2 - 280*a*c^4*x - 80*c^4)*sqrt(-a^2*x^2 + 1))/(a^8*x^7)","A",0
646,1,153,0,0.657152," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","\frac{240 \, a^{5} c^{3} x^{5} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 15 \, a^{5} c^{3} x^{5} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + 40 \, a^{5} c^{3} x^{5} + {\left(40 \, a^{5} c^{3} x^{5} - 152 \, a^{4} c^{3} x^{4} - 55 \, a^{3} c^{3} x^{3} + 24 \, a^{2} c^{3} x^{2} + 30 \, a c^{3} x + 8 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{40 \, a^{6} x^{5}}"," ",0,"1/40*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 15*a^5*c^3*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 40*a^5*c^3*x^5 + (40*a^5*c^3*x^5 - 152*a^4*c^3*x^4 - 55*a^3*c^3*x^3 + 24*a^2*c^3*x^2 + 30*a*c^3*x + 8*c^3)*sqrt(-a^2*x^2 + 1))/(a^6*x^5)","A",0
647,1,131,0,0.582696," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","\frac{36 \, a^{3} c^{2} x^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 3 \, a^{3} c^{2} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + 6 \, a^{3} c^{2} x^{3} + {\left(6 \, a^{3} c^{2} x^{3} - 16 \, a^{2} c^{2} x^{2} - 9 \, a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}}"," ",0,"1/6*(36*a^3*c^2*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 3*a^3*c^2*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 6*a^3*c^2*x^3 + (6*a^3*c^2*x^3 - 16*a^2*c^2*x^2 - 9*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)","A",0
648,1,82,0,0.600091," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2),x, algorithm=""fricas"")","\frac{6 \, a c x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 3 \, a c x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + a c x + \sqrt{-a^{2} x^{2} + 1} {\left(a c x + c\right)}}{a^{2} x}"," ",0,"(6*a*c*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 3*a*c*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) + a*c*x + sqrt(-a^2*x^2 + 1)*(a*c*x + c))/(a^2*x)","A",0
649,1,101,0,0.561300," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),x, algorithm=""fricas"")","\frac{14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{2} x^{2} - 19 \, a x + 14\right)} \sqrt{-a^{2} x^{2} + 1} + 14}{3 \, {\left(a^{3} c x^{2} - 2 \, a^{2} c x + a c\right)}}"," ",0,"1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^3*c*x^2 - 2*a^2*c*x + a*c)","A",0
650,1,143,0,0.608563," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","\frac{24 \, a^{3} x^{3} - 72 \, a^{2} x^{2} + 72 \, a x + 30 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(5 \, a^{3} x^{3} - 39 \, a^{2} x^{2} + 57 \, a x - 24\right)} \sqrt{-a^{2} x^{2} + 1} - 24}{5 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}"," ",0,"1/5*(24*a^3*x^3 - 72*a^2*x^2 + 72*a*x + 30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (5*a^3*x^3 - 39*a^2*x^2 + 57*a*x - 24)*sqrt(-a^2*x^2 + 1) - 24)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)","A",0
651,1,212,0,0.528501," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","\frac{176 \, a^{5} x^{5} - 528 \, a^{4} x^{4} + 352 \, a^{3} x^{3} + 352 \, a^{2} x^{2} - 528 \, a x + 210 \, {\left(a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(35 \, a^{5} x^{5} - 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} + 125 \, a^{2} x^{2} - 423 \, a x + 176\right)} \sqrt{-a^{2} x^{2} + 1} + 176}{35 \, {\left(a^{6} c^{3} x^{5} - 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"1/35*(176*a^5*x^5 - 528*a^4*x^4 + 352*a^3*x^3 + 352*a^2*x^2 - 528*a*x + 210*(a^5*x^5 - 3*a^4*x^4 + 2*a^3*x^3 + 2*a^2*x^2 - 3*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (35*a^5*x^5 - 286*a^4*x^4 + 368*a^3*x^3 + 125*a^2*x^2 - 423*a*x + 176)*sqrt(-a^2*x^2 + 1) + 176)/(a^6*c^3*x^5 - 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 - 3*a^2*c^3*x + a*c^3)","A",0
652,1,281,0,0.723122," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","\frac{1664 \, a^{7} x^{7} - 4992 \, a^{6} x^{6} + 1664 \, a^{5} x^{5} + 8320 \, a^{4} x^{4} - 8320 \, a^{3} x^{3} - 1664 \, a^{2} x^{2} + 4992 \, a x + 1890 \, {\left(a^{7} x^{7} - 3 \, a^{6} x^{6} + a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} - a^{2} x^{2} + 3 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(315 \, a^{7} x^{7} - 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} + 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} + 339 \, a^{2} x^{2} + 4047 \, a x - 1664\right)} \sqrt{-a^{2} x^{2} + 1} - 1664}{315 \, {\left(a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"1/315*(1664*a^7*x^7 - 4992*a^6*x^6 + 1664*a^5*x^5 + 8320*a^4*x^4 - 8320*a^3*x^3 - 1664*a^2*x^2 + 4992*a*x + 1890*(a^7*x^7 - 3*a^6*x^6 + a^5*x^5 + 5*a^4*x^4 - 5*a^3*x^3 - a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (315*a^7*x^7 - 2669*a^6*x^6 + 2967*a^5*x^5 + 4029*a^4*x^4 - 7399*a^3*x^3 + 339*a^2*x^2 + 4047*a*x - 1664)*sqrt(-a^2*x^2 + 1) - 1664)/(a^8*c^4*x^7 - 3*a^7*c^4*x^6 + a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 - a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)","A",0
653,1,111,0,0.561516," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^5,x, algorithm=""fricas"")","\frac{630 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log\left(x\right) - 1890 \, a^{8} c^{5} x^{8} + 2520 \, a^{7} c^{5} x^{7} + 2940 \, a^{6} c^{5} x^{6} - 1764 \, a^{4} c^{5} x^{4} - 840 \, a^{3} c^{5} x^{3} + 270 \, a^{2} c^{5} x^{2} + 315 \, a c^{5} x + 70 \, c^{5}}{630 \, a^{10} x^{9}}"," ",0,"1/630*(630*a^10*c^5*x^10 + 2520*a^9*c^5*x^9*log(x) - 1890*a^8*c^5*x^8 + 2520*a^7*c^5*x^7 + 2940*a^6*c^5*x^6 - 1764*a^4*c^5*x^4 - 840*a^3*c^5*x^3 + 270*a^2*c^5*x^2 + 315*a*c^5*x + 70*c^5)/(a^10*x^9)","A",0
654,1,100,0,0.477798," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","\frac{105 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log\left(x\right) - 420 \, a^{6} c^{4} x^{6} + 210 \, a^{5} c^{4} x^{5} + 350 \, a^{4} c^{4} x^{4} + 105 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x - 15 \, c^{4}}{105 \, a^{8} x^{7}}"," ",0,"1/105*(105*a^8*c^4*x^8 + 420*a^7*c^4*x^7*log(x) - 420*a^6*c^4*x^6 + 210*a^5*c^4*x^5 + 350*a^4*c^4*x^4 + 105*a^3*c^4*x^3 - 84*a^2*c^4*x^2 - 70*a*c^4*x - 15*c^4)/(a^8*x^7)","A",0
655,1,67,0,0.637145," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","\frac{15 \, a^{6} c^{3} x^{6} + 60 \, a^{5} c^{3} x^{5} \log\left(x\right) - 75 \, a^{4} c^{3} x^{4} + 25 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x + 3 \, c^{3}}{15 \, a^{6} x^{5}}"," ",0,"1/15*(15*a^6*c^3*x^6 + 60*a^5*c^3*x^5*log(x) - 75*a^4*c^3*x^4 + 25*a^2*c^3*x^2 + 15*a*c^3*x + 3*c^3)/(a^6*x^5)","A",0
656,1,56,0,0.675148," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","\frac{3 \, a^{4} c^{2} x^{4} + 12 \, a^{3} c^{2} x^{3} \log\left(x\right) - 18 \, a^{2} c^{2} x^{2} - 6 \, a c^{2} x - c^{2}}{3 \, a^{4} x^{3}}"," ",0,"1/3*(3*a^4*c^2*x^4 + 12*a^3*c^2*x^3*log(x) - 18*a^2*c^2*x^2 - 6*a*c^2*x - c^2)/(a^4*x^3)","A",0
657,1,35,0,0.598757," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2),x, algorithm=""fricas"")","\frac{a^{2} c x^{2} + 8 \, a c x \log\left(a x - 1\right) - 4 \, a c x \log\left(x\right) + c}{a^{2} x}"," ",0,"(a^2*c*x^2 + 8*a*c*x*log(a*x - 1) - 4*a*c*x*log(x) + c)/(a^2*x)","A",0
658,1,64,0,1.004597," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(c-c/a^2/x^2),x, algorithm=""fricas"")","\frac{a^{3} x^{3} - 2 \, a^{2} x^{2} - 4 \, a x + 4 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) + 4}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}"," ",0,"(a^3*x^3 - 2*a^2*x^2 - 4*a*x + 4*(a^2*x^2 - 2*a*x + 1)*log(a*x - 1) + 4)/(a^3*c*x^2 - 2*a^2*c*x + a*c)","A",0
659,1,100,0,0.546381," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","\frac{3 \, a^{4} x^{4} - 9 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 27 \, a x + 12 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right)} \log\left(a x - 1\right) - 13}{3 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}"," ",0,"1/3*(3*a^4*x^4 - 9*a^3*x^3 - 9*a^2*x^2 + 27*a*x + 12*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(a*x - 1) - 13)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)","A",0
660,1,163,0,0.496993," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","\frac{96 \, a^{5} x^{5} - 384 \, a^{4} x^{4} - 90 \, a^{3} x^{3} + 1320 \, a^{2} x^{2} - 1402 \, a x - 3 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \log\left(a x + 1\right) + 387 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \log\left(a x - 1\right) + 448}{96 \, {\left(a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"1/96*(96*a^5*x^5 - 384*a^4*x^4 - 90*a^3*x^3 + 1320*a^2*x^2 - 1402*a*x - 3*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(a*x + 1) + 387*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(a*x - 1) + 448)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)","A",0
661,1,207,0,0.655929," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","\frac{960 \, a^{7} x^{7} - 3840 \, a^{6} x^{6} - 2730 \, a^{5} x^{5} + 18600 \, a^{4} x^{4} - 13600 \, a^{3} x^{3} - 9800 \, a^{2} x^{2} + 15082 \, a x - 75 \, {\left(a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right)} \log\left(a x + 1\right) + 3915 \, {\left(a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right)} \log\left(a x - 1\right) - 4768}{960 \, {\left(a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"1/960*(960*a^7*x^7 - 3840*a^6*x^6 - 2730*a^5*x^5 + 18600*a^4*x^4 - 13600*a^3*x^3 - 9800*a^2*x^2 + 15082*a*x - 75*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(a*x + 1) + 3915*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(a*x - 1) - 4768)/(a^7*c^4*x^6 - 4*a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^3*c^4*x^2 + 4*a^2*c^4*x - a*c^4)","A",0
662,1,176,0,0.838756," ","integrate((c-c/a^2/x^2)^4/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{3360 \, a^{7} c^{4} x^{7} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 3675 \, a^{7} c^{4} x^{7} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 1680 \, a^{7} c^{4} x^{7} - {\left(1680 \, a^{7} c^{4} x^{7} + 2816 \, a^{6} c^{4} x^{6} + 3045 \, a^{5} c^{4} x^{5} - 1952 \, a^{4} c^{4} x^{4} - 1330 \, a^{3} c^{4} x^{3} + 1056 \, a^{2} c^{4} x^{2} + 280 \, a c^{4} x - 240 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{1680 \, a^{8} x^{7}}"," ",0,"-1/1680*(3360*a^7*c^4*x^7*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 3675*a^7*c^4*x^7*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 1680*a^7*c^4*x^7 - (1680*a^7*c^4*x^7 + 2816*a^6*c^4*x^6 + 3045*a^5*c^4*x^5 - 1952*a^4*c^4*x^4 - 1330*a^3*c^4*x^3 + 1056*a^2*c^4*x^2 + 280*a*c^4*x - 240*c^4)*sqrt(-a^2*x^2 + 1))/(a^8*x^7)","A",0
663,1,154,0,1.151530," ","integrate((c-c/a^2/x^2)^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{240 \, a^{5} c^{3} x^{5} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 225 \, a^{5} c^{3} x^{5} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 120 \, a^{5} c^{3} x^{5} - {\left(120 \, a^{5} c^{3} x^{5} + 184 \, a^{4} c^{3} x^{4} + 135 \, a^{3} c^{3} x^{3} - 88 \, a^{2} c^{3} x^{2} - 30 \, a c^{3} x + 24 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, a^{6} x^{5}}"," ",0,"-1/120*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 225*a^5*c^3*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 120*a^5*c^3*x^5 - (120*a^5*c^3*x^5 + 184*a^4*c^3*x^4 + 135*a^3*c^3*x^3 - 88*a^2*c^3*x^2 - 30*a*c^3*x + 24*c^3)*sqrt(-a^2*x^2 + 1))/(a^6*x^5)","A",0
664,1,132,0,0.553379," ","integrate((c-c/a^2/x^2)^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{12 \, a^{3} c^{2} x^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 9 \, a^{3} c^{2} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 6 \, a^{3} c^{2} x^{3} - {\left(6 \, a^{3} c^{2} x^{3} + 8 \, a^{2} c^{2} x^{2} + 3 \, a c^{2} x - 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}}"," ",0,"-1/6*(12*a^3*c^2*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 9*a^3*c^2*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 6*a^3*c^2*x^3 - (6*a^3*c^2*x^3 + 8*a^2*c^2*x^2 + 3*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)","A",0
665,1,85,0,0.456201," ","integrate((c-c/a^2/x^2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, a c x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - a c x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - a c x - \sqrt{-a^{2} x^{2} + 1} {\left(a c x + c\right)}}{a^{2} x}"," ",0,"-(2*a*c*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - a*c*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) - a*c*x - sqrt(-a^2*x^2 + 1)*(a*c*x + c))/(a^2*x)","A",0
666,1,66,0,0.420183," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2),x, algorithm=""fricas"")","\frac{2 \, a x - 2 \, {\left(a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a x + 2\right)} + 2}{a^{2} c x + a c}"," ",0,"(2*a*x - 2*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x + 2) + 2)/(a^2*c*x + a*c)","A",0
667,1,141,0,0.565814," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","\frac{8 \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 8 \, a x - 6 \, {\left(a^{3} x^{3} + a^{2} x^{2} - a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{3} x^{3} + 7 \, a^{2} x^{2} - 5 \, a x - 8\right)} \sqrt{-a^{2} x^{2} + 1} - 8}{3 \, {\left(a^{4} c^{2} x^{3} + a^{3} c^{2} x^{2} - a^{2} c^{2} x - a c^{2}\right)}}"," ",0,"1/3*(8*a^3*x^3 + 8*a^2*x^2 - 8*a*x - 6*(a^3*x^3 + a^2*x^2 - a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^3*x^3 + 7*a^2*x^2 - 5*a*x - 8)*sqrt(-a^2*x^2 + 1) - 8)/(a^4*c^2*x^3 + a^3*c^2*x^2 - a^2*c^2*x - a*c^2)","A",0
668,1,208,0,0.502699," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","\frac{48 \, a^{5} x^{5} + 48 \, a^{4} x^{4} - 96 \, a^{3} x^{3} - 96 \, a^{2} x^{2} + 48 \, a x - 30 \, {\left(a^{5} x^{5} + a^{4} x^{4} - 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(15 \, a^{5} x^{5} + 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} - 87 \, a^{2} x^{2} + 33 \, a x + 48\right)} \sqrt{-a^{2} x^{2} + 1} + 48}{15 \, {\left(a^{6} c^{3} x^{5} + a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"1/15*(48*a^5*x^5 + 48*a^4*x^4 - 96*a^3*x^3 - 96*a^2*x^2 + 48*a*x - 30*(a^5*x^5 + a^4*x^4 - 2*a^3*x^3 - 2*a^2*x^2 + a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^5*x^5 + 38*a^4*x^4 - 52*a^3*x^3 - 87*a^2*x^2 + 33*a*x + 48)*sqrt(-a^2*x^2 + 1) + 48)/(a^6*c^3*x^5 + a^5*c^3*x^4 - 2*a^4*c^3*x^3 - 2*a^3*c^3*x^2 + a^2*c^3*x + a*c^3)","A",0
669,1,281,0,1.091099," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","\frac{384 \, a^{7} x^{7} + 384 \, a^{6} x^{6} - 1152 \, a^{5} x^{5} - 1152 \, a^{4} x^{4} + 1152 \, a^{3} x^{3} + 1152 \, a^{2} x^{2} - 384 \, a x - 210 \, {\left(a^{7} x^{7} + a^{6} x^{6} - 3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(105 \, a^{7} x^{7} + 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} - 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} + 1065 \, a^{2} x^{2} - 279 \, a x - 384\right)} \sqrt{-a^{2} x^{2} + 1} - 384}{105 \, {\left(a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"1/105*(384*a^7*x^7 + 384*a^6*x^6 - 1152*a^5*x^5 - 1152*a^4*x^4 + 1152*a^3*x^3 + 1152*a^2*x^2 - 384*a*x - 210*(a^7*x^7 + a^6*x^6 - 3*a^5*x^5 - 3*a^4*x^4 + 3*a^3*x^3 + 3*a^2*x^2 - a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (105*a^7*x^7 + 281*a^6*x^6 - 559*a^5*x^5 - 965*a^4*x^4 + 715*a^3*x^3 + 1065*a^2*x^2 - 279*a*x - 384)*sqrt(-a^2*x^2 + 1) - 384)/(a^8*c^4*x^7 + a^7*c^4*x^6 - 3*a^6*c^4*x^5 - 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 + 3*a^3*c^4*x^2 - a^2*c^4*x - a*c^4)","A",0
670,1,89,0,0.473539," ","integrate((c-c/a^2/x^2)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{210 \, a^{8} c^{4} x^{8} - 420 \, a^{7} c^{4} x^{7} \log\left(x\right) + 420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}}"," ",0,"-1/210*(210*a^8*c^4*x^8 - 420*a^7*c^4*x^7*log(x) + 420*a^6*c^4*x^6 - 630*a^5*c^4*x^5 + 315*a^3*c^4*x^3 - 84*a^2*c^4*x^2 - 70*a*c^4*x + 30*c^4)/(a^8*x^7)","A",0
671,1,78,0,0.929665," ","integrate((c-c/a^2/x^2)^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{30 \, a^{6} c^{3} x^{6} - 60 \, a^{5} c^{3} x^{5} \log\left(x\right) + 30 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}}"," ",0,"-1/30*(30*a^6*c^3*x^6 - 60*a^5*c^3*x^5*log(x) + 30*a^4*c^3*x^4 - 60*a^3*c^3*x^3 + 10*a^2*c^3*x^2 + 15*a*c^3*x - 6*c^3)/(a^6*x^5)","A",0
672,1,43,0,0.520827," ","integrate((c-c/a^2/x^2)^2/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{3 \, a^{4} c^{2} x^{4} - 6 \, a^{3} c^{2} x^{3} \log\left(x\right) - 3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}}"," ",0,"-1/3*(3*a^4*c^2*x^4 - 6*a^3*c^2*x^3*log(x) - 3*a*c^2*x + c^2)/(a^4*x^3)","A",0
673,1,27,0,0.423611," ","integrate((c-c/a^2/x^2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{a^{2} c x^{2} - 2 \, a c x \log\left(x\right) - c}{a^{2} x}"," ",0,"-(a^2*c*x^2 - 2*a*c*x*log(x) - c)/(a^2*x)","A",0
674,1,39,0,0.702570," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2),x, algorithm=""fricas"")","-\frac{a^{2} x^{2} + a x - 2 \, {\left(a x + 1\right)} \log\left(a x + 1\right) - 1}{a^{2} c x + a c}"," ",0,"-(a^2*x^2 + a*x - 2*(a*x + 1)*log(a*x + 1) - 1)/(a^2*c*x + a*c)","A",0
675,1,92,0,0.585221," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","-\frac{8 \, a^{3} x^{3} + 16 \, a^{2} x^{2} - 6 \, a x - 17 \, {\left(a^{2} x^{2} + 2 \, a x + 1\right)} \log\left(a x + 1\right) + {\left(a^{2} x^{2} + 2 \, a x + 1\right)} \log\left(a x - 1\right) - 12}{8 \, {\left(a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"-1/8*(8*a^3*x^3 + 16*a^2*x^2 - 6*a*x - 17*(a^2*x^2 + 2*a*x + 1)*log(a*x + 1) + (a^2*x^2 + 2*a*x + 1)*log(a*x - 1) - 12)/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2)","A",0
676,1,137,0,0.636709," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","-\frac{12 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 30 \, a^{3} x^{3} - 48 \, a^{2} x^{2} + 14 \, a x - 27 \, {\left(a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right)} \log\left(a x + 1\right) + 3 \, {\left(a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right)} \log\left(a x - 1\right) + 22}{12 \, {\left(a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/12*(12*a^5*x^5 + 24*a^4*x^4 - 30*a^3*x^3 - 48*a^2*x^2 + 14*a*x - 27*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*log(a*x + 1) + 3*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*log(a*x - 1) + 22)/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x - a*c^3)","A",0
677,1,233,0,0.492146," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","-\frac{384 \, a^{7} x^{7} + 768 \, a^{6} x^{6} - 1638 \, a^{5} x^{5} - 2508 \, a^{4} x^{4} + 1732 \, a^{3} x^{3} + 2516 \, a^{2} x^{2} - 550 \, a x - 909 \, {\left(a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right)} \log\left(a x + 1\right) + 141 \, {\left(a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right)} \log\left(a x - 1\right) - 800}{384 \, {\left(a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/384*(384*a^7*x^7 + 768*a^6*x^6 - 1638*a^5*x^5 - 2508*a^4*x^4 + 1732*a^3*x^3 + 2516*a^2*x^2 - 550*a*x - 909*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*log(a*x + 1) + 141*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*log(a*x - 1) - 800)/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)","A",0
678,1,176,0,0.519134," ","integrate((c-c/a^2/x^2)^4/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\frac{3360 \, a^{7} c^{4} x^{7} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 525 \, a^{7} c^{4} x^{7} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 560 \, a^{7} c^{4} x^{7} - {\left(560 \, a^{7} c^{4} x^{7} + 2496 \, a^{6} c^{4} x^{6} - 525 \, a^{5} c^{4} x^{5} - 992 \, a^{4} c^{4} x^{4} + 770 \, a^{3} c^{4} x^{3} + 96 \, a^{2} c^{4} x^{2} - 280 \, a c^{4} x + 80 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{560 \, a^{8} x^{7}}"," ",0,"1/560*(3360*a^7*c^4*x^7*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 525*a^7*c^4*x^7*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 560*a^7*c^4*x^7 - (560*a^7*c^4*x^7 + 2496*a^6*c^4*x^6 - 525*a^5*c^4*x^5 - 992*a^4*c^4*x^4 + 770*a^3*c^4*x^3 + 96*a^2*c^4*x^2 - 280*a*c^4*x + 80*c^4)*sqrt(-a^2*x^2 + 1))/(a^8*x^7)","A",0
679,1,154,0,0.614701," ","integrate((c-c/a^2/x^2)^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\frac{240 \, a^{5} c^{3} x^{5} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 15 \, a^{5} c^{3} x^{5} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 40 \, a^{5} c^{3} x^{5} - {\left(40 \, a^{5} c^{3} x^{5} + 152 \, a^{4} c^{3} x^{4} - 55 \, a^{3} c^{3} x^{3} - 24 \, a^{2} c^{3} x^{2} + 30 \, a c^{3} x - 8 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{40 \, a^{6} x^{5}}"," ",0,"1/40*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 15*a^5*c^3*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 40*a^5*c^3*x^5 - (40*a^5*c^3*x^5 + 152*a^4*c^3*x^4 - 55*a^3*c^3*x^3 - 24*a^2*c^3*x^2 + 30*a*c^3*x - 8*c^3)*sqrt(-a^2*x^2 + 1))/(a^6*x^5)","A",0
680,1,132,0,0.507180," ","integrate((c-c/a^2/x^2)^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\frac{36 \, a^{3} c^{2} x^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 3 \, a^{3} c^{2} x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - 6 \, a^{3} c^{2} x^{3} - {\left(6 \, a^{3} c^{2} x^{3} + 16 \, a^{2} c^{2} x^{2} - 9 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}}"," ",0,"1/6*(36*a^3*c^2*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*a^3*c^2*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 6*a^3*c^2*x^3 - (6*a^3*c^2*x^3 + 16*a^2*c^2*x^2 - 9*a*c^2*x + 2*c^2)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)","A",0
681,1,86,0,0.714384," ","integrate((c-c/a^2/x^2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\frac{6 \, a c x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + 3 \, a c x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - a c x - \sqrt{-a^{2} x^{2} + 1} {\left(a c x - c\right)}}{a^{2} x}"," ",0,"(6*a*c*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*a*c*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) - a*c*x - sqrt(-a^2*x^2 + 1)*(a*c*x - c))/(a^2*x)","A",0
682,1,101,0,0.631836," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),x, algorithm=""fricas"")","-\frac{14 \, a^{2} x^{2} + 28 \, a x - 18 \, {\left(a^{2} x^{2} + 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{2} x^{2} + 19 \, a x + 14\right)} \sqrt{-a^{2} x^{2} + 1} + 14}{3 \, {\left(a^{3} c x^{2} + 2 \, a^{2} c x + a c\right)}}"," ",0,"-1/3*(14*a^2*x^2 + 28*a*x - 18*(a^2*x^2 + 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 + 19*a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^3*c*x^2 + 2*a^2*c*x + a*c)","A",0
683,1,142,0,0.875007," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","-\frac{24 \, a^{3} x^{3} + 72 \, a^{2} x^{2} + 72 \, a x - 30 \, {\left(a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(5 \, a^{3} x^{3} + 39 \, a^{2} x^{2} + 57 \, a x + 24\right)} \sqrt{-a^{2} x^{2} + 1} + 24}{5 \, {\left(a^{4} c^{2} x^{3} + 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"-1/5*(24*a^3*x^3 + 72*a^2*x^2 + 72*a*x - 30*(a^3*x^3 + 3*a^2*x^2 + 3*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (5*a^3*x^3 + 39*a^2*x^2 + 57*a*x + 24)*sqrt(-a^2*x^2 + 1) + 24)/(a^4*c^2*x^3 + 3*a^3*c^2*x^2 + 3*a^2*c^2*x + a*c^2)","A",0
684,1,213,0,0.565680," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^3,x, algorithm=""fricas"")","-\frac{176 \, a^{5} x^{5} + 528 \, a^{4} x^{4} + 352 \, a^{3} x^{3} - 352 \, a^{2} x^{2} - 528 \, a x - 210 \, {\left(a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(35 \, a^{5} x^{5} + 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} - 125 \, a^{2} x^{2} - 423 \, a x - 176\right)} \sqrt{-a^{2} x^{2} + 1} - 176}{35 \, {\left(a^{6} c^{3} x^{5} + 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/35*(176*a^5*x^5 + 528*a^4*x^4 + 352*a^3*x^3 - 352*a^2*x^2 - 528*a*x - 210*(a^5*x^5 + 3*a^4*x^4 + 2*a^3*x^3 - 2*a^2*x^2 - 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (35*a^5*x^5 + 286*a^4*x^4 + 368*a^3*x^3 - 125*a^2*x^2 - 423*a*x - 176)*sqrt(-a^2*x^2 + 1) - 176)/(a^6*c^3*x^5 + 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^3*c^3*x^2 - 3*a^2*c^3*x - a*c^3)","A",0
685,1,278,0,0.488494," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^4,x, algorithm=""fricas"")","-\frac{1664 \, a^{7} x^{7} + 4992 \, a^{6} x^{6} + 1664 \, a^{5} x^{5} - 8320 \, a^{4} x^{4} - 8320 \, a^{3} x^{3} + 1664 \, a^{2} x^{2} + 4992 \, a x - 1890 \, {\left(a^{7} x^{7} + 3 \, a^{6} x^{6} + a^{5} x^{5} - 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} + a^{2} x^{2} + 3 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(315 \, a^{7} x^{7} + 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} - 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} - 339 \, a^{2} x^{2} + 4047 \, a x + 1664\right)} \sqrt{-a^{2} x^{2} + 1} + 1664}{315 \, {\left(a^{8} c^{4} x^{7} + 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} + a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/315*(1664*a^7*x^7 + 4992*a^6*x^6 + 1664*a^5*x^5 - 8320*a^4*x^4 - 8320*a^3*x^3 + 1664*a^2*x^2 + 4992*a*x - 1890*(a^7*x^7 + 3*a^6*x^6 + a^5*x^5 - 5*a^4*x^4 - 5*a^3*x^3 + a^2*x^2 + 3*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (315*a^7*x^7 + 2669*a^6*x^6 + 2967*a^5*x^5 - 4029*a^4*x^4 - 7399*a^3*x^3 - 339*a^2*x^2 + 4047*a*x + 1664)*sqrt(-a^2*x^2 + 1) + 1664)/(a^8*c^4*x^7 + 3*a^7*c^4*x^6 + a^6*c^4*x^5 - 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 + a^3*c^4*x^2 + 3*a^2*c^4*x + a*c^4)","A",0
686,1,606,0,0.699741," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^(9/2),x, algorithm=""fricas"")","\left[\frac{420 \, {\left(a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - {\left(840 \, a^{9} c^{4} x^{9} + 3360 \, a^{7} c^{4} x^{7} + 1680 \, a^{6} c^{4} x^{6} - 1680 \, a^{5} c^{4} x^{5} - {\left(840 \, a^{9} + 3360 \, a^{7} + 1680 \, a^{6} - 1680 \, a^{5} - 1260 \, a^{4} + 672 \, a^{3} + 560 \, a^{2} - 120 \, a - 105\right)} c^{4} x^{8} - 1260 \, a^{4} c^{4} x^{4} + 672 \, a^{3} c^{4} x^{3} + 560 \, a^{2} c^{4} x^{2} - 120 \, a c^{4} x - 105 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{840 \, {\left(a^{10} x^{9} - a^{8} x^{7}\right)}}, -\frac{840 \, {\left(a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + {\left(840 \, a^{9} c^{4} x^{9} + 3360 \, a^{7} c^{4} x^{7} + 1680 \, a^{6} c^{4} x^{6} - 1680 \, a^{5} c^{4} x^{5} - {\left(840 \, a^{9} + 3360 \, a^{7} + 1680 \, a^{6} - 1680 \, a^{5} - 1260 \, a^{4} + 672 \, a^{3} + 560 \, a^{2} - 120 \, a - 105\right)} c^{4} x^{8} - 1260 \, a^{4} c^{4} x^{4} + 672 \, a^{3} c^{4} x^{3} + 560 \, a^{2} c^{4} x^{2} - 120 \, a c^{4} x - 105 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{840 \, {\left(a^{10} x^{9} - a^{8} x^{7}\right)}}\right]"," ",0,"[1/840*(420*(a^9*c^4*x^9 - a^7*c^4*x^7)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (840*a^9*c^4*x^9 + 3360*a^7*c^4*x^7 + 1680*a^6*c^4*x^6 - 1680*a^5*c^4*x^5 - (840*a^9 + 3360*a^7 + 1680*a^6 - 1680*a^5 - 1260*a^4 + 672*a^3 + 560*a^2 - 120*a - 105)*c^4*x^8 - 1260*a^4*c^4*x^4 + 672*a^3*c^4*x^3 + 560*a^2*c^4*x^2 - 120*a*c^4*x - 105*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^10*x^9 - a^8*x^7), -1/840*(840*(a^9*c^4*x^9 - a^7*c^4*x^7)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (840*a^9*c^4*x^9 + 3360*a^7*c^4*x^7 + 1680*a^6*c^4*x^6 - 1680*a^5*c^4*x^5 - (840*a^9 + 3360*a^7 + 1680*a^6 - 1680*a^5 - 1260*a^4 + 672*a^3 + 560*a^2 - 120*a - 105)*c^4*x^8 - 1260*a^4*c^4*x^4 + 672*a^3*c^4*x^3 + 560*a^2*c^4*x^2 - 120*a*c^4*x - 105*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^10*x^9 - a^8*x^7)]","A",0
687,1,542,0,0.787306," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^(7/2),x, algorithm=""fricas"")","\left[\frac{30 \, {\left(a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - {\left(60 \, a^{7} c^{3} x^{7} + 180 \, a^{5} c^{3} x^{5} + 90 \, a^{4} c^{3} x^{4} - {\left(60 \, a^{7} + 180 \, a^{5} + 90 \, a^{4} - 60 \, a^{3} - 45 \, a^{2} + 12 \, a + 10\right)} c^{3} x^{6} - 60 \, a^{3} c^{3} x^{3} - 45 \, a^{2} c^{3} x^{2} + 12 \, a c^{3} x + 10 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{8} x^{7} - a^{6} x^{5}\right)}}, -\frac{60 \, {\left(a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + {\left(60 \, a^{7} c^{3} x^{7} + 180 \, a^{5} c^{3} x^{5} + 90 \, a^{4} c^{3} x^{4} - {\left(60 \, a^{7} + 180 \, a^{5} + 90 \, a^{4} - 60 \, a^{3} - 45 \, a^{2} + 12 \, a + 10\right)} c^{3} x^{6} - 60 \, a^{3} c^{3} x^{3} - 45 \, a^{2} c^{3} x^{2} + 12 \, a c^{3} x + 10 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{8} x^{7} - a^{6} x^{5}\right)}}\right]"," ",0,"[1/60*(30*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (60*a^7*c^3*x^7 + 180*a^5*c^3*x^5 + 90*a^4*c^3*x^4 - (60*a^7 + 180*a^5 + 90*a^4 - 60*a^3 - 45*a^2 + 12*a + 10)*c^3*x^6 - 60*a^3*c^3*x^3 - 45*a^2*c^3*x^2 + 12*a*c^3*x + 10*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5), -1/60*(60*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (60*a^7*c^3*x^7 + 180*a^5*c^3*x^5 + 90*a^4*c^3*x^4 - (60*a^7 + 180*a^5 + 90*a^4 - 60*a^3 - 45*a^2 + 12*a + 10)*c^3*x^6 - 60*a^3*c^3*x^3 - 45*a^2*c^3*x^2 + 12*a*c^3*x + 10*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5)]","A",0
688,1,478,0,1.710519," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - {\left(12 \, a^{5} c^{2} x^{5} + 24 \, a^{3} c^{2} x^{3} - {\left(12 \, a^{5} + 24 \, a^{3} + 12 \, a^{2} - 4 \, a - 3\right)} c^{2} x^{4} + 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x - 3 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{12 \, {\left(a^{6} x^{5} - a^{4} x^{3}\right)}}, -\frac{12 \, {\left(a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + {\left(12 \, a^{5} c^{2} x^{5} + 24 \, a^{3} c^{2} x^{3} - {\left(12 \, a^{5} + 24 \, a^{3} + 12 \, a^{2} - 4 \, a - 3\right)} c^{2} x^{4} + 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x - 3 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{12 \, {\left(a^{6} x^{5} - a^{4} x^{3}\right)}}\right]"," ",0,"[1/12*(6*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (12*a^5*c^2*x^5 + 24*a^3*c^2*x^3 - (12*a^5 + 24*a^3 + 12*a^2 - 4*a - 3)*c^2*x^4 + 12*a^2*c^2*x^2 - 4*a*c^2*x - 3*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3), -1/12*(12*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (12*a^5*c^2*x^5 + 24*a^3*c^2*x^3 - (12*a^5 + 24*a^3 + 12*a^2 - 4*a - 3)*c^2*x^4 + 12*a^2*c^2*x^2 - 4*a*c^2*x - 3*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3)]","A",0
689,1,373,0,1.533268," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{3} c x^{3} - a c x\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - {\left(2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + 2 \, a + 1\right)} c x^{2} + 2 \, a c x + c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{3} - a^{2} x\right)}}, -\frac{2 \, {\left(a^{3} c x^{3} - a c x\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + {\left(2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + 2 \, a + 1\right)} c x^{2} + 2 \, a c x + c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{3} - a^{2} x\right)}}\right]"," ",0,"[1/2*((a^3*c*x^3 - a*c*x)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (2*a^3*c*x^3 - (2*a^3 + 2*a + 1)*c*x^2 + 2*a*c*x + c)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3 - a^2*x), -1/2*(2*(a^3*c*x^3 - a*c*x)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (2*a^3*c*x^3 - (2*a^3 + 2*a + 1)*c*x^2 + 2*a*c*x + c)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3 - a^2*x)]","A",0
690,1,320,0,0.689102," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - 2 \, {\left(a^{2} x^{2} - a^{2} x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{3} x^{2} - a\right)}}, -\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + {\left(a^{2} x^{2} - a^{2} x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} x^{2} - a}\right]"," ",0,"[1/2*((a^2*x^2 - 1)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - 2*(a^2*x^2 - a^2*x)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*x^2 - a), -((a^2*x^2 - 1)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (a^2*x^2 - a^2*x)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*x^2 - a)]","B",0
691,1,366,0,1.412552," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x - {\left(a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right)}{2 \, {\left(a^{3} c x^{2} - a c\right)}}, -\frac{\sqrt{-a^{2} x^{2} + 1} a^{2} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right)}{a^{3} c x^{2} - a c}\right]"," ",0,"[-1/2*(2*sqrt(-a^2*x^2 + 1)*a^2*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + (a^2*x^2 - 1)*sqrt(-c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x - (a^5*x^5 - 4*a^4*x^4 + 6*a^3*x^3 - 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)))/(a^3*c*x^2 - a*c), -(sqrt(-a^2*x^2 + 1)*a^2*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + (a^2*x^2 - 1)*sqrt(c)*arctan((a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 - 2*a^2*c*x^2 - a*c*x + 2*c)))/(a^3*c*x^2 - a*c)]","B",0
692,0,0,0,0.706716," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} a^{4} x^{4} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*a^4*x^4*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^5*c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)","F",0
693,0,0,0,0.651089," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} a^{6} x^{6} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{7} c^{3} x^{7} - a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} - a c^{3} x + c^{3}}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*a^6*x^6*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^7*c^3*x^7 - a^6*c^3*x^6 - 3*a^5*c^3*x^5 + 3*a^4*c^3*x^4 + 3*a^3*c^3*x^3 - 3*a^2*c^3*x^2 - a*c^3*x + c^3), x)","F",0
694,0,0,0,0.785044," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} a^{8} x^{8} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{9} c^{4} x^{9} - a^{8} c^{4} x^{8} - 4 \, a^{7} c^{4} x^{7} + 4 \, a^{6} c^{4} x^{6} + 6 \, a^{5} c^{4} x^{5} - 6 \, a^{4} c^{4} x^{4} - 4 \, a^{3} c^{4} x^{3} + 4 \, a^{2} c^{4} x^{2} + a c^{4} x - c^{4}}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*a^8*x^8*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^9*c^4*x^9 - a^8*c^4*x^8 - 4*a^7*c^4*x^7 + 4*a^6*c^4*x^6 + 6*a^5*c^4*x^5 - 6*a^4*c^4*x^4 - 4*a^3*c^4*x^3 + 4*a^2*c^4*x^2 + a*c^4*x - c^4), x)","F",0
695,1,482,0,0.770151," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(9/2),x, algorithm=""fricas"")","\left[\frac{53760 \, a^{7} \sqrt{-c} c^{4} x^{7} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + 25725 \, a^{7} \sqrt{-c} c^{4} x^{7} \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - 2 \, {\left(13440 \, a^{8} c^{4} x^{8} - 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} + 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} - 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} + 3840 \, a c^{4} x + 1680 \, c^{4}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{26880 \, a^{8} x^{7}}, -\frac{25725 \, a^{7} c^{\frac{9}{2}} x^{7} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 13440 \, a^{7} c^{\frac{9}{2}} x^{7} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) + {\left(13440 \, a^{8} c^{4} x^{8} - 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} + 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} - 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} + 3840 \, a c^{4} x + 1680 \, c^{4}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{13440 \, a^{8} x^{7}}\right]"," ",0,"[1/26880*(53760*a^7*sqrt(-c)*c^4*x^7*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 25725*a^7*sqrt(-c)*c^4*x^7*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(13440*a^8*c^4*x^8 - 45056*a^7*c^4*x^7 + 14595*a^6*c^4*x^6 + 31232*a^5*c^4*x^5 + 770*a^4*c^4*x^4 - 16896*a^3*c^4*x^3 - 4760*a^2*c^4*x^2 + 3840*a*c^4*x + 1680*c^4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7), -1/13440*(25725*a^7*c^(9/2)*x^7*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 13440*a^7*c^(9/2)*x^7*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + (13440*a^8*c^4*x^8 - 45056*a^7*c^4*x^7 + 14595*a^6*c^4*x^6 + 31232*a^5*c^4*x^5 + 770*a^4*c^4*x^4 - 16896*a^3*c^4*x^3 - 4760*a^2*c^4*x^2 + 3840*a*c^4*x + 1680*c^4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7)]","A",0
696,1,438,0,0.635252," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(7/2),x, algorithm=""fricas"")","\left[\frac{960 \, a^{5} \sqrt{-c} c^{3} x^{5} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + 375 \, a^{5} \sqrt{-c} c^{3} x^{5} \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - 2 \, {\left(240 \, a^{6} c^{3} x^{6} - 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} + 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} - 96 \, a c^{3} x - 40 \, c^{3}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{480 \, a^{6} x^{5}}, -\frac{375 \, a^{5} c^{\frac{7}{2}} x^{5} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 240 \, a^{5} c^{\frac{7}{2}} x^{5} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) + {\left(240 \, a^{6} c^{3} x^{6} - 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} + 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} - 96 \, a c^{3} x - 40 \, c^{3}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{240 \, a^{6} x^{5}}\right]"," ",0,"[1/480*(960*a^5*sqrt(-c)*c^3*x^5*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 375*a^5*sqrt(-c)*c^3*x^5*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(240*a^6*c^3*x^6 - 736*a^5*c^3*x^5 + 105*a^4*c^3*x^4 + 352*a^3*c^3*x^3 + 70*a^2*c^3*x^2 - 96*a*c^3*x - 40*c^3)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5), -1/240*(375*a^5*c^(7/2)*x^5*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 240*a^5*c^(7/2)*x^5*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + (240*a^6*c^3*x^6 - 736*a^5*c^3*x^5 + 105*a^4*c^3*x^4 + 352*a^3*c^3*x^3 + 70*a^2*c^3*x^2 - 96*a*c^3*x - 40*c^3)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5)]","A",0
697,1,394,0,0.496032," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{96 \, a^{3} \sqrt{-c} c^{2} x^{3} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + 27 \, a^{3} \sqrt{-c} c^{2} x^{3} \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - 2 \, {\left(24 \, a^{4} c^{2} x^{4} - 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} + 16 \, a c^{2} x + 6 \, c^{2}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, a^{4} x^{3}}, -\frac{27 \, a^{3} c^{\frac{5}{2}} x^{3} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 24 \, a^{3} c^{\frac{5}{2}} x^{3} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) + {\left(24 \, a^{4} c^{2} x^{4} - 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} + 16 \, a c^{2} x + 6 \, c^{2}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, a^{4} x^{3}}\right]"," ",0,"[1/48*(96*a^3*sqrt(-c)*c^2*x^3*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 27*a^3*sqrt(-c)*c^2*x^3*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(24*a^4*c^2*x^4 - 64*a^3*c^2*x^3 - 3*a^2*c^2*x^2 + 16*a*c^2*x + 6*c^2)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3), -1/24*(27*a^3*c^(5/2)*x^3*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 24*a^3*c^(5/2)*x^3*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + (24*a^4*c^2*x^4 - 64*a^3*c^2*x^3 - 3*a^2*c^2*x^2 + 16*a*c^2*x + 6*c^2)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3)]","A",0
698,1,316,0,1.394455," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{8 \, a \sqrt{-c} c x \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + a \sqrt{-c} c x \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - 2 \, {\left(2 \, a^{2} c x^{2} - 4 \, a c x - c\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, a^{2} x}, -\frac{a c^{\frac{3}{2}} x \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 2 \, a c^{\frac{3}{2}} x \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) + {\left(2 \, a^{2} c x^{2} - 4 \, a c x - c\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{2} x}\right]"," ",0,"[1/4*(8*a*sqrt(-c)*c*x*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + a*sqrt(-c)*c*x*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(2*a^2*c*x^2 - 4*a*c*x - c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x), -1/2*(a*c^(3/2)*x*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 2*a*c^(3/2)*x*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + (2*a^2*c*x^2 - 4*a*c*x - c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x)]","A",0
699,1,270,0,0.560565," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 4 \, \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - \sqrt{-c} \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right)}{2 \, a}, -\frac{a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - \sqrt{c} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right)}{a}\right]"," ",0,"[-1/2*(2*a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 4*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - sqrt(-c)*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2))/a, -(a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - sqrt(c)*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a]","A",0
700,1,216,0,1.549812," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a x - 1\right)} \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - {\left(a^{2} x^{2} - 3 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x - a c}, \frac{2 \, {\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - {\left(a^{2} x^{2} - 3 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x - a c}\right]"," ",0,"[((a*x - 1)*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (a^2*x^2 - 3*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*c*x - a*c), (2*(a*x - 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (a^2*x^2 - 3*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*c*x - a*c)]","A",0
701,1,281,0,0.982438," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - {\left(3 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 10 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}, \frac{6 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - {\left(3 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 10 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}\right]"," ",0,"[1/3*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (3*a^3*x^3 - 14*a^2*x^2 + 10*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), 1/3*(6*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (3*a^3*x^3 - 14*a^2*x^2 + 10*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]","A",0
702,1,353,0,0.626684," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - {\left(15 \, a^{5} x^{5} - 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 82 \, a^{2} x^{2} - 56 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left(a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right)}}, \frac{30 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - {\left(15 \, a^{5} x^{5} - 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 82 \, a^{2} x^{2} - 56 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left(a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right)}}\right]"," ",0,"[1/15*(15*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (15*a^5*x^5 - 76*a^4*x^4 + 32*a^3*x^3 + 82*a^2*x^2 - 56*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^2*c^3*x - a*c^3), 1/15*(30*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (15*a^5*x^5 - 76*a^4*x^4 + 32*a^3*x^3 + 82*a^2*x^2 - 56*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^2*c^3*x - a*c^3)]","A",0
703,1,497,0,0.858575," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(7/2),x, algorithm=""fricas"")","\left[\frac{105 \, {\left(a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - {\left(105 \, a^{7} x^{7} - 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} + 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} - 654 \, a^{2} x^{2} + 432 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left(a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}, \frac{210 \, {\left(a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - {\left(105 \, a^{7} x^{7} - 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} + 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} - 654 \, a^{2} x^{2} + 432 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left(a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}\right]"," ",0,"[1/105*(105*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (105*a^7*x^7 - 562*a^6*x^6 + 74*a^5*x^5 + 1226*a^4*x^4 - 636*a^3*x^3 - 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4), 1/105*(210*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (105*a^7*x^7 - 562*a^6*x^6 + 74*a^5*x^5 + 1226*a^4*x^4 - 636*a^3*x^3 - 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)]","A",0
704,1,569,0,0.988297," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(9/2),x, algorithm=""fricas"")","\left[\frac{315 \, {\left(a^{8} x^{8} - 2 \, a^{7} x^{7} - 2 \, a^{6} x^{6} + 6 \, a^{5} x^{5} - 6 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + 2 \, a x - 1\right)} \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - {\left(315 \, a^{9} x^{9} - 1756 \, a^{8} x^{8} - 268 \, a^{7} x^{7} + 5784 \, a^{6} x^{6} - 2060 \, a^{5} x^{5} - 6200 \, a^{4} x^{4} + 3372 \, a^{3} x^{3} + 2186 \, a^{2} x^{2} - 1408 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{315 \, {\left(a^{9} c^{5} x^{8} - 2 \, a^{8} c^{5} x^{7} - 2 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{4} c^{5} x^{3} + 2 \, a^{3} c^{5} x^{2} + 2 \, a^{2} c^{5} x - a c^{5}\right)}}, \frac{630 \, {\left(a^{8} x^{8} - 2 \, a^{7} x^{7} - 2 \, a^{6} x^{6} + 6 \, a^{5} x^{5} - 6 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + 2 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - {\left(315 \, a^{9} x^{9} - 1756 \, a^{8} x^{8} - 268 \, a^{7} x^{7} + 5784 \, a^{6} x^{6} - 2060 \, a^{5} x^{5} - 6200 \, a^{4} x^{4} + 3372 \, a^{3} x^{3} + 2186 \, a^{2} x^{2} - 1408 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{315 \, {\left(a^{9} c^{5} x^{8} - 2 \, a^{8} c^{5} x^{7} - 2 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{4} c^{5} x^{3} + 2 \, a^{3} c^{5} x^{2} + 2 \, a^{2} c^{5} x - a c^{5}\right)}}\right]"," ",0,"[1/315*(315*(a^8*x^8 - 2*a^7*x^7 - 2*a^6*x^6 + 6*a^5*x^5 - 6*a^3*x^3 + 2*a^2*x^2 + 2*a*x - 1)*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (315*a^9*x^9 - 1756*a^8*x^8 - 268*a^7*x^7 + 5784*a^6*x^6 - 2060*a^5*x^5 - 6200*a^4*x^4 + 3372*a^3*x^3 + 2186*a^2*x^2 - 1408*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^9*c^5*x^8 - 2*a^8*c^5*x^7 - 2*a^7*c^5*x^6 + 6*a^6*c^5*x^5 - 6*a^4*c^5*x^3 + 2*a^3*c^5*x^2 + 2*a^2*c^5*x - a*c^5), 1/315*(630*(a^8*x^8 - 2*a^7*x^7 - 2*a^6*x^6 + 6*a^5*x^5 - 6*a^3*x^3 + 2*a^2*x^2 + 2*a*x - 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (315*a^9*x^9 - 1756*a^8*x^8 - 268*a^7*x^7 + 5784*a^6*x^6 - 2060*a^5*x^5 - 6200*a^4*x^4 + 3372*a^3*x^3 + 2186*a^2*x^2 - 1408*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^9*c^5*x^8 - 2*a^8*c^5*x^7 - 2*a^7*c^5*x^6 + 6*a^6*c^5*x^5 - 6*a^4*c^5*x^3 + 2*a^3*c^5*x^2 + 2*a^2*c^5*x - a*c^5)]","A",0
705,1,542,0,0.793128," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(9/2),x, algorithm=""fricas"")","\left[\frac{420 \, {\left(a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) + {\left(280 \, a^{9} c^{4} x^{9} + 1120 \, a^{6} c^{4} x^{6} + 560 \, a^{5} c^{4} x^{5} - {\left(280 \, a^{9} + 1120 \, a^{6} + 560 \, a^{5} - 420 \, a^{4} - 448 \, a^{3} + 120 \, a + 35\right)} c^{4} x^{8} - 420 \, a^{4} c^{4} x^{4} - 448 \, a^{3} c^{4} x^{3} + 120 \, a c^{4} x + 35 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{280 \, {\left(a^{10} x^{9} - a^{8} x^{7}\right)}}, \frac{840 \, {\left(a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + {\left(280 \, a^{9} c^{4} x^{9} + 1120 \, a^{6} c^{4} x^{6} + 560 \, a^{5} c^{4} x^{5} - {\left(280 \, a^{9} + 1120 \, a^{6} + 560 \, a^{5} - 420 \, a^{4} - 448 \, a^{3} + 120 \, a + 35\right)} c^{4} x^{8} - 420 \, a^{4} c^{4} x^{4} - 448 \, a^{3} c^{4} x^{3} + 120 \, a c^{4} x + 35 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{280 \, {\left(a^{10} x^{9} - a^{8} x^{7}\right)}}\right]"," ",0,"[1/280*(420*(a^9*c^4*x^9 - a^7*c^4*x^7)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (280*a^9*c^4*x^9 + 1120*a^6*c^4*x^6 + 560*a^5*c^4*x^5 - (280*a^9 + 1120*a^6 + 560*a^5 - 420*a^4 - 448*a^3 + 120*a + 35)*c^4*x^8 - 420*a^4*c^4*x^4 - 448*a^3*c^4*x^3 + 120*a*c^4*x + 35*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^10*x^9 - a^8*x^7), 1/280*(840*(a^9*c^4*x^9 - a^7*c^4*x^7)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (280*a^9*c^4*x^9 + 1120*a^6*c^4*x^6 + 560*a^5*c^4*x^5 - (280*a^9 + 1120*a^6 + 560*a^5 - 420*a^4 - 448*a^3 + 120*a + 35)*c^4*x^8 - 420*a^4*c^4*x^4 - 448*a^3*c^4*x^3 + 120*a*c^4*x + 35*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^10*x^9 - a^8*x^7)]","A",0
706,1,542,0,0.772983," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(7/2),x, algorithm=""fricas"")","\left[\frac{90 \, {\left(a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) + {\left(60 \, a^{7} c^{3} x^{7} - 60 \, a^{5} c^{3} x^{5} + 150 \, a^{4} c^{3} x^{4} - {\left(60 \, a^{7} - 60 \, a^{5} + 150 \, a^{4} + 100 \, a^{3} - 15 \, a^{2} - 36 \, a - 10\right)} c^{3} x^{6} + 100 \, a^{3} c^{3} x^{3} - 15 \, a^{2} c^{3} x^{2} - 36 \, a c^{3} x - 10 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{8} x^{7} - a^{6} x^{5}\right)}}, \frac{180 \, {\left(a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + {\left(60 \, a^{7} c^{3} x^{7} - 60 \, a^{5} c^{3} x^{5} + 150 \, a^{4} c^{3} x^{4} - {\left(60 \, a^{7} - 60 \, a^{5} + 150 \, a^{4} + 100 \, a^{3} - 15 \, a^{2} - 36 \, a - 10\right)} c^{3} x^{6} + 100 \, a^{3} c^{3} x^{3} - 15 \, a^{2} c^{3} x^{2} - 36 \, a c^{3} x - 10 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{8} x^{7} - a^{6} x^{5}\right)}}\right]"," ",0,"[1/60*(90*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (60*a^7*c^3*x^7 - 60*a^5*c^3*x^5 + 150*a^4*c^3*x^4 - (60*a^7 - 60*a^5 + 150*a^4 + 100*a^3 - 15*a^2 - 36*a - 10)*c^3*x^6 + 100*a^3*c^3*x^3 - 15*a^2*c^3*x^2 - 36*a*c^3*x - 10*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5), 1/60*(180*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (60*a^7*c^3*x^7 - 60*a^5*c^3*x^5 + 150*a^4*c^3*x^4 - (60*a^7 - 60*a^5 + 150*a^4 + 100*a^3 - 15*a^2 - 36*a - 10)*c^3*x^6 + 100*a^3*c^3*x^3 - 15*a^2*c^3*x^2 - 36*a*c^3*x - 10*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5)]","A",0
707,1,474,0,1.107015," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) + {\left(4 \, a^{5} c^{2} x^{5} - 8 \, a^{3} c^{2} x^{3} - {\left(4 \, a^{5} - 8 \, a^{3} + 4 \, a^{2} + 4 \, a + 1\right)} c^{2} x^{4} + 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x + c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{6} x^{5} - a^{4} x^{3}\right)}}, \frac{12 \, {\left(a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + {\left(4 \, a^{5} c^{2} x^{5} - 8 \, a^{3} c^{2} x^{3} - {\left(4 \, a^{5} - 8 \, a^{3} + 4 \, a^{2} + 4 \, a + 1\right)} c^{2} x^{4} + 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x + c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{6} x^{5} - a^{4} x^{3}\right)}}\right]"," ",0,"[1/4*(6*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (4*a^5*c^2*x^5 - 8*a^3*c^2*x^3 - (4*a^5 - 8*a^3 + 4*a^2 + 4*a + 1)*c^2*x^4 + 4*a^2*c^2*x^2 + 4*a*c^2*x + c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3), 1/4*(12*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (4*a^5*c^2*x^5 - 8*a^3*c^2*x^3 - (4*a^5 - 8*a^3 + 4*a^2 + 4*a + 1)*c^2*x^4 + 4*a^2*c^2*x^2 + 4*a*c^2*x + c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3)]","A",0
708,1,378,0,0.595714," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} c x^{3} - a c x\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) + {\left(2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - 6 \, a - 1\right)} c x^{2} - 6 \, a c x - c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{3} - a^{2} x\right)}}, \frac{6 \, {\left(a^{3} c x^{3} - a c x\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + {\left(2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - 6 \, a - 1\right)} c x^{2} - 6 \, a c x - c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{3} - a^{2} x\right)}}\right]"," ",0,"[1/2*(3*(a^3*c*x^3 - a*c*x)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (2*a^3*c*x^3 - (2*a^3 - 6*a - 1)*c*x^2 - 6*a*c*x - c)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3 - a^2*x), 1/2*(6*(a^3*c*x^3 - a*c*x)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (2*a^3*c*x^3 - (2*a^3 - 6*a - 1)*c*x^2 - 6*a*c*x - c)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3 - a^2*x)]","A",0
709,0,0,0,3.932666," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^2 - 2*a*x + 1), x)","F",0
710,1,440,0,2.047988," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) - 2 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} c x^{3} - a^{3} c x^{2} - a^{2} c x + a c\right)}}, \frac{3 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right) + {\left(a^{3} x^{3} - 3 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{4} c x^{3} - a^{3} c x^{2} - a^{2} c x + a c}\right]"," ",0,"[-1/2*(3*(a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(-c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^5*x^5 - 4*a^4*x^4 + 6*a^3*x^3 - 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) - 2*(a^3*x^3 - 3*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*c*x^3 - a^3*c*x^2 - a^2*c*x + a*c), (3*(a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(c)*arctan((a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 - 2*a^2*c*x^2 - a*c*x + 2*c)) + (a^3*x^3 - 3*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*c*x^3 - a^3*c*x^2 - a^2*c*x + a*c)]","A",0
711,1,477,0,2.468006," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) - {\left(2 \, a^{4} x^{4} - 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{5} c^{2} x^{4} - 2 \, a^{4} c^{2} x^{3} + 2 \, a^{2} c^{2} x - a c^{2}\right)}}, \frac{6 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right) + {\left(2 \, a^{4} x^{4} - 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{5} c^{2} x^{4} - 2 \, a^{4} c^{2} x^{3} + 2 \, a^{2} c^{2} x - a c^{2}\right)}}\right]"," ",0,"[-1/2*(3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*sqrt(-c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^5*x^5 - 4*a^4*x^4 + 6*a^3*x^3 - 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) - (2*a^4*x^4 - 9*a^3*x^3 + 6*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^2*x^4 - 2*a^4*c^2*x^3 + 2*a^2*c^2*x - a*c^2), 1/2*(6*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*sqrt(c)*arctan((a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 - 2*a^2*c*x^2 - a*c*x + 2*c)) + (2*a^4*x^4 - 9*a^3*x^3 + 6*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^2*x^4 - 2*a^4*c^2*x^3 + 2*a^2*c^2*x - a*c^2)]","A",0
712,0,0,0,2.624662," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} a^{6} x^{6} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{7} c^{3} x^{7} - 3 \, a^{6} c^{3} x^{6} + a^{5} c^{3} x^{5} + 5 \, a^{4} c^{3} x^{4} - 5 \, a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2} + 3 \, a c^{3} x - c^{3}}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*a^6*x^6*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^7*c^3*x^7 - 3*a^6*c^3*x^6 + a^5*c^3*x^5 + 5*a^4*c^3*x^4 - 5*a^3*c^3*x^3 - a^2*c^3*x^2 + 3*a*c^3*x - c^3), x)","F",0
713,0,0,0,2.793947," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} a^{8} x^{8} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{9} c^{4} x^{9} - 3 \, a^{8} c^{4} x^{8} + 8 \, a^{6} c^{4} x^{6} - 6 \, a^{5} c^{4} x^{5} - 6 \, a^{4} c^{4} x^{4} + 8 \, a^{3} c^{4} x^{3} - 3 \, a c^{4} x + c^{4}}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*a^8*x^8*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^9*c^4*x^9 - 3*a^8*c^4*x^8 + 8*a^6*c^4*x^6 - 6*a^5*c^4*x^5 - 6*a^4*c^4*x^4 + 8*a^3*c^4*x^3 - 3*a*c^4*x + c^4), x)","F",0
714,1,606,0,2.278886," ","integrate((c-c/a^2/x^2)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{420 \, {\left(a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) + {\left(840 \, a^{9} c^{4} x^{9} + 3360 \, a^{7} c^{4} x^{7} - 1680 \, a^{6} c^{4} x^{6} - 1680 \, a^{5} c^{4} x^{5} - {\left(840 \, a^{9} + 3360 \, a^{7} - 1680 \, a^{6} - 1680 \, a^{5} + 1260 \, a^{4} + 672 \, a^{3} - 560 \, a^{2} - 120 \, a + 105\right)} c^{4} x^{8} + 1260 \, a^{4} c^{4} x^{4} + 672 \, a^{3} c^{4} x^{3} - 560 \, a^{2} c^{4} x^{2} - 120 \, a c^{4} x + 105 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{840 \, {\left(a^{10} x^{9} - a^{8} x^{7}\right)}}, -\frac{840 \, {\left(a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - {\left(840 \, a^{9} c^{4} x^{9} + 3360 \, a^{7} c^{4} x^{7} - 1680 \, a^{6} c^{4} x^{6} - 1680 \, a^{5} c^{4} x^{5} - {\left(840 \, a^{9} + 3360 \, a^{7} - 1680 \, a^{6} - 1680 \, a^{5} + 1260 \, a^{4} + 672 \, a^{3} - 560 \, a^{2} - 120 \, a + 105\right)} c^{4} x^{8} + 1260 \, a^{4} c^{4} x^{4} + 672 \, a^{3} c^{4} x^{3} - 560 \, a^{2} c^{4} x^{2} - 120 \, a c^{4} x + 105 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{840 \, {\left(a^{10} x^{9} - a^{8} x^{7}\right)}}\right]"," ",0,"[1/840*(420*(a^9*c^4*x^9 - a^7*c^4*x^7)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (840*a^9*c^4*x^9 + 3360*a^7*c^4*x^7 - 1680*a^6*c^4*x^6 - 1680*a^5*c^4*x^5 - (840*a^9 + 3360*a^7 - 1680*a^6 - 1680*a^5 + 1260*a^4 + 672*a^3 - 560*a^2 - 120*a + 105)*c^4*x^8 + 1260*a^4*c^4*x^4 + 672*a^3*c^4*x^3 - 560*a^2*c^4*x^2 - 120*a*c^4*x + 105*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^10*x^9 - a^8*x^7), -1/840*(840*(a^9*c^4*x^9 - a^7*c^4*x^7)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (840*a^9*c^4*x^9 + 3360*a^7*c^4*x^7 - 1680*a^6*c^4*x^6 - 1680*a^5*c^4*x^5 - (840*a^9 + 3360*a^7 - 1680*a^6 - 1680*a^5 + 1260*a^4 + 672*a^3 - 560*a^2 - 120*a + 105)*c^4*x^8 + 1260*a^4*c^4*x^4 + 672*a^3*c^4*x^3 - 560*a^2*c^4*x^2 - 120*a*c^4*x + 105*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^10*x^9 - a^8*x^7)]","A",0
715,1,542,0,2.263308," ","integrate((c-c/a^2/x^2)^(7/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{30 \, {\left(a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) + {\left(60 \, a^{7} c^{3} x^{7} + 180 \, a^{5} c^{3} x^{5} - 90 \, a^{4} c^{3} x^{4} - {\left(60 \, a^{7} + 180 \, a^{5} - 90 \, a^{4} - 60 \, a^{3} + 45 \, a^{2} + 12 \, a - 10\right)} c^{3} x^{6} - 60 \, a^{3} c^{3} x^{3} + 45 \, a^{2} c^{3} x^{2} + 12 \, a c^{3} x - 10 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{8} x^{7} - a^{6} x^{5}\right)}}, -\frac{60 \, {\left(a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - {\left(60 \, a^{7} c^{3} x^{7} + 180 \, a^{5} c^{3} x^{5} - 90 \, a^{4} c^{3} x^{4} - {\left(60 \, a^{7} + 180 \, a^{5} - 90 \, a^{4} - 60 \, a^{3} + 45 \, a^{2} + 12 \, a - 10\right)} c^{3} x^{6} - 60 \, a^{3} c^{3} x^{3} + 45 \, a^{2} c^{3} x^{2} + 12 \, a c^{3} x - 10 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{8} x^{7} - a^{6} x^{5}\right)}}\right]"," ",0,"[1/60*(30*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (60*a^7*c^3*x^7 + 180*a^5*c^3*x^5 - 90*a^4*c^3*x^4 - (60*a^7 + 180*a^5 - 90*a^4 - 60*a^3 + 45*a^2 + 12*a - 10)*c^3*x^6 - 60*a^3*c^3*x^3 + 45*a^2*c^3*x^2 + 12*a*c^3*x - 10*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5), -1/60*(60*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (60*a^7*c^3*x^7 + 180*a^5*c^3*x^5 - 90*a^4*c^3*x^4 - (60*a^7 + 180*a^5 - 90*a^4 - 60*a^3 + 45*a^2 + 12*a - 10)*c^3*x^6 - 60*a^3*c^3*x^3 + 45*a^2*c^3*x^2 + 12*a*c^3*x - 10*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5)]","A",0
716,1,478,0,0.746647," ","integrate((c-c/a^2/x^2)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) + {\left(12 \, a^{5} c^{2} x^{5} + 24 \, a^{3} c^{2} x^{3} - {\left(12 \, a^{5} + 24 \, a^{3} - 12 \, a^{2} - 4 \, a + 3\right)} c^{2} x^{4} - 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x + 3 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{12 \, {\left(a^{6} x^{5} - a^{4} x^{3}\right)}}, -\frac{12 \, {\left(a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - {\left(12 \, a^{5} c^{2} x^{5} + 24 \, a^{3} c^{2} x^{3} - {\left(12 \, a^{5} + 24 \, a^{3} - 12 \, a^{2} - 4 \, a + 3\right)} c^{2} x^{4} - 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x + 3 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{12 \, {\left(a^{6} x^{5} - a^{4} x^{3}\right)}}\right]"," ",0,"[1/12*(6*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (12*a^5*c^2*x^5 + 24*a^3*c^2*x^3 - (12*a^5 + 24*a^3 - 12*a^2 - 4*a + 3)*c^2*x^4 - 12*a^2*c^2*x^2 - 4*a*c^2*x + 3*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3), -1/12*(12*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (12*a^5*c^2*x^5 + 24*a^3*c^2*x^3 - (12*a^5 + 24*a^3 - 12*a^2 - 4*a + 3)*c^2*x^4 - 12*a^2*c^2*x^2 - 4*a*c^2*x + 3*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3)]","A",0
717,1,377,0,0.772053," ","integrate((c-c/a^2/x^2)^(3/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{3} c x^{3} - a c x\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) + {\left(2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + 2 \, a - 1\right)} c x^{2} + 2 \, a c x - c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{3} - a^{2} x\right)}}, -\frac{2 \, {\left(a^{3} c x^{3} - a c x\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - {\left(2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + 2 \, a - 1\right)} c x^{2} + 2 \, a c x - c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{3} - a^{2} x\right)}}\right]"," ",0,"[1/2*((a^3*c*x^3 - a*c*x)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (2*a^3*c*x^3 - (2*a^3 + 2*a - 1)*c*x^2 + 2*a*c*x - c)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3 - a^2*x), -1/2*(2*(a^3*c*x^3 - a*c*x)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (2*a^3*c*x^3 - (2*a^3 + 2*a - 1)*c*x^2 + 2*a*c*x - c)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3 - a^2*x)]","A",0
718,1,321,0,1.238551," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) + 2 \, {\left(a^{2} x^{2} - a^{2} x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{3} x^{2} - a\right)}}, -\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - {\left(a^{2} x^{2} - a^{2} x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} x^{2} - a}\right]"," ",0,"[1/2*((a^2*x^2 - 1)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + 2*(a^2*x^2 - a^2*x)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*x^2 - a), -((a^2*x^2 - 1)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (a^2*x^2 - a^2*x)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*x^2 - a)]","B",0
719,1,367,0,0.656167," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left(a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right)}{2 \, {\left(a^{3} c x^{2} - a c\right)}}, \frac{\sqrt{-a^{2} x^{2} + 1} a^{2} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right)}{a^{3} c x^{2} - a c}\right]"," ",0,"[1/2*(2*sqrt(-a^2*x^2 + 1)*a^2*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - (a^2*x^2 - 1)*sqrt(-c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x - (a^5*x^5 + 4*a^4*x^4 + 6*a^3*x^3 + 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)))/(a^3*c*x^2 - a*c), (sqrt(-a^2*x^2 + 1)*a^2*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - (a^2*x^2 - 1)*sqrt(c)*arctan((a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 + 2*a^2*c*x^2 - a*c*x - 2*c)))/(a^3*c*x^2 - a*c)]","B",0
720,0,0,0,1.701431," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} a^{4} x^{4} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{5} c^{2} x^{5} + a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + a c^{2} x + c^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*a^4*x^4*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^5*c^2*x^5 + a^4*c^2*x^4 - 2*a^3*c^2*x^3 - 2*a^2*c^2*x^2 + a*c^2*x + c^2), x)","F",0
721,0,0,0,1.230688," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} a^{6} x^{6} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{7} c^{3} x^{7} + a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} + 3 \, a^{2} c^{3} x^{2} - a c^{3} x - c^{3}}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*a^6*x^6*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^7*c^3*x^7 + a^6*c^3*x^6 - 3*a^5*c^3*x^5 - 3*a^4*c^3*x^4 + 3*a^3*c^3*x^3 + 3*a^2*c^3*x^2 - a*c^3*x - c^3), x)","F",0
722,0,0,0,0.629972," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} a^{8} x^{8} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{9} c^{4} x^{9} + a^{8} c^{4} x^{8} - 4 \, a^{7} c^{4} x^{7} - 4 \, a^{6} c^{4} x^{6} + 6 \, a^{5} c^{4} x^{5} + 6 \, a^{4} c^{4} x^{4} - 4 \, a^{3} c^{4} x^{3} - 4 \, a^{2} c^{4} x^{2} + a c^{4} x + c^{4}}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*a^8*x^8*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^9*c^4*x^9 + a^8*c^4*x^8 - 4*a^7*c^4*x^7 - 4*a^6*c^4*x^6 + 6*a^5*c^4*x^5 + 6*a^4*c^4*x^4 - 4*a^3*c^4*x^3 - 4*a^2*c^4*x^2 + a*c^4*x + c^4), x)","F",0
723,1,482,0,0.603449," ","integrate((c-c/a^2/x^2)^(9/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{53760 \, a^{7} \sqrt{-c} c^{4} x^{7} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 25725 \, a^{7} \sqrt{-c} c^{4} x^{7} \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) + 2 \, {\left(13440 \, a^{8} c^{4} x^{8} + 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} - 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} + 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} - 3840 \, a c^{4} x + 1680 \, c^{4}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{26880 \, a^{8} x^{7}}, -\frac{25725 \, a^{7} c^{\frac{9}{2}} x^{7} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 13440 \, a^{7} c^{\frac{9}{2}} x^{7} \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) + {\left(13440 \, a^{8} c^{4} x^{8} + 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} - 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} + 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} - 3840 \, a c^{4} x + 1680 \, c^{4}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{13440 \, a^{8} x^{7}}\right]"," ",0,"[-1/26880*(53760*a^7*sqrt(-c)*c^4*x^7*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 25725*a^7*sqrt(-c)*c^4*x^7*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) + 2*(13440*a^8*c^4*x^8 + 45056*a^7*c^4*x^7 + 14595*a^6*c^4*x^6 - 31232*a^5*c^4*x^5 + 770*a^4*c^4*x^4 + 16896*a^3*c^4*x^3 - 4760*a^2*c^4*x^2 - 3840*a*c^4*x + 1680*c^4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7), -1/13440*(25725*a^7*c^(9/2)*x^7*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 13440*a^7*c^(9/2)*x^7*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + (13440*a^8*c^4*x^8 + 45056*a^7*c^4*x^7 + 14595*a^6*c^4*x^6 - 31232*a^5*c^4*x^5 + 770*a^4*c^4*x^4 + 16896*a^3*c^4*x^3 - 4760*a^2*c^4*x^2 - 3840*a*c^4*x + 1680*c^4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7)]","A",0
724,1,438,0,1.216299," ","integrate((c-c/a^2/x^2)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{960 \, a^{5} \sqrt{-c} c^{3} x^{5} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 375 \, a^{5} \sqrt{-c} c^{3} x^{5} \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) + 2 \, {\left(240 \, a^{6} c^{3} x^{6} + 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} - 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} + 96 \, a c^{3} x - 40 \, c^{3}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{480 \, a^{6} x^{5}}, -\frac{375 \, a^{5} c^{\frac{7}{2}} x^{5} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 240 \, a^{5} c^{\frac{7}{2}} x^{5} \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) + {\left(240 \, a^{6} c^{3} x^{6} + 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} - 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} + 96 \, a c^{3} x - 40 \, c^{3}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{240 \, a^{6} x^{5}}\right]"," ",0,"[-1/480*(960*a^5*sqrt(-c)*c^3*x^5*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 375*a^5*sqrt(-c)*c^3*x^5*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) + 2*(240*a^6*c^3*x^6 + 736*a^5*c^3*x^5 + 105*a^4*c^3*x^4 - 352*a^3*c^3*x^3 + 70*a^2*c^3*x^2 + 96*a*c^3*x - 40*c^3)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5), -1/240*(375*a^5*c^(7/2)*x^5*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 240*a^5*c^(7/2)*x^5*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + (240*a^6*c^3*x^6 + 736*a^5*c^3*x^5 + 105*a^4*c^3*x^4 - 352*a^3*c^3*x^3 + 70*a^2*c^3*x^2 + 96*a*c^3*x - 40*c^3)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5)]","A",0
725,1,394,0,1.654019," ","integrate((c-c/a^2/x^2)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{96 \, a^{3} \sqrt{-c} c^{2} x^{3} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 27 \, a^{3} \sqrt{-c} c^{2} x^{3} \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) + 2 \, {\left(24 \, a^{4} c^{2} x^{4} + 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 16 \, a c^{2} x + 6 \, c^{2}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, a^{4} x^{3}}, -\frac{27 \, a^{3} c^{\frac{5}{2}} x^{3} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 24 \, a^{3} c^{\frac{5}{2}} x^{3} \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) + {\left(24 \, a^{4} c^{2} x^{4} + 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 16 \, a c^{2} x + 6 \, c^{2}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, a^{4} x^{3}}\right]"," ",0,"[-1/48*(96*a^3*sqrt(-c)*c^2*x^3*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 27*a^3*sqrt(-c)*c^2*x^3*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) + 2*(24*a^4*c^2*x^4 + 64*a^3*c^2*x^3 - 3*a^2*c^2*x^2 - 16*a*c^2*x + 6*c^2)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3), -1/24*(27*a^3*c^(5/2)*x^3*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 24*a^3*c^(5/2)*x^3*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + (24*a^4*c^2*x^4 + 64*a^3*c^2*x^3 - 3*a^2*c^2*x^2 - 16*a*c^2*x + 6*c^2)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3)]","A",0
726,1,317,0,0.639877," ","integrate((c-c/a^2/x^2)^(3/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{8 \, a \sqrt{-c} c x \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - a \sqrt{-c} c x \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) + 2 \, {\left(2 \, a^{2} c x^{2} + 4 \, a c x - c\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, a^{2} x}, -\frac{a c^{\frac{3}{2}} x \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - 2 \, a c^{\frac{3}{2}} x \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) + {\left(2 \, a^{2} c x^{2} + 4 \, a c x - c\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{2} x}\right]"," ",0,"[-1/4*(8*a*sqrt(-c)*c*x*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - a*sqrt(-c)*c*x*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) + 2*(2*a^2*c*x^2 + 4*a*c*x - c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x), -1/2*(a*c^(3/2)*x*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 2*a*c^(3/2)*x*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + (2*a^2*c*x^2 + 4*a*c*x - c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x)]","A",0
727,1,270,0,0.692858," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + 4 \, \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - \sqrt{-c} \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right)}{2 \, a}, -\frac{a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - \sqrt{c} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - \sqrt{c} \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right)}{a}\right]"," ",0,"[-1/2*(2*a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + 4*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - sqrt(-c)*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2))/a, -(a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - sqrt(c)*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a]","A",0
728,1,214,0,0.596518," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a x + 1\right)} \sqrt{c} \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - {\left(a^{2} x^{2} + 3 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x + a c}, -\frac{2 \, {\left(a x + 1\right)} \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + {\left(a^{2} x^{2} + 3 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x + a c}\right]"," ",0,"[((a*x + 1)*sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (a^2*x^2 + 3*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*c*x + a*c), -(2*(a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (a^2*x^2 + 3*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*c*x + a*c)]","A",0
729,1,280,0,0.699863," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} x^{2} + 2 \, a x + 1\right)} \sqrt{c} \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - {\left(3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 10 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left(a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right)}}, -\frac{6 \, {\left(a^{2} x^{2} + 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + {\left(3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 10 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left(a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right)}}\right]"," ",0,"[1/3*(3*(a^2*x^2 + 2*a*x + 1)*sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (3*a^3*x^3 + 14*a^2*x^2 + 10*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2), -1/3*(6*(a^2*x^2 + 2*a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (3*a^3*x^3 + 14*a^2*x^2 + 10*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2)]","A",0
730,1,352,0,0.569660," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right)} \sqrt{c} \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - {\left(15 \, a^{5} x^{5} + 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} - 82 \, a^{2} x^{2} - 56 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left(a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right)}}, -\frac{30 \, {\left(a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + {\left(15 \, a^{5} x^{5} + 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} - 82 \, a^{2} x^{2} - 56 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left(a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right)}}\right]"," ",0,"[1/15*(15*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (15*a^5*x^5 + 76*a^4*x^4 + 32*a^3*x^3 - 82*a^2*x^2 - 56*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x - a*c^3), -1/15*(30*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (15*a^5*x^5 + 76*a^4*x^4 + 32*a^3*x^3 - 82*a^2*x^2 - 56*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x - a*c^3)]","A",0
731,1,496,0,0.779337," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(7/2),x, algorithm=""fricas"")","\left[\frac{105 \, {\left(a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right)} \sqrt{c} \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - {\left(105 \, a^{7} x^{7} + 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} - 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} + 654 \, a^{2} x^{2} + 432 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left(a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right)}}, -\frac{210 \, {\left(a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + {\left(105 \, a^{7} x^{7} + 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} - 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} + 654 \, a^{2} x^{2} + 432 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left(a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right)}}\right]"," ",0,"[1/105*(105*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (105*a^7*x^7 + 562*a^6*x^6 + 74*a^5*x^5 - 1226*a^4*x^4 - 636*a^3*x^3 + 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4), -1/105*(210*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (105*a^7*x^7 + 562*a^6*x^6 + 74*a^5*x^5 - 1226*a^4*x^4 - 636*a^3*x^3 + 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)]","A",0
732,1,544,0,0.786990," ","integrate((c-c/a^2/x^2)^(9/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{420 \, {\left(a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - {\left(280 \, a^{9} c^{4} x^{9} - 1120 \, a^{6} c^{4} x^{6} + 560 \, a^{5} c^{4} x^{5} - {\left(280 \, a^{9} - 1120 \, a^{6} + 560 \, a^{5} + 420 \, a^{4} - 448 \, a^{3} + 120 \, a - 35\right)} c^{4} x^{8} + 420 \, a^{4} c^{4} x^{4} - 448 \, a^{3} c^{4} x^{3} + 120 \, a c^{4} x - 35 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{280 \, {\left(a^{10} x^{9} - a^{8} x^{7}\right)}}, \frac{840 \, {\left(a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - {\left(280 \, a^{9} c^{4} x^{9} - 1120 \, a^{6} c^{4} x^{6} + 560 \, a^{5} c^{4} x^{5} - {\left(280 \, a^{9} - 1120 \, a^{6} + 560 \, a^{5} + 420 \, a^{4} - 448 \, a^{3} + 120 \, a - 35\right)} c^{4} x^{8} + 420 \, a^{4} c^{4} x^{4} - 448 \, a^{3} c^{4} x^{3} + 120 \, a c^{4} x - 35 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{280 \, {\left(a^{10} x^{9} - a^{8} x^{7}\right)}}\right]"," ",0,"[1/280*(420*(a^9*c^4*x^9 - a^7*c^4*x^7)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (280*a^9*c^4*x^9 - 1120*a^6*c^4*x^6 + 560*a^5*c^4*x^5 - (280*a^9 - 1120*a^6 + 560*a^5 + 420*a^4 - 448*a^3 + 120*a - 35)*c^4*x^8 + 420*a^4*c^4*x^4 - 448*a^3*c^4*x^3 + 120*a*c^4*x - 35*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^10*x^9 - a^8*x^7), 1/280*(840*(a^9*c^4*x^9 - a^7*c^4*x^7)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (280*a^9*c^4*x^9 - 1120*a^6*c^4*x^6 + 560*a^5*c^4*x^5 - (280*a^9 - 1120*a^6 + 560*a^5 + 420*a^4 - 448*a^3 + 120*a - 35)*c^4*x^8 + 420*a^4*c^4*x^4 - 448*a^3*c^4*x^3 + 120*a*c^4*x - 35*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^10*x^9 - a^8*x^7)]","A",0
733,1,544,0,0.713676," ","integrate((c-c/a^2/x^2)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{90 \, {\left(a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - {\left(60 \, a^{7} c^{3} x^{7} - 60 \, a^{5} c^{3} x^{5} - 150 \, a^{4} c^{3} x^{4} - {\left(60 \, a^{7} - 60 \, a^{5} - 150 \, a^{4} + 100 \, a^{3} + 15 \, a^{2} - 36 \, a + 10\right)} c^{3} x^{6} + 100 \, a^{3} c^{3} x^{3} + 15 \, a^{2} c^{3} x^{2} - 36 \, a c^{3} x + 10 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{8} x^{7} - a^{6} x^{5}\right)}}, \frac{180 \, {\left(a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - {\left(60 \, a^{7} c^{3} x^{7} - 60 \, a^{5} c^{3} x^{5} - 150 \, a^{4} c^{3} x^{4} - {\left(60 \, a^{7} - 60 \, a^{5} - 150 \, a^{4} + 100 \, a^{3} + 15 \, a^{2} - 36 \, a + 10\right)} c^{3} x^{6} + 100 \, a^{3} c^{3} x^{3} + 15 \, a^{2} c^{3} x^{2} - 36 \, a c^{3} x + 10 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{8} x^{7} - a^{6} x^{5}\right)}}\right]"," ",0,"[1/60*(90*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (60*a^7*c^3*x^7 - 60*a^5*c^3*x^5 - 150*a^4*c^3*x^4 - (60*a^7 - 60*a^5 - 150*a^4 + 100*a^3 + 15*a^2 - 36*a + 10)*c^3*x^6 + 100*a^3*c^3*x^3 + 15*a^2*c^3*x^2 - 36*a*c^3*x + 10*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5), 1/60*(180*(a^7*c^3*x^7 - a^5*c^3*x^5)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (60*a^7*c^3*x^7 - 60*a^5*c^3*x^5 - 150*a^4*c^3*x^4 - (60*a^7 - 60*a^5 - 150*a^4 + 100*a^3 + 15*a^2 - 36*a + 10)*c^3*x^6 + 100*a^3*c^3*x^3 + 15*a^2*c^3*x^2 - 36*a*c^3*x + 10*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7 - a^6*x^5)]","A",0
734,1,480,0,0.610913," ","integrate((c-c/a^2/x^2)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - {\left(4 \, a^{5} c^{2} x^{5} - 8 \, a^{3} c^{2} x^{3} - {\left(4 \, a^{5} - 8 \, a^{3} - 4 \, a^{2} + 4 \, a - 1\right)} c^{2} x^{4} - 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x - c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{6} x^{5} - a^{4} x^{3}\right)}}, \frac{12 \, {\left(a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - {\left(4 \, a^{5} c^{2} x^{5} - 8 \, a^{3} c^{2} x^{3} - {\left(4 \, a^{5} - 8 \, a^{3} - 4 \, a^{2} + 4 \, a - 1\right)} c^{2} x^{4} - 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x - c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{6} x^{5} - a^{4} x^{3}\right)}}\right]"," ",0,"[1/4*(6*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (4*a^5*c^2*x^5 - 8*a^3*c^2*x^3 - (4*a^5 - 8*a^3 - 4*a^2 + 4*a - 1)*c^2*x^4 - 4*a^2*c^2*x^2 + 4*a*c^2*x - c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3), 1/4*(12*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (4*a^5*c^2*x^5 - 8*a^3*c^2*x^3 - (4*a^5 - 8*a^3 - 4*a^2 + 4*a - 1)*c^2*x^4 - 4*a^2*c^2*x^2 + 4*a*c^2*x - c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3)]","A",0
735,1,376,0,0.772067," ","integrate((c-c/a^2/x^2)^(3/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} c x^{3} - a c x\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - {\left(2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - 6 \, a + 1\right)} c x^{2} - 6 \, a c x + c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{3} - a^{2} x\right)}}, \frac{6 \, {\left(a^{3} c x^{3} - a c x\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - {\left(2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - 6 \, a + 1\right)} c x^{2} - 6 \, a c x + c\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{3} - a^{2} x\right)}}\right]"," ",0,"[1/2*(3*(a^3*c*x^3 - a*c*x)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (2*a^3*c*x^3 - (2*a^3 - 6*a + 1)*c*x^2 - 6*a*c*x + c)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3 - a^2*x), 1/2*(6*(a^3*c*x^3 - a*c*x)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (2*a^3*c*x^3 - (2*a^3 - 6*a + 1)*c*x^2 - 6*a*c*x + c)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3 - a^2*x)]","A",0
736,0,0,0,0.600171," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{2} + 2 \, a x + 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(a*x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^2 + 2*a*x + 1), x)","F",0
737,1,439,0,1.733717," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{3} x^{3} + a^{2} x^{2} - a x - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left(a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right) + 2 \, {\left(a^{3} x^{3} + 3 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} c x^{3} + a^{3} c x^{2} - a^{2} c x - a c\right)}}, \frac{3 \, {\left(a^{3} x^{3} + a^{2} x^{2} - a x - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right) - {\left(a^{3} x^{3} + 3 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{4} c x^{3} + a^{3} c x^{2} - a^{2} c x - a c}\right]"," ",0,"[-1/2*(3*(a^3*x^3 + a^2*x^2 - a*x - 1)*sqrt(-c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x + (a^5*x^5 + 4*a^4*x^4 + 6*a^3*x^3 + 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)) + 2*(a^3*x^3 + 3*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*c*x^3 + a^3*c*x^2 - a^2*c*x - a*c), (3*(a^3*x^3 + a^2*x^2 - a*x - 1)*sqrt(c)*arctan((a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 + 2*a^2*c*x^2 - a*c*x - 2*c)) - (a^3*x^3 + 3*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*c*x^3 + a^3*c*x^2 - a^2*c*x - a*c)]","A",0
738,1,477,0,0.660459," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left(a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right) + {\left(2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{5} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{3} - 2 \, a^{2} c^{2} x - a c^{2}\right)}}, \frac{6 \, {\left(a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right) - {\left(2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 6 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{5} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{3} - 2 \, a^{2} c^{2} x - a c^{2}\right)}}\right]"," ",0,"[-1/2*(3*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*sqrt(-c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x + (a^5*x^5 + 4*a^4*x^4 + 6*a^3*x^3 + 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)) + (2*a^4*x^4 + 9*a^3*x^3 + 6*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^2*x^4 + 2*a^4*c^2*x^3 - 2*a^2*c^2*x - a*c^2), 1/2*(6*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*sqrt(c)*arctan((a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 + 2*a^2*c*x^2 - a*c*x - 2*c)) - (2*a^4*x^4 + 9*a^3*x^3 + 6*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^2*x^4 + 2*a^4*c^2*x^3 - 2*a^2*c^2*x - a*c^2)]","A",0
739,0,0,0,0.858609," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} a^{6} x^{6} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{7} c^{3} x^{7} + 3 \, a^{6} c^{3} x^{6} + a^{5} c^{3} x^{5} - 5 \, a^{4} c^{3} x^{4} - 5 \, a^{3} c^{3} x^{3} + a^{2} c^{3} x^{2} + 3 \, a c^{3} x + c^{3}}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*a^6*x^6*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^7*c^3*x^7 + 3*a^6*c^3*x^6 + a^5*c^3*x^5 - 5*a^4*c^3*x^4 - 5*a^3*c^3*x^3 + a^2*c^3*x^2 + 3*a*c^3*x + c^3), x)","F",0
740,0,0,0,1.726250," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} a^{8} x^{8} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{9} c^{4} x^{9} + 3 \, a^{8} c^{4} x^{8} - 8 \, a^{6} c^{4} x^{6} - 6 \, a^{5} c^{4} x^{5} + 6 \, a^{4} c^{4} x^{4} + 8 \, a^{3} c^{4} x^{3} - 3 \, a c^{4} x - c^{4}}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*a^8*x^8*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^9*c^4*x^9 + 3*a^8*c^4*x^8 - 8*a^6*c^4*x^6 - 6*a^5*c^4*x^5 + 6*a^4*c^4*x^4 + 8*a^3*c^4*x^3 - 3*a*c^4*x - c^4), x)","F",0
741,1,78,0,1.040833," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a m x^{2} + {\left(m + 1\right)} x\right)} x^{m} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{{\left(a^{2} m^{2} + a^{2} m\right)} x^{2} - m^{2} - m}"," ",0,"-sqrt(-a^2*x^2 + 1)*(a*m*x^2 + (m + 1)*x)*x^m*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/((a^2*m^2 + a^2*m)*x^2 - m^2 - m)","A",0
742,1,58,0,0.604518," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, a x^{4} + 3 \, x^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/6*(2*a*x^4 + 3*x^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^2 - 1)","A",0
743,1,57,0,0.635060," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + 2 \, x^{2}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/2*sqrt(-a^2*x^2 + 1)*(a*x^3 + 2*x^2)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^2 - 1)","A",0
744,1,320,0,0.696111," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - 2 \, {\left(a^{2} x^{2} - a^{2} x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{3} x^{2} - a\right)}}, -\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + {\left(a^{2} x^{2} - a^{2} x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} x^{2} - a}\right]"," ",0,"[1/2*((a^2*x^2 - 1)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - 2*(a^2*x^2 - a^2*x)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*x^2 - a), -((a^2*x^2 - 1)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (a^2*x^2 - a^2*x)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*x^2 - a)]","B",0
745,1,294,0,0.881704," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} {\left(x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{2} x^{2} - 1\right)}}, -\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{2} - 1}\right]"," ",0,"[1/2*((a^2*x^2 - 1)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - 2*sqrt(-a^2*x^2 + 1)*(x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^2 - 1), -((a^2*x^2 - 1)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + sqrt(-a^2*x^2 + 1)*(x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^2 - 1)]","B",0
746,1,63,0,0.556489," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^(1/2)/x^2,x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a + 1\right)} x^{2} - 2 \, a x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{2} x^{3} - x\right)}}"," ",0,"-1/2*sqrt(-a^2*x^2 + 1)*((2*a + 1)*x^2 - 2*a*x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^3 - x)","A",0
747,1,222,0,0.542924," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(6 \, a^{4} x^{4} + 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 32 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right)}{48 \, a^{4}}, -\frac{{\left(6 \, a^{4} x^{4} + 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 32 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right)}{24 \, a^{4}}\right]"," ",0,"[-1/48*(2*(6*a^4*x^4 + 16*a^3*x^3 + 21*a^2*x^2 + 32*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 21*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a^4, -1/24*((6*a^4*x^4 + 16*a^3*x^3 + 21*a^2*x^2 + 32*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 21*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)))/a^4]","A",0
748,1,204,0,0.661680," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{3} x^{3} + 3 \, a^{2} x^{2} + 5 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right)}{6 \, a^{3}}, -\frac{{\left(a^{3} x^{3} + 3 \, a^{2} x^{2} + 5 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right)}{3 \, a^{3}}\right]"," ",0,"[-1/6*(2*(a^3*x^3 + 3*a^2*x^2 + 5*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 3*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a^3, -1/3*((a^3*x^3 + 3*a^2*x^2 + 5*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 3*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)))/a^3]","A",0
749,1,188,0,0.584221," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{2} x^{2} + 4 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right)}{4 \, a^{2}}, -\frac{{\left(a^{2} x^{2} + 4 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right)}{2 \, a^{2}}\right]"," ",0,"[-1/4*(2*(a^2*x^2 + 4*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 3*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a^2, -1/2*((a^2*x^2 + 4*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 3*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)))/a^2]","A",0
750,1,270,0,0.634324," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 4 \, \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - \sqrt{-c} \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right)}{2 \, a}, -\frac{a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - \sqrt{c} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right)}{a}\right]"," ",0,"[-1/2*(2*a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 4*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - sqrt(-c)*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2))/a, -(a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - sqrt(c)*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a]","A",0
751,1,255,0,1.492845," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x,x, algorithm=""fricas"")","\left[\sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + \sqrt{-c} \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}, 2 \, \sqrt{c} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + \frac{1}{2} \, \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}\right]"," ",0,"[sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + sqrt(-c)*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - sqrt((a^2*c*x^2 - c)/(a^2*x^2)), 2*sqrt(c)*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 1/2*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - sqrt((a^2*c*x^2 - c)/(a^2*x^2))]","A",0
752,1,177,0,0.598564," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^2,x, algorithm=""fricas"")","\left[\frac{3 \, a \sqrt{-c} x \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - 2 \, {\left(4 \, a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, x}, \frac{3 \, a \sqrt{c} x \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - {\left(4 \, a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, x}\right]"," ",0,"[1/4*(3*a*sqrt(-c)*x*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(4*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x, 1/2*(3*a*sqrt(c)*x*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (4*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x]","A",0
753,1,201,0,0.756691," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^3,x, algorithm=""fricas"")","\left[\frac{3 \, a^{2} \sqrt{-c} x^{2} \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - 2 \, {\left(5 \, a^{2} x^{2} + 3 \, a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, x^{2}}, \frac{3 \, a^{2} \sqrt{c} x^{2} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - {\left(5 \, a^{2} x^{2} + 3 \, a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, x^{2}}\right]"," ",0,"[1/6*(3*a^2*sqrt(-c)*x^2*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(5*a^2*x^2 + 3*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^2, 1/3*(3*a^2*sqrt(c)*x^2*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (5*a^2*x^2 + 3*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^2]","A",0
754,1,217,0,0.757514," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^4,x, algorithm=""fricas"")","\left[\frac{21 \, a^{3} \sqrt{-c} x^{3} \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - 2 \, {\left(32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, x^{3}}, \frac{21 \, a^{3} \sqrt{c} x^{3} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - {\left(32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, x^{3}}\right]"," ",0,"[1/48*(21*a^3*sqrt(-c)*x^3*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(32*a^3*x^3 + 21*a^2*x^2 + 16*a*x + 6)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^3, 1/24*(21*a^3*sqrt(c)*x^3*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (32*a^3*x^3 + 21*a^2*x^2 + 16*a*x + 6)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^3]","A",0
755,1,233,0,0.893363," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^5,x, algorithm=""fricas"")","\left[\frac{15 \, a^{4} \sqrt{-c} x^{4} \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - 2 \, {\left(24 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 10 \, a x + 4\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{40 \, x^{4}}, \frac{15 \, a^{4} \sqrt{c} x^{4} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - {\left(24 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 10 \, a x + 4\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{20 \, x^{4}}\right]"," ",0,"[1/40*(15*a^4*sqrt(-c)*x^4*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(24*a^4*x^4 + 15*a^3*x^3 + 12*a^2*x^2 + 10*a*x + 4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^4, 1/20*(15*a^4*sqrt(c)*x^4*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (24*a^4*x^4 + 15*a^3*x^3 + 12*a^2*x^2 + 10*a*x + 4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^4]","A",0
756,1,419,0,0.972120," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x - {\left(a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) + {\left(a^{5} x^{5} + 4 \, a^{4} x^{4} + 8 \, a^{3} x^{3} + 16 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{6} x^{2} - a^{4}\right)}}, \frac{16 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right) + {\left(a^{5} x^{5} + 4 \, a^{4} x^{4} + 8 \, a^{3} x^{3} + 16 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{6} x^{2} - a^{4}\right)}}\right]"," ",0,"[1/4*(8*(a^2*x^2 - 1)*sqrt(-c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x - (a^5*x^5 - 4*a^4*x^4 + 6*a^3*x^3 - 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) + (a^5*x^5 + 4*a^4*x^4 + 8*a^3*x^3 + 16*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^2 - a^4), 1/4*(16*(a^2*x^2 - 1)*sqrt(c)*arctan((a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 - 2*a^2*c*x^2 - a*c*x + 2*c)) + (a^5*x^5 + 4*a^4*x^4 + 8*a^3*x^3 + 16*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^2 - a^4)]","A",0
757,1,405,0,1.936157," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x - {\left(a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) + {\left(2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 24 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left(a^{5} x^{2} - a^{3}\right)}}, \frac{24 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right) + {\left(2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 24 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left(a^{5} x^{2} - a^{3}\right)}}\right]"," ",0,"[1/6*(12*(a^2*x^2 - 1)*sqrt(-c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x - (a^5*x^5 - 4*a^4*x^4 + 6*a^3*x^3 - 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) + (2*a^4*x^4 + 9*a^3*x^3 + 24*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*x^2 - a^3), 1/6*(24*(a^2*x^2 - 1)*sqrt(c)*arctan((a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 - 2*a^2*c*x^2 - a*c*x + 2*c)) + (2*a^4*x^4 + 9*a^3*x^3 + 24*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*x^2 - a^3)]","A",0
758,1,387,0,1.423208," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x - {\left(a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) + {\left(a^{3} x^{3} + 6 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{2} - a^{2}\right)}}, \frac{8 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} - 2 \, a^{2} c x^{2} - a c x + 2 \, c}\right) + {\left(a^{3} x^{3} + 6 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{2} - a^{2}\right)}}\right]"," ",0,"[1/2*(4*(a^2*x^2 - 1)*sqrt(-c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x - (a^5*x^5 - 4*a^4*x^4 + 6*a^3*x^3 - 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) + (a^3*x^3 + 6*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^2 - a^2), 1/2*(8*(a^2*x^2 - 1)*sqrt(c)*arctan((a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 - 2*a^2*c*x^2 - a*c*x + 2*c)) + (a^3*x^3 + 6*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^2 - a^2)]","A",0
759,0,0,0,0.478155," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^2 - 2*a*x + 1), x)","F",0
760,0,0,0,1.465812," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{3} - 2 \, a x^{2} + x}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^3 - 2*a*x^2 + x), x)","F",0
761,1,492,0,1.182004," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{3} x^{3} - a x\right)} \sqrt{-c} \log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x - {\left(4 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - {\left(4 \, a^{4} - 6 \, a^{3} + 4 \, a^{2} - a\right)} x^{5} + 4 \, a^{2} x^{2} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right) - \sqrt{-a^{2} x^{2} + 1} {\left({\left(6 \, a + 1\right)} x^{2} - 6 \, a x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{2} x^{3} - x\right)}}, \frac{8 \, {\left(a^{3} x^{3} - a x\right)} \sqrt{c} \arctan\left(\frac{{\left(2 \, a^{2} x^{2} - {\left(2 \, a^{3} - 2 \, a^{2} + a\right)} x^{3} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right) - \sqrt{-a^{2} x^{2} + 1} {\left({\left(6 \, a + 1\right)} x^{2} - 6 \, a x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{2} x^{3} - x\right)}}\right]"," ",0,"[1/2*(4*(a^3*x^3 - a*x)*sqrt(-c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x - (4*a^4*x^4 - 6*a^3*x^3 - (4*a^4 - 6*a^3 + 4*a^2 - a)*x^5 + 4*a^2*x^2 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) - sqrt(-a^2*x^2 + 1)*((6*a + 1)*x^2 - 6*a*x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^3 - x), 1/2*(8*(a^3*x^3 - a*x)*sqrt(c)*arctan((2*a^2*x^2 - (2*a^3 - 2*a^2 + a)*x^3 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) - sqrt(-a^2*x^2 + 1)*((6*a + 1)*x^2 - 6*a*x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^3 - x)]","A",0
762,1,530,0,3.918880," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x^3,x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{-c} \log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x - {\left(4 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - {\left(4 \, a^{4} - 6 \, a^{3} + 4 \, a^{2} - a\right)} x^{5} + 4 \, a^{2} x^{2} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right) + {\left(24 \, a^{2} x^{2} - {\left(24 \, a^{2} + 9 \, a + 2\right)} x^{3} + 9 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left(a^{2} x^{4} - x^{2}\right)}}, \frac{24 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{c} \arctan\left(\frac{{\left(2 \, a^{2} x^{2} - {\left(2 \, a^{3} - 2 \, a^{2} + a\right)} x^{3} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right) + {\left(24 \, a^{2} x^{2} - {\left(24 \, a^{2} + 9 \, a + 2\right)} x^{3} + 9 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left(a^{2} x^{4} - x^{2}\right)}}\right]"," ",0,"[1/6*(12*(a^4*x^4 - a^2*x^2)*sqrt(-c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x - (4*a^4*x^4 - 6*a^3*x^3 - (4*a^4 - 6*a^3 + 4*a^2 - a)*x^5 + 4*a^2*x^2 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) + (24*a^2*x^2 - (24*a^2 + 9*a + 2)*x^3 + 9*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^4 - x^2), 1/6*(24*(a^4*x^4 - a^2*x^2)*sqrt(c)*arctan((2*a^2*x^2 - (2*a^3 - 2*a^2 + a)*x^3 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) + (24*a^2*x^2 - (24*a^2 + 9*a + 2)*x^3 + 9*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^4 - x^2)]","A",0
763,1,556,0,0.965907," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x^4,x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{-c} \log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x - {\left(4 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - {\left(4 \, a^{4} - 6 \, a^{3} + 4 \, a^{2} - a\right)} x^{5} + 4 \, a^{2} x^{2} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right) + {\left(16 \, a^{3} x^{3} - {\left(16 \, a^{3} + 8 \, a^{2} + 4 \, a + 1\right)} x^{4} + 8 \, a^{2} x^{2} + 4 \, a x + 1\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{2} x^{5} - x^{3}\right)}}, \frac{16 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{c} \arctan\left(\frac{{\left(2 \, a^{2} x^{2} - {\left(2 \, a^{3} - 2 \, a^{2} + a\right)} x^{3} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right) + {\left(16 \, a^{3} x^{3} - {\left(16 \, a^{3} + 8 \, a^{2} + 4 \, a + 1\right)} x^{4} + 8 \, a^{2} x^{2} + 4 \, a x + 1\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{2} x^{5} - x^{3}\right)}}\right]"," ",0,"[1/4*(8*(a^5*x^5 - a^3*x^3)*sqrt(-c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x - (4*a^4*x^4 - 6*a^3*x^3 - (4*a^4 - 6*a^3 + 4*a^2 - a)*x^5 + 4*a^2*x^2 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) + (16*a^3*x^3 - (16*a^3 + 8*a^2 + 4*a + 1)*x^4 + 8*a^2*x^2 + 4*a*x + 1)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^5 - x^3), 1/4*(16*(a^5*x^5 - a^3*x^3)*sqrt(c)*arctan((2*a^2*x^2 - (2*a^3 - 2*a^2 + a)*x^3 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) + (16*a^3*x^3 - (16*a^3 + 8*a^2 + 4*a + 1)*x^4 + 8*a^2*x^2 + 4*a*x + 1)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^5 - x^3)]","A",0
764,1,582,0,0.939921," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x^5,x, algorithm=""fricas"")","\left[\frac{120 \, {\left(a^{6} x^{6} - a^{4} x^{4}\right)} \sqrt{-c} \log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x - {\left(4 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - {\left(4 \, a^{4} - 6 \, a^{3} + 4 \, a^{2} - a\right)} x^{5} + 4 \, a^{2} x^{2} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right) + {\left(240 \, a^{4} x^{4} + 120 \, a^{3} x^{3} - {\left(240 \, a^{4} + 120 \, a^{3} + 80 \, a^{2} + 45 \, a + 12\right)} x^{5} + 80 \, a^{2} x^{2} + 45 \, a x + 12\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{2} x^{6} - x^{4}\right)}}, \frac{240 \, {\left(a^{6} x^{6} - a^{4} x^{4}\right)} \sqrt{c} \arctan\left(\frac{{\left(2 \, a^{2} x^{2} - {\left(2 \, a^{3} - 2 \, a^{2} + a\right)} x^{3} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right) + {\left(240 \, a^{4} x^{4} + 120 \, a^{3} x^{3} - {\left(240 \, a^{4} + 120 \, a^{3} + 80 \, a^{2} + 45 \, a + 12\right)} x^{5} + 80 \, a^{2} x^{2} + 45 \, a x + 12\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{2} x^{6} - x^{4}\right)}}\right]"," ",0,"[1/60*(120*(a^6*x^6 - a^4*x^4)*sqrt(-c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x - (4*a^4*x^4 - 6*a^3*x^3 - (4*a^4 - 6*a^3 + 4*a^2 - a)*x^5 + 4*a^2*x^2 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) + (240*a^4*x^4 + 120*a^3*x^3 - (240*a^4 + 120*a^3 + 80*a^2 + 45*a + 12)*x^5 + 80*a^2*x^2 + 45*a*x + 12)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^6 - x^4), 1/60*(240*(a^6*x^6 - a^4*x^4)*sqrt(c)*arctan((2*a^2*x^2 - (2*a^3 - 2*a^2 + a)*x^3 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) + (240*a^4*x^4 + 120*a^3*x^3 - (240*a^4 + 120*a^3 + 80*a^2 + 45*a + 12)*x^5 + 80*a^2*x^2 + 45*a*x + 12)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^6 - x^4)]","A",0
765,1,78,0,0.831671," ","integrate(x^m*(c-c/a^2/x^2)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a m x^{2} - {\left(m + 1\right)} x\right)} x^{m} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{{\left(a^{2} m^{2} + a^{2} m\right)} x^{2} - m^{2} - m}"," ",0,"sqrt(-a^2*x^2 + 1)*(a*m*x^2 - (m + 1)*x)*x^m*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/((a^2*m^2 + a^2*m)*x^2 - m^2 - m)","A",0
766,1,58,0,2.437601," ","integrate(x^2*(c-c/a^2/x^2)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{{\left(2 \, a x^{4} - 3 \, x^{3}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/6*(2*a*x^4 - 3*x^3)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^2 - 1)","A",0
767,1,57,0,1.159914," ","integrate(x*(c-c/a^2/x^2)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} - 2 \, x^{2}\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/2*sqrt(-a^2*x^2 + 1)*(a*x^3 - 2*x^2)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^2 - 1)","A",0
768,1,321,0,0.846887," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) + 2 \, {\left(a^{2} x^{2} - a^{2} x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{3} x^{2} - a\right)}}, -\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - {\left(a^{2} x^{2} - a^{2} x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} x^{2} - a}\right]"," ",0,"[1/2*((a^2*x^2 - 1)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + 2*(a^2*x^2 - a^2*x)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*x^2 - a), -((a^2*x^2 - 1)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (a^2*x^2 - a^2*x)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*x^2 - a)]","B",0
769,1,295,0,1.107992," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left(a x^{5} - a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right) - 2 \, \sqrt{-a^{2} x^{2} + 1} {\left(x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{2} x^{2} - 1\right)}}, \frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + a x\right)} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - \sqrt{-a^{2} x^{2} + 1} {\left(x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{2} - 1}\right]"," ",0,"[1/2*((a^2*x^2 - 1)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - 2*sqrt(-a^2*x^2 + 1)*(x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^2 - 1), ((a^2*x^2 - 1)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - sqrt(-a^2*x^2 + 1)*(x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^2 - 1)]","B",0
770,1,63,0,1.047560," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a - 1\right)} x^{2} - 2 \, a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{2} x^{3} - x\right)}}"," ",0,"1/2*sqrt(-a^2*x^2 + 1)*((2*a - 1)*x^2 - 2*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^3 - x)","A",0
771,1,222,0,3.200876," ","integrate(x^3*(c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(6 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 32 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right)}{48 \, a^{4}}, -\frac{{\left(6 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 32 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right)}{24 \, a^{4}}\right]"," ",0,"[-1/48*(2*(6*a^4*x^4 - 16*a^3*x^3 + 21*a^2*x^2 - 32*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 21*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a^4, -1/24*((6*a^4*x^4 - 16*a^3*x^3 + 21*a^2*x^2 - 32*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 21*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)))/a^4]","A",0
772,1,204,0,2.049500," ","integrate(x^2*(c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 5 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt{c} \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right)}{6 \, a^{3}}, -\frac{{\left(a^{3} x^{3} - 3 \, a^{2} x^{2} + 5 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + 3 \, \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right)}{3 \, a^{3}}\right]"," ",0,"[-1/6*(2*(a^3*x^3 - 3*a^2*x^2 + 5*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 3*sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a^3, -1/3*((a^3*x^3 - 3*a^2*x^2 + 5*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + 3*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)))/a^3]","A",0
773,1,188,0,0.745359," ","integrate(x*(c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{2} x^{2} - 4 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right)}{4 \, a^{2}}, -\frac{{\left(a^{2} x^{2} - 4 \, a x\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right)}{2 \, a^{2}}\right]"," ",0,"[-1/4*(2*(a^2*x^2 - 4*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 3*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a^2, -1/2*((a^2*x^2 - 4*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 3*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)))/a^2]","A",0
774,1,270,0,0.741800," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + 4 \, \sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - \sqrt{-c} \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right)}{2 \, a}, -\frac{a x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - \sqrt{c} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) - \sqrt{c} \log\left(2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right)}{a}\right]"," ",0,"[-1/2*(2*a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + 4*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - sqrt(-c)*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2))/a, -(a*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - sqrt(c)*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a]","A",0
775,1,255,0,1.568078," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x,x, algorithm=""fricas"")","\left[\sqrt{-c} \arctan\left(\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + \sqrt{-c} \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}, -2 \, \sqrt{c} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + \frac{1}{2} \, \sqrt{c} \log\left(2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right) - \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}\right]"," ",0,"[sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + sqrt(-c)*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - sqrt((a^2*c*x^2 - c)/(a^2*x^2)), -2*sqrt(c)*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 1/2*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - sqrt((a^2*c*x^2 - c)/(a^2*x^2))]","A",0
776,1,176,0,0.513943," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^2,x, algorithm=""fricas"")","\left[\frac{3 \, a \sqrt{-c} x \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) + 2 \, {\left(4 \, a x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, x}, \frac{3 \, a \sqrt{c} x \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + {\left(4 \, a x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, x}\right]"," ",0,"[1/4*(3*a*sqrt(-c)*x*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) + 2*(4*a*x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x, 1/2*(3*a*sqrt(c)*x*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (4*a*x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x]","A",0
777,1,200,0,0.723196," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^3,x, algorithm=""fricas"")","\left[\frac{3 \, a^{2} \sqrt{-c} x^{2} \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - 2 \, {\left(5 \, a^{2} x^{2} - 3 \, a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, x^{2}}, -\frac{3 \, a^{2} \sqrt{c} x^{2} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + {\left(5 \, a^{2} x^{2} - 3 \, a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, x^{2}}\right]"," ",0,"[1/6*(3*a^2*sqrt(-c)*x^2*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(5*a^2*x^2 - 3*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^2, -1/3*(3*a^2*sqrt(c)*x^2*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (5*a^2*x^2 - 3*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^2]","A",0
778,1,216,0,1.576497," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm=""fricas"")","\left[\frac{21 \, a^{3} \sqrt{-c} x^{3} \log\left(-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) + 2 \, {\left(32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, x^{3}}, \frac{21 \, a^{3} \sqrt{c} x^{3} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + {\left(32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, x^{3}}\right]"," ",0,"[1/48*(21*a^3*sqrt(-c)*x^3*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) + 2*(32*a^3*x^3 - 21*a^2*x^2 + 16*a*x - 6)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^3, 1/24*(21*a^3*sqrt(c)*x^3*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (32*a^3*x^3 - 21*a^2*x^2 + 16*a*x - 6)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^3]","A",0
779,1,232,0,0.568353," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x, algorithm=""fricas"")","\left[\frac{15 \, a^{4} \sqrt{-c} x^{4} \log\left(-\frac{a^{2} c x^{2} + 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right) - 2 \, {\left(24 \, a^{4} x^{4} - 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 10 \, a x + 4\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{40 \, x^{4}}, -\frac{15 \, a^{4} \sqrt{c} x^{4} \arctan\left(\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right) + {\left(24 \, a^{4} x^{4} - 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 10 \, a x + 4\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{20 \, x^{4}}\right]"," ",0,"[1/40*(15*a^4*sqrt(-c)*x^4*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(24*a^4*x^4 - 15*a^3*x^3 + 12*a^2*x^2 - 10*a*x + 4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^4, -1/20*(15*a^4*sqrt(c)*x^4*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (24*a^4*x^4 - 15*a^3*x^3 + 12*a^2*x^2 - 10*a*x + 4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^4]","A",0
780,1,419,0,0.669990," ","integrate(x^3*(c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left(a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right) - {\left(a^{5} x^{5} - 4 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 16 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{6} x^{2} - a^{4}\right)}}, -\frac{16 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right) + {\left(a^{5} x^{5} - 4 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 16 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{6} x^{2} - a^{4}\right)}}\right]"," ",0,"[1/4*(8*(a^2*x^2 - 1)*sqrt(-c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x + (a^5*x^5 + 4*a^4*x^4 + 6*a^3*x^3 + 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)) - (a^5*x^5 - 4*a^4*x^4 + 8*a^3*x^3 - 16*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^2 - a^4), -1/4*(16*(a^2*x^2 - 1)*sqrt(c)*arctan((a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 + 2*a^2*c*x^2 - a*c*x - 2*c)) + (a^5*x^5 - 4*a^4*x^4 + 8*a^3*x^3 - 16*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^2 - a^4)]","A",0
781,1,407,0,0.687731," ","integrate(x^2*(c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left(a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right) - {\left(2 \, a^{4} x^{4} - 9 \, a^{3} x^{3} + 24 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left(a^{5} x^{2} - a^{3}\right)}}, \frac{24 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right) - {\left(2 \, a^{4} x^{4} - 9 \, a^{3} x^{3} + 24 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left(a^{5} x^{2} - a^{3}\right)}}\right]"," ",0,"[1/6*(12*(a^2*x^2 - 1)*sqrt(-c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x - (a^5*x^5 + 4*a^4*x^4 + 6*a^3*x^3 + 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)) - (2*a^4*x^4 - 9*a^3*x^3 + 24*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*x^2 - a^3), 1/6*(24*(a^2*x^2 - 1)*sqrt(c)*arctan((a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 + 2*a^2*c*x^2 - a*c*x - 2*c)) - (2*a^4*x^4 - 9*a^3*x^3 + 24*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*x^2 - a^3)]","A",0
782,1,387,0,0.604281," ","integrate(x*(c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left(a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right) - {\left(a^{3} x^{3} - 6 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{2} - a^{2}\right)}}, -\frac{8 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \arctan\left(\frac{{\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right) + {\left(a^{3} x^{3} - 6 \, a^{2} x^{2}\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{4} x^{2} - a^{2}\right)}}\right]"," ",0,"[1/2*(4*(a^2*x^2 - 1)*sqrt(-c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x + (a^5*x^5 + 4*a^4*x^4 + 6*a^3*x^3 + 4*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)) - (a^3*x^3 - 6*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^2 - a^2), -1/2*(8*(a^2*x^2 - 1)*sqrt(c)*arctan((a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 + 2*a^2*c*x^2 - a*c*x - 2*c)) + (a^3*x^3 - 6*a^2*x^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^2 - a^2)]","A",0
783,0,0,0,0.698516," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{2} + 2 \, a x + 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(a*x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^2 + 2*a*x + 1), x)","F",0
784,0,0,0,0.618316," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{3} + 2 \, a x^{2} + x}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(a*x - 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^3 + 2*a*x^2 + x), x)","F",0
785,1,485,0,0.618634," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{3} x^{3} - a x\right)} \sqrt{-c} \log\left(\frac{4 \, a^{5} c x^{5} + {\left(2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right)} c x^{6} + {\left(4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right)} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x - {\left(4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - {\left(4 \, a^{4} + 6 \, a^{3} + 4 \, a^{2} + a\right)} x^{5} + 4 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right) + \sqrt{-a^{2} x^{2} + 1} {\left({\left(6 \, a - 1\right)} x^{2} - 6 \, a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{2} x^{3} - x\right)}}, -\frac{8 \, {\left(a^{3} x^{3} - a x\right)} \sqrt{c} \arctan\left(-\frac{{\left(2 \, a^{2} x^{2} + {\left(2 \, a^{3} + 2 \, a^{2} + a\right)} x^{3} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + a^{2}\right)} c x^{4} + {\left(a^{2} + 2 \, a + 1\right)} c x^{2} - 2 \, a c x - c}\right) - \sqrt{-a^{2} x^{2} + 1} {\left({\left(6 \, a - 1\right)} x^{2} - 6 \, a x + 1\right)} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left(a^{2} x^{3} - x\right)}}\right]"," ",0,"[1/2*(4*(a^3*x^3 - a*x)*sqrt(-c)*log((4*a^5*c*x^5 + (2*a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*c*x^6 + (4*a^4 - 4*a^3 - 6*a^2 - 4*a - 1)*c*x^4 - 5*a^2*c*x^2 - 4*a*c*x - (4*a^4*x^4 + 6*a^3*x^3 - (4*a^4 + 6*a^3 + 4*a^2 + a)*x^5 + 4*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^4*x^6 + 2*a^3*x^5 - 2*a*x^3 - x^2)) + sqrt(-a^2*x^2 + 1)*((6*a - 1)*x^2 - 6*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^3 - x), -1/2*(8*(a^3*x^3 - a*x)*sqrt(c)*arctan(-(2*a^2*x^2 + (2*a^3 + 2*a^2 + a)*x^3 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c*x^3 - (2*a^3 + a^2)*c*x^4 + (a^2 + 2*a + 1)*c*x^2 - 2*a*c*x - c)) - sqrt(-a^2*x^2 + 1)*((6*a - 1)*x^2 - 6*a*x + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^3 - x)]","A",0
786,1,523,0,0.791368," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^3,x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{-c} \log\left(\frac{4 \, a^{5} c x^{5} + {\left(2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right)} c x^{6} + {\left(4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right)} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - {\left(4 \, a^{4} + 6 \, a^{3} + 4 \, a^{2} + a\right)} x^{5} + 4 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right) + {\left(24 \, a^{2} x^{2} - {\left(24 \, a^{2} - 9 \, a + 2\right)} x^{3} - 9 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left(a^{2} x^{4} - x^{2}\right)}}, \frac{24 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{c} \arctan\left(-\frac{{\left(2 \, a^{2} x^{2} + {\left(2 \, a^{3} + 2 \, a^{2} + a\right)} x^{3} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + a^{2}\right)} c x^{4} + {\left(a^{2} + 2 \, a + 1\right)} c x^{2} - 2 \, a c x - c}\right) + {\left(24 \, a^{2} x^{2} - {\left(24 \, a^{2} - 9 \, a + 2\right)} x^{3} - 9 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left(a^{2} x^{4} - x^{2}\right)}}\right]"," ",0,"[1/6*(12*(a^4*x^4 - a^2*x^2)*sqrt(-c)*log((4*a^5*c*x^5 + (2*a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*c*x^6 + (4*a^4 - 4*a^3 - 6*a^2 - 4*a - 1)*c*x^4 - 5*a^2*c*x^2 - 4*a*c*x + (4*a^4*x^4 + 6*a^3*x^3 - (4*a^4 + 6*a^3 + 4*a^2 + a)*x^5 + 4*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^4*x^6 + 2*a^3*x^5 - 2*a*x^3 - x^2)) + (24*a^2*x^2 - (24*a^2 - 9*a + 2)*x^3 - 9*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^4 - x^2), 1/6*(24*(a^4*x^4 - a^2*x^2)*sqrt(c)*arctan(-(2*a^2*x^2 + (2*a^3 + 2*a^2 + a)*x^3 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c*x^3 - (2*a^3 + a^2)*c*x^4 + (a^2 + 2*a + 1)*c*x^2 - 2*a*c*x - c)) + (24*a^2*x^2 - (24*a^2 - 9*a + 2)*x^3 - 9*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^4 - x^2)]","A",0
787,1,551,0,0.681941," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^4,x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{-c} \log\left(\frac{4 \, a^{5} c x^{5} + {\left(2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right)} c x^{6} + {\left(4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right)} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x - {\left(4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - {\left(4 \, a^{4} + 6 \, a^{3} + 4 \, a^{2} + a\right)} x^{5} + 4 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right) - {\left(16 \, a^{3} x^{3} - {\left(16 \, a^{3} - 8 \, a^{2} + 4 \, a - 1\right)} x^{4} - 8 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{2} x^{5} - x^{3}\right)}}, -\frac{16 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{c} \arctan\left(-\frac{{\left(2 \, a^{2} x^{2} + {\left(2 \, a^{3} + 2 \, a^{2} + a\right)} x^{3} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + a^{2}\right)} c x^{4} + {\left(a^{2} + 2 \, a + 1\right)} c x^{2} - 2 \, a c x - c}\right) + {\left(16 \, a^{3} x^{3} - {\left(16 \, a^{3} - 8 \, a^{2} + 4 \, a - 1\right)} x^{4} - 8 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left(a^{2} x^{5} - x^{3}\right)}}\right]"," ",0,"[1/4*(8*(a^5*x^5 - a^3*x^3)*sqrt(-c)*log((4*a^5*c*x^5 + (2*a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*c*x^6 + (4*a^4 - 4*a^3 - 6*a^2 - 4*a - 1)*c*x^4 - 5*a^2*c*x^2 - 4*a*c*x - (4*a^4*x^4 + 6*a^3*x^3 - (4*a^4 + 6*a^3 + 4*a^2 + a)*x^5 + 4*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^4*x^6 + 2*a^3*x^5 - 2*a*x^3 - x^2)) - (16*a^3*x^3 - (16*a^3 - 8*a^2 + 4*a - 1)*x^4 - 8*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^5 - x^3), -1/4*(16*(a^5*x^5 - a^3*x^3)*sqrt(c)*arctan(-(2*a^2*x^2 + (2*a^3 + 2*a^2 + a)*x^3 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c*x^3 - (2*a^3 + a^2)*c*x^4 + (a^2 + 2*a + 1)*c*x^2 - 2*a*c*x - c)) + (16*a^3*x^3 - (16*a^3 - 8*a^2 + 4*a - 1)*x^4 - 8*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^5 - x^3)]","A",0
788,1,575,0,1.045499," ","integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm=""fricas"")","\left[\frac{120 \, {\left(a^{6} x^{6} - a^{4} x^{4}\right)} \sqrt{-c} \log\left(\frac{4 \, a^{5} c x^{5} + {\left(2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right)} c x^{6} + {\left(4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right)} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - {\left(4 \, a^{4} + 6 \, a^{3} + 4 \, a^{2} + a\right)} x^{5} + 4 \, a^{2} x^{2} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right) + {\left(240 \, a^{4} x^{4} - 120 \, a^{3} x^{3} - {\left(240 \, a^{4} - 120 \, a^{3} + 80 \, a^{2} - 45 \, a + 12\right)} x^{5} + 80 \, a^{2} x^{2} - 45 \, a x + 12\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{2} x^{6} - x^{4}\right)}}, \frac{240 \, {\left(a^{6} x^{6} - a^{4} x^{4}\right)} \sqrt{c} \arctan\left(-\frac{{\left(2 \, a^{2} x^{2} + {\left(2 \, a^{3} + 2 \, a^{2} + a\right)} x^{3} + a x\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + a^{2}\right)} c x^{4} + {\left(a^{2} + 2 \, a + 1\right)} c x^{2} - 2 \, a c x - c}\right) + {\left(240 \, a^{4} x^{4} - 120 \, a^{3} x^{3} - {\left(240 \, a^{4} - 120 \, a^{3} + 80 \, a^{2} - 45 \, a + 12\right)} x^{5} + 80 \, a^{2} x^{2} - 45 \, a x + 12\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left(a^{2} x^{6} - x^{4}\right)}}\right]"," ",0,"[1/60*(120*(a^6*x^6 - a^4*x^4)*sqrt(-c)*log((4*a^5*c*x^5 + (2*a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*c*x^6 + (4*a^4 - 4*a^3 - 6*a^2 - 4*a - 1)*c*x^4 - 5*a^2*c*x^2 - 4*a*c*x + (4*a^4*x^4 + 6*a^3*x^3 - (4*a^4 + 6*a^3 + 4*a^2 + a)*x^5 + 4*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^4*x^6 + 2*a^3*x^5 - 2*a*x^3 - x^2)) + (240*a^4*x^4 - 120*a^3*x^3 - (240*a^4 - 120*a^3 + 80*a^2 - 45*a + 12)*x^5 + 80*a^2*x^2 - 45*a*x + 12)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^6 - x^4), 1/60*(240*(a^6*x^6 - a^4*x^4)*sqrt(c)*arctan(-(2*a^2*x^2 + (2*a^3 + 2*a^2 + a)*x^3 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c*x^3 - (2*a^3 + a^2)*c*x^4 + (a^2 + 2*a + 1)*c*x^2 - 2*a*c*x - c)) + (240*a^4*x^4 - 120*a^3*x^3 - (240*a^4 - 120*a^3 + 80*a^2 - 45*a + 12)*x^5 + 80*a^2*x^2 - 45*a*x + 12)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^6 - x^4)]","A",0
789,0,0,0,0.708830," ","integrate((c-c/a^2/x^2)^p/exp(2*p*arctanh(a*x)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right)^{p}}{\left(\frac{a x + 1}{a x - 1}\right)^{p}}, x\right)"," ",0,"integral(((a^2*c*x^2 - c)/(a^2*x^2))^p/((a*x + 1)/(a*x - 1))^p, x)","F",0
790,0,0,0,0.756675," ","integrate(exp(2*p*arctanh(a*x))*(c-c/a^2/x^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\frac{a x + 1}{a x - 1}\right)^{p} \left(\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right)^{p}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^p*((a^2*c*x^2 - c)/(a^2*x^2))^p, x)","F",0
791,0,0,0,0.796167," ","integrate(exp(n*arctanh(a*x))*(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} x^{4}}, x\right)"," ",0,"integral((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*x^4), x)","F",0
792,0,0,0,1.373983," ","integrate(exp(n*arctanh(a*x))*(c-c/a^2/x^2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} c x^{2} - c\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} x^{2}}, x\right)"," ",0,"integral((a^2*c*x^2 - c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*x^2), x)","F",0
793,0,0,0,0.602718," ","integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{a^{2} x^{2} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(a^2*x^2*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)","F",0
794,0,0,0,0.985198," ","integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{a^{4} x^{4} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right)"," ",0,"integral(a^4*x^4*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)","F",0
795,0,0,0,1.104589," ","integrate(exp(n*arctanh(a*x))*(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} c x^{2} - c\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{2}}, x\right)"," ",0,"integral((a^2*c*x^2 - c)*((a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*x^2), x)","F",0
796,0,0,0,0.574608," ","integrate(exp(n*arctanh(a*x))*(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)), x)","F",0
797,0,0,0,0.701156," ","integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{a^{2} x^{2} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(a^2*x^2*((a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c), x)","F",0
798,0,0,0,0.735405," ","integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{a^{4} x^{4} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right)"," ",0,"integral(a^4*x^4*((a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)","F",0
799,0,0,0,0.592566," ","integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{a^{6} x^{6} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, x\right)"," ",0,"integral(a^6*x^6*((a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)","F",0
800,0,0,0,0.526045," ","integrate(exp(n*arctanh(a*x))*(c-c/a^2/x^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n} \left(\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right)^{p}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n)*((a^2*c*x^2 - c)/(a^2*x^2))^p, x)","F",0
801,0,0,0,0.705697," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} x^{2} + 2 \, a x + 1\right)} \left(\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right)^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral((a^2*x^2 + 2*a*x + 1)*((a^2*c*x^2 - c)/(a^2*x^2))^p/(a^2*x^2 - 2*a*x + 1), x)","F",0
802,0,0,0,1.525100," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} \left(\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right)^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*((a^2*c*x^2 - c)/(a^2*x^2))^p/(a^2*x^2 - 2*a*x + 1), x)","F",0
803,0,0,0,0.646194," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x + 1\right)} \left(\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right)^{p}}{a x - 1}, x\right)"," ",0,"integral(-(a*x + 1)*((a^2*c*x^2 - c)/(a^2*x^2))^p/(a*x - 1), x)","F",0
804,0,0,0,0.991199," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} \left(\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right)^{p}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*((a^2*c*x^2 - c)/(a^2*x^2))^p/(a*x - 1), x)","F",0
805,0,0,0,0.497193," ","integrate((c-c/a^2/x^2)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} \left(\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right)^{p}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*((a^2*c*x^2 - c)/(a^2*x^2))^p/(a*x + 1), x)","F",0
806,0,0,0,1.066633," ","integrate((c-c/a^2/x^2)^p/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x - 1\right)} \left(\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right)^{p}}{a x + 1}, x\right)"," ",0,"integral(-(a*x - 1)*((a^2*c*x^2 - c)/(a^2*x^2))^p/(a*x + 1), x)","F",0
807,0,0,0,1.431595," ","integrate((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)} \left(\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right)^{p}}{a^{2} x^{2} + 2 \, a x + 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(a*x - 1)*((a^2*c*x^2 - c)/(a^2*x^2))^p/(a^2*x^2 + 2*a*x + 1), x)","F",0
808,0,0,0,0.648561," ","integrate((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-x^{2} + 1} \sqrt{x + 1} x \sin\left(x\right)}{x - 1}, x\right)"," ",0,"integral(-sqrt(-x^2 + 1)*sqrt(x + 1)*x*sin(x)/(x - 1), x)","F",0
809,0,0,0,2.479659," ","integrate((1+x)^(3/2)/(-x^2+1)^(1/2)*sin(x),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-x^{2} + 1} \sqrt{x + 1} \sin\left(x\right)}{x - 1}, x\right)"," ",0,"integral(-sqrt(-x^2 + 1)*sqrt(x + 1)*sin(x)/(x - 1), x)","F",0
810,0,0,0,0.454170," ","integrate((1+x)/(-x^2+1)^(1/2)*(1-x)^(1/2)*x*sin(x),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-x^{2} + 1} x \sqrt{-x + 1} \sin\left(x\right)}{x - 1}, x\right)"," ",0,"integral(-sqrt(-x^2 + 1)*x*sqrt(-x + 1)*sin(x)/(x - 1), x)","F",0
811,0,0,0,0.673600," ","integrate((1+x)/(-x^2+1)^(1/2)*(1-x)^(1/2)*sin(x),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-x^{2} + 1} \sqrt{-x + 1} \sin\left(x\right)}{x - 1}, x\right)"," ",0,"integral(-sqrt(-x^2 + 1)*sqrt(-x + 1)*sin(x)/(x - 1), x)","F",0
812,0,0,0,0.731129," ","integrate((1+x)^(5/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(x^{2} + x\right)} \sqrt{-x^{2} + 1} \sqrt{x + 1} \sin\left(x\right)}{x - 1}, x\right)"," ",0,"integral(-(x^2 + x)*sqrt(-x^2 + 1)*sqrt(x + 1)*sin(x)/(x - 1), x)","F",0
813,0,0,0,0.540835," ","integrate((1+x)^(5/2)/(-x^2+1)^(1/2)*sin(x),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-x^{2} + 1} {\left(x + 1\right)}^{\frac{3}{2}} \sin\left(x\right)}{x - 1}, x\right)"," ",0,"integral(-sqrt(-x^2 + 1)*(x + 1)^(3/2)*sin(x)/(x - 1), x)","F",0
814,0,0,0,0.623849," ","integrate((1+x)/(-x^2+1)^(1/2)*(1-x)^(3/2)*x*sin(x),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-x^{2} + 1} x \sqrt{-x + 1} \sin\left(x\right), x\right)"," ",0,"integral(sqrt(-x^2 + 1)*x*sqrt(-x + 1)*sin(x), x)","F",0
815,0,0,0,0.752533," ","integrate((1+x)/(-x^2+1)^(1/2)*(1-x)^(3/2)*sin(x),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-x^{2} + 1} \sqrt{-x + 1} \sin\left(x\right), x\right)"," ",0,"integral(sqrt(-x^2 + 1)*sqrt(-x + 1)*sin(x), x)","F",0
816,0,0,0,0.721717," ","integrate((1+x)^(1/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-x^{2} + 1} \sqrt{x + 1} x \sin\left(x\right)}{x^{2} - 1}, x\right)"," ",0,"integral(-sqrt(-x^2 + 1)*sqrt(x + 1)*x*sin(x)/(x^2 - 1), x)","F",0
817,0,0,0,1.997757," ","integrate((1+x)^(1/2)/(-x^2+1)^(1/2)*sin(x),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-x^{2} + 1} \sqrt{x + 1} \sin\left(x\right)}{x^{2} - 1}, x\right)"," ",0,"integral(-sqrt(-x^2 + 1)*sqrt(x + 1)*sin(x)/(x^2 - 1), x)","F",0
818,1,144,0,0.595206," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x^3,x, algorithm=""fricas"")","\frac{3 \, {\left(8 \, a^{3} - 12 \, a^{2} + 12 \, a - 3\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) - {\left(6 \, b^{3} x^{3} - 2 \, {\left(3 \, a - 4\right)} b^{2} x^{2} - 6 \, a^{3} + {\left(6 \, a^{2} - 20 \, a + 9\right)} b x + 44 \, a^{2} - 39 \, a + 16\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{24 \, b^{4}}"," ",0,"1/24*(3*(8*a^3 - 12*a^2 + 12*a - 3)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - (6*b^3*x^3 - 2*(3*a - 4)*b^2*x^2 - 6*a^3 + (6*a^2 - 20*a + 9)*b*x + 44*a^2 - 39*a + 16)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/b^4","A",0
819,1,116,0,1.162366," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x^2,x, algorithm=""fricas"")","-\frac{3 \, {\left(2 \, a^{2} - 2 \, a + 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + {\left(2 \, b^{2} x^{2} - {\left(2 \, a - 3\right)} b x + 2 \, a^{2} - 9 \, a + 4\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, b^{3}}"," ",0,"-1/6*(3*(2*a^2 - 2*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (2*b^2*x^2 - (2*a - 3)*b*x + 2*a^2 - 9*a + 4)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/b^3","A",0
820,1,92,0,0.477905," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x,x, algorithm=""fricas"")","\frac{{\left(2 \, a - 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x - a + 2\right)}}{2 \, b^{2}}"," ",0,"1/2*((2*a - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x - a + 2))/b^2","A",0
821,1,76,0,0.653776," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} + \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right)}{b}"," ",0,"-(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1) + arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)))/b","B",0
822,1,306,0,0.697644," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{1}{2} \, \sqrt{-\frac{a + 1}{a - 1}} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 4 \, a^{2} - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{3} + {\left(a^{2} - a\right)} b x - a^{2} - a + 1\right)} \sqrt{-\frac{a + 1}{a - 1}} + 2}{x^{2}}\right) - \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right), \sqrt{\frac{a + 1}{a - 1}} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{\frac{a + 1}{a - 1}}}{{\left(a + 1\right)} b^{2} x^{2} + a^{3} + 2 \, {\left(a^{2} + a\right)} b x + a^{2} - a - 1}\right) - \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right)\right]"," ",0,"[1/2*sqrt(-(a + 1)/(a - 1))*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2 - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^2 - a)*b*x - a^2 - a + 1)*sqrt(-(a + 1)/(a - 1)) + 2)/x^2) - arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)), sqrt((a + 1)/(a - 1))*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt((a + 1)/(a - 1))/((a + 1)*b^2*x^2 + a^3 + 2*(a^2 + a)*b*x + a^2 - a - 1)) - arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1))]","B",0
823,1,282,0,1.120326," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^2,x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + 1} b x \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{2} - 1\right)}}{2 \, {\left(a^{3} - a^{2} - a + 1\right)} x}, -\frac{\sqrt{a^{2} - 1} b x \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{2} - 1\right)}}{{\left(a^{3} - a^{2} - a + 1\right)} x}\right]"," ",0,"[-1/2*(sqrt(-a^2 + 1)*b*x*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1))/((a^3 - a^2 - a + 1)*x), -(sqrt(a^2 - 1)*b*x*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1))/((a^3 - a^2 - a + 1)*x)]","A",0
824,1,359,0,0.722114," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^3,x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + 1} {\left(2 \, a + 1\right)} b^{2} x^{2} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) - 2 \, {\left(a^{4} - {\left(a^{3} + 2 \, a^{2} - a - 2\right)} b x - 2 \, a^{2} + 1\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left(a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right)} x^{2}}, \frac{\sqrt{a^{2} - 1} {\left(2 \, a + 1\right)} b^{2} x^{2} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) + {\left(a^{4} - {\left(a^{3} + 2 \, a^{2} - a - 2\right)} b x - 2 \, a^{2} + 1\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left(a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right)} x^{2}}\right]"," ",0,"[-1/4*(sqrt(-a^2 + 1)*(2*a + 1)*b^2*x^2*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) - 2*(a^4 - (a^3 + 2*a^2 - a - 2)*b*x - 2*a^2 + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*x^2), 1/2*(sqrt(a^2 - 1)*(2*a + 1)*b^2*x^2*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + (a^4 - (a^3 + 2*a^2 - a - 2)*b*x - 2*a^2 + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*x^2)]","A",0
825,1,488,0,1.760491," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^4,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(2 \, a^{2} + 2 \, a + 1\right)} \sqrt{-a^{2} + 1} b^{3} x^{3} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) - 2 \, {\left(2 \, a^{6} + {\left(2 \, a^{4} + 9 \, a^{3} + 2 \, a^{2} - 9 \, a - 4\right)} b^{2} x^{2} - 6 \, a^{4} - {\left(2 \, a^{5} + 3 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} + 2 \, a + 3\right)} b x + 6 \, a^{2} - 2\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{12 \, {\left(a^{7} - a^{6} - 3 \, a^{5} + 3 \, a^{4} + 3 \, a^{3} - 3 \, a^{2} - a + 1\right)} x^{3}}, -\frac{3 \, {\left(2 \, a^{2} + 2 \, a + 1\right)} \sqrt{a^{2} - 1} b^{3} x^{3} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) - {\left(2 \, a^{6} + {\left(2 \, a^{4} + 9 \, a^{3} + 2 \, a^{2} - 9 \, a - 4\right)} b^{2} x^{2} - 6 \, a^{4} - {\left(2 \, a^{5} + 3 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} + 2 \, a + 3\right)} b x + 6 \, a^{2} - 2\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left(a^{7} - a^{6} - 3 \, a^{5} + 3 \, a^{4} + 3 \, a^{3} - 3 \, a^{2} - a + 1\right)} x^{3}}\right]"," ",0,"[-1/12*(3*(2*a^2 + 2*a + 1)*sqrt(-a^2 + 1)*b^3*x^3*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) - 2*(2*a^6 + (2*a^4 + 9*a^3 + 2*a^2 - 9*a - 4)*b^2*x^2 - 6*a^4 - (2*a^5 + 3*a^4 - 4*a^3 - 6*a^2 + 2*a + 3)*b*x + 6*a^2 - 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 - a^6 - 3*a^5 + 3*a^4 + 3*a^3 - 3*a^2 - a + 1)*x^3), -1/6*(3*(2*a^2 + 2*a + 1)*sqrt(a^2 - 1)*b^3*x^3*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) - (2*a^6 + (2*a^4 + 9*a^3 + 2*a^2 - 9*a - 4)*b^2*x^2 - 6*a^4 - (2*a^5 + 3*a^4 - 4*a^3 - 6*a^2 + 2*a + 3)*b*x + 6*a^2 - 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 - a^6 - 3*a^5 + 3*a^4 + 3*a^3 - 3*a^2 - a + 1)*x^3)]","A",0
826,1,93,0,0.606091," ","integrate((b*x+a+1)^2/(1-(b*x+a)^2)*x^4,x, algorithm=""fricas"")","-\frac{6 \, b^{5} x^{5} + 15 \, b^{4} x^{4} - 20 \, {\left(a - 1\right)} b^{3} x^{3} + 30 \, {\left(a^{2} - 2 \, a + 1\right)} b^{2} x^{2} - 60 \, {\left(a^{3} - 3 \, a^{2} + 3 \, a - 1\right)} b x + 60 \, {\left(a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right)} \log\left(b x + a - 1\right)}{30 \, b^{5}}"," ",0,"-1/30*(6*b^5*x^5 + 15*b^4*x^4 - 20*(a - 1)*b^3*x^3 + 30*(a^2 - 2*a + 1)*b^2*x^2 - 60*(a^3 - 3*a^2 + 3*a - 1)*b*x + 60*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(b*x + a - 1))/b^5","A",0
827,1,67,0,0.704052," ","integrate((b*x+a+1)^2/(1-(b*x+a)^2)*x^3,x, algorithm=""fricas"")","-\frac{3 \, b^{4} x^{4} + 8 \, b^{3} x^{3} - 12 \, {\left(a - 1\right)} b^{2} x^{2} + 24 \, {\left(a^{2} - 2 \, a + 1\right)} b x - 24 \, {\left(a^{3} - 3 \, a^{2} + 3 \, a - 1\right)} \log\left(b x + a - 1\right)}{12 \, b^{4}}"," ",0,"-1/12*(3*b^4*x^4 + 8*b^3*x^3 - 12*(a - 1)*b^2*x^2 + 24*(a^2 - 2*a + 1)*b*x - 24*(a^3 - 3*a^2 + 3*a - 1)*log(b*x + a - 1))/b^4","A",0
828,1,45,0,0.967645," ","integrate((b*x+a+1)^2/(1-(b*x+a)^2)*x^2,x, algorithm=""fricas"")","-\frac{b^{3} x^{3} + 3 \, b^{2} x^{2} - 6 \, {\left(a - 1\right)} b x + 6 \, {\left(a^{2} - 2 \, a + 1\right)} \log\left(b x + a - 1\right)}{3 \, b^{3}}"," ",0,"-1/3*(b^3*x^3 + 3*b^2*x^2 - 6*(a - 1)*b*x + 6*(a^2 - 2*a + 1)*log(b*x + a - 1))/b^3","A",0
829,1,29,0,0.565897," ","integrate((b*x+a+1)^2/(1-(b*x+a)^2)*x,x, algorithm=""fricas"")","-\frac{b^{2} x^{2} + 4 \, b x - 4 \, {\left(a - 1\right)} \log\left(b x + a - 1\right)}{2 \, b^{2}}"," ",0,"-1/2*(b^2*x^2 + 4*b*x - 4*(a - 1)*log(b*x + a - 1))/b^2","A",0
830,1,18,0,1.116026," ","integrate((b*x+a+1)^2/(1-(b*x+a)^2),x, algorithm=""fricas"")","-\frac{b x + 2 \, \log\left(b x + a - 1\right)}{b}"," ",0,"-(b*x + 2*log(b*x + a - 1))/b","A",0
831,1,23,0,0.458954," ","integrate((b*x+a+1)^2/(1-(b*x+a)^2)/x,x, algorithm=""fricas"")","-\frac{{\left(a + 1\right)} \log\left(x\right) - 2 \, \log\left(b x + a - 1\right)}{a - 1}"," ",0,"-((a + 1)*log(x) - 2*log(b*x + a - 1))/(a - 1)","A",0
832,1,39,0,1.176064," ","integrate((b*x+a+1)^2/(1-(b*x+a)^2)/x^2,x, algorithm=""fricas"")","-\frac{2 \, b x \log\left(b x + a - 1\right) - 2 \, b x \log\left(x\right) - a^{2} + 1}{{\left(a^{2} - 2 \, a + 1\right)} x}"," ",0,"-(2*b*x*log(b*x + a - 1) - 2*b*x*log(x) - a^2 + 1)/((a^2 - 2*a + 1)*x)","A",0
833,1,65,0,1.129689," ","integrate((b*x+a+1)^2/(1-(b*x+a)^2)/x^3,x, algorithm=""fricas"")","\frac{4 \, b^{2} x^{2} \log\left(b x + a - 1\right) - 4 \, b^{2} x^{2} \log\left(x\right) + a^{3} - 4 \, {\left(a - 1\right)} b x - a^{2} - a + 1}{2 \, {\left(a^{3} - 3 \, a^{2} + 3 \, a - 1\right)} x^{2}}"," ",0,"1/2*(4*b^2*x^2*log(b*x + a - 1) - 4*b^2*x^2*log(x) + a^3 - 4*(a - 1)*b*x - a^2 - a + 1)/((a^3 - 3*a^2 + 3*a - 1)*x^2)","A",0
834,1,88,0,1.409863," ","integrate((b*x+a+1)^2/(1-(b*x+a)^2)/x^4,x, algorithm=""fricas"")","-\frac{6 \, b^{3} x^{3} \log\left(b x + a - 1\right) - 6 \, b^{3} x^{3} \log\left(x\right) - 6 \, {\left(a - 1\right)} b^{2} x^{2} - a^{4} + 2 \, a^{3} + 3 \, {\left(a^{2} - 2 \, a + 1\right)} b x - 2 \, a + 1}{3 \, {\left(a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right)} x^{3}}"," ",0,"-1/3*(6*b^3*x^3*log(b*x + a - 1) - 6*b^3*x^3*log(x) - 6*(a - 1)*b^2*x^2 - a^4 + 2*a^3 + 3*(a^2 - 2*a + 1)*b*x - 2*a + 1)/((a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*x^3)","A",0
835,1,192,0,0.733881," ","integrate((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)*x^3,x, algorithm=""fricas"")","-\frac{3 \, {\left(8 \, a^{4} - 44 \, a^{3} + {\left(8 \, a^{3} - 36 \, a^{2} + 44 \, a - 17\right)} b x + 80 \, a^{2} - 61 \, a + 17\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) - {\left(2 \, b^{4} x^{4} + 6 \, b^{3} x^{3} - {\left(10 \, a - 11\right)} b^{2} x^{2} - 2 \, a^{4} + 78 \, a^{3} + {\left(22 \, a^{2} - 54 \, a + 29\right)} b x - 233 \, a^{2} + 237 \, a - 80\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{8 \, {\left(b^{5} x + {\left(a - 1\right)} b^{4}\right)}}"," ",0,"-1/8*(3*(8*a^4 - 44*a^3 + (8*a^3 - 36*a^2 + 44*a - 17)*b*x + 80*a^2 - 61*a + 17)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - (2*b^4*x^4 + 6*b^3*x^3 - (10*a - 11)*b^2*x^2 - 2*a^4 + 78*a^3 + (22*a^2 - 54*a + 29)*b*x - 233*a^2 + 237*a - 80)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^5*x + (a - 1)*b^4)","A",0
836,1,159,0,0.743073," ","integrate((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)*x^2,x, algorithm=""fricas"")","\frac{3 \, {\left(6 \, a^{3} + {\left(6 \, a^{2} - 18 \, a + 11\right)} b x - 24 \, a^{2} + 29 \, a - 11\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + {\left(2 \, b^{3} x^{3} + 7 \, b^{2} x^{2} + 2 \, a^{3} - {\left(16 \, a - 19\right)} b x - 53 \, a^{2} + 103 \, a - 52\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left(b^{4} x + {\left(a - 1\right)} b^{3}\right)}}"," ",0,"1/6*(3*(6*a^3 + (6*a^2 - 18*a + 11)*b*x - 24*a^2 + 29*a - 11)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (2*b^3*x^3 + 7*b^2*x^2 + 2*a^3 - (16*a - 19)*b*x - 53*a^2 + 103*a - 52)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^4*x + (a - 1)*b^3)","A",0
837,1,131,0,0.727978," ","integrate((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)*x,x, algorithm=""fricas"")","-\frac{3 \, {\left({\left(2 \, a - 3\right)} b x + 2 \, a^{2} - 5 \, a + 3\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) - {\left(b^{2} x^{2} - a^{2} + 5 \, b x + 15 \, a - 14\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left(b^{3} x + {\left(a - 1\right)} b^{2}\right)}}"," ",0,"-1/2*(3*((2*a - 3)*b*x + 2*a^2 - 5*a + 3)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - (b^2*x^2 - a^2 + 5*b*x + 15*a - 14)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^3*x + (a - 1)*b^2)","A",0
838,1,100,0,0.582113," ","integrate((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2),x, algorithm=""fricas"")","\frac{3 \, {\left(b x + a - 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a - 5\right)}}{b^{2} x + {\left(a - 1\right)} b}"," ",0,"(3*(b*x + a - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a - 5))/(b^2*x + (a - 1)*b)","A",0
839,1,439,0,0.702443," ","integrate((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left({\left(a + 1\right)} b x + a^{2} - 1\right)} \sqrt{-\frac{a + 1}{a - 1}} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 4 \, a^{2} + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{3} + {\left(a^{2} - a\right)} b x - a^{2} - a + 1\right)} \sqrt{-\frac{a + 1}{a - 1}} + 2}{x^{2}}\right) + 2 \, {\left({\left(a - 1\right)} b x + a^{2} - 2 \, a + 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + 8 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left({\left(a - 1\right)} b x + a^{2} - 2 \, a + 1\right)}}, -\frac{{\left({\left(a + 1\right)} b x + a^{2} - 1\right)} \sqrt{\frac{a + 1}{a - 1}} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{\frac{a + 1}{a - 1}}}{{\left(a + 1\right)} b^{2} x^{2} + a^{3} + 2 \, {\left(a^{2} + a\right)} b x + a^{2} - a - 1}\right) - {\left({\left(a - 1\right)} b x + a^{2} - 2 \, a + 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) - 4 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{{\left(a - 1\right)} b x + a^{2} - 2 \, a + 1}\right]"," ",0,"[1/2*(((a + 1)*b*x + a^2 - 1)*sqrt(-(a + 1)/(a - 1))*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2 + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^2 - a)*b*x - a^2 - a + 1)*sqrt(-(a + 1)/(a - 1)) + 2)/x^2) + 2*((a - 1)*b*x + a^2 - 2*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + 8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a - 1)*b*x + a^2 - 2*a + 1), -(((a + 1)*b*x + a^2 - 1)*sqrt((a + 1)/(a - 1))*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt((a + 1)/(a - 1))/((a + 1)*b^2*x^2 + a^3 + 2*(a^2 + a)*b*x + a^2 - a - 1)) - ((a - 1)*b*x + a^2 - 2*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - 4*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a - 1)*b*x + a^2 - 2*a + 1)]","B",0
840,1,370,0,0.804466," ","integrate((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)/x^2,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(b^{2} x^{2} + {\left(a - 1\right)} b x\right)} \sqrt{-\frac{a + 1}{a - 1}} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 4 \, a^{2} - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{3} + {\left(a^{2} - a\right)} b x - a^{2} - a + 1\right)} \sqrt{-\frac{a + 1}{a - 1}} + 2}{x^{2}}\right) - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left({\left(a + 5\right)} b x + a^{2} - 1\right)}}{2 \, {\left({\left(a^{2} - 2 \, a + 1\right)} b x^{2} + {\left(a^{3} - 3 \, a^{2} + 3 \, a - 1\right)} x\right)}}, \frac{3 \, {\left(b^{2} x^{2} + {\left(a - 1\right)} b x\right)} \sqrt{\frac{a + 1}{a - 1}} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{\frac{a + 1}{a - 1}}}{{\left(a + 1\right)} b^{2} x^{2} + a^{3} + 2 \, {\left(a^{2} + a\right)} b x + a^{2} - a - 1}\right) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left({\left(a + 5\right)} b x + a^{2} - 1\right)}}{{\left(a^{2} - 2 \, a + 1\right)} b x^{2} + {\left(a^{3} - 3 \, a^{2} + 3 \, a - 1\right)} x}\right]"," ",0,"[1/2*(3*(b^2*x^2 + (a - 1)*b*x)*sqrt(-(a + 1)/(a - 1))*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2 - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^2 - a)*b*x - a^2 - a + 1)*sqrt(-(a + 1)/(a - 1)) + 2)/x^2) - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*((a + 5)*b*x + a^2 - 1))/((a^2 - 2*a + 1)*b*x^2 + (a^3 - 3*a^2 + 3*a - 1)*x), (3*(b^2*x^2 + (a - 1)*b*x)*sqrt((a + 1)/(a - 1))*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt((a + 1)/(a - 1))/((a + 1)*b^2*x^2 + a^3 + 2*(a^2 + a)*b*x + a^2 - a - 1)) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*((a + 5)*b*x + a^2 - 1))/((a^2 - 2*a + 1)*b*x^2 + (a^3 - 3*a^2 + 3*a - 1)*x)]","A",0
841,1,523,0,1.551023," ","integrate((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)/x^3,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(2 \, a + 3\right)} b^{3} x^{3} + {\left(2 \, a^{2} + a - 3\right)} b^{2} x^{2}\right)} \sqrt{-a^{2} + 1} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) + 2 \, {\left(a^{5} - {\left(a^{3} + 14 \, a^{2} - a - 14\right)} b^{2} x^{2} - a^{4} - 2 \, a^{3} - 5 \, {\left(a^{3} - a^{2} - a + 1\right)} b x + 2 \, a^{2} + a - 1\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left({\left(a^{5} - 3 \, a^{4} + 2 \, a^{3} + 2 \, a^{2} - 3 \, a + 1\right)} b x^{3} + {\left(a^{6} - 4 \, a^{5} + 5 \, a^{4} - 5 \, a^{2} + 4 \, a - 1\right)} x^{2}\right)}}, -\frac{3 \, {\left({\left(2 \, a + 3\right)} b^{3} x^{3} + {\left(2 \, a^{2} + a - 3\right)} b^{2} x^{2}\right)} \sqrt{a^{2} - 1} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) + {\left(a^{5} - {\left(a^{3} + 14 \, a^{2} - a - 14\right)} b^{2} x^{2} - a^{4} - 2 \, a^{3} - 5 \, {\left(a^{3} - a^{2} - a + 1\right)} b x + 2 \, a^{2} + a - 1\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left({\left(a^{5} - 3 \, a^{4} + 2 \, a^{3} + 2 \, a^{2} - 3 \, a + 1\right)} b x^{3} + {\left(a^{6} - 4 \, a^{5} + 5 \, a^{4} - 5 \, a^{2} + 4 \, a - 1\right)} x^{2}\right)}}\right]"," ",0,"[-1/4*(3*((2*a + 3)*b^3*x^3 + (2*a^2 + a - 3)*b^2*x^2)*sqrt(-a^2 + 1)*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 2*(a^5 - (a^3 + 14*a^2 - a - 14)*b^2*x^2 - a^4 - 2*a^3 - 5*(a^3 - a^2 - a + 1)*b*x + 2*a^2 + a - 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 - 3*a^4 + 2*a^3 + 2*a^2 - 3*a + 1)*b*x^3 + (a^6 - 4*a^5 + 5*a^4 - 5*a^2 + 4*a - 1)*x^2), -1/2*(3*((2*a + 3)*b^3*x^3 + (2*a^2 + a - 3)*b^2*x^2)*sqrt(a^2 - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + (a^5 - (a^3 + 14*a^2 - a - 14)*b^2*x^2 - a^4 - 2*a^3 - 5*(a^3 - a^2 - a + 1)*b*x + 2*a^2 + a - 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 - 3*a^4 + 2*a^3 + 2*a^2 - 3*a + 1)*b*x^3 + (a^6 - 4*a^5 + 5*a^4 - 5*a^2 + 4*a - 1)*x^2)]","A",0
842,1,704,0,0.876992," ","integrate((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)/x^4,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(6 \, a^{2} + 18 \, a + 11\right)} b^{4} x^{4} + {\left(6 \, a^{3} + 12 \, a^{2} - 7 \, a - 11\right)} b^{3} x^{3}\right)} \sqrt{-a^{2} + 1} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) + 2 \, {\left(2 \, a^{7} + {\left(2 \, a^{4} + 51 \, a^{3} + 50 \, a^{2} - 51 \, a - 52\right)} b^{3} x^{3} - 2 \, a^{6} - 6 \, a^{5} + {\left(16 \, a^{4} + 3 \, a^{3} - 35 \, a^{2} - 3 \, a + 19\right)} b^{2} x^{2} + 6 \, a^{4} + 6 \, a^{3} - 7 \, {\left(a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right)} b x - 6 \, a^{2} - 2 \, a + 2\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{12 \, {\left({\left(a^{7} - 3 \, a^{6} + a^{5} + 5 \, a^{4} - 5 \, a^{3} - a^{2} + 3 \, a - 1\right)} b x^{4} + {\left(a^{8} - 4 \, a^{7} + 4 \, a^{6} + 4 \, a^{5} - 10 \, a^{4} + 4 \, a^{3} + 4 \, a^{2} - 4 \, a + 1\right)} x^{3}\right)}}, \frac{3 \, {\left({\left(6 \, a^{2} + 18 \, a + 11\right)} b^{4} x^{4} + {\left(6 \, a^{3} + 12 \, a^{2} - 7 \, a - 11\right)} b^{3} x^{3}\right)} \sqrt{a^{2} - 1} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) - {\left(2 \, a^{7} + {\left(2 \, a^{4} + 51 \, a^{3} + 50 \, a^{2} - 51 \, a - 52\right)} b^{3} x^{3} - 2 \, a^{6} - 6 \, a^{5} + {\left(16 \, a^{4} + 3 \, a^{3} - 35 \, a^{2} - 3 \, a + 19\right)} b^{2} x^{2} + 6 \, a^{4} + 6 \, a^{3} - 7 \, {\left(a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right)} b x - 6 \, a^{2} - 2 \, a + 2\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left({\left(a^{7} - 3 \, a^{6} + a^{5} + 5 \, a^{4} - 5 \, a^{3} - a^{2} + 3 \, a - 1\right)} b x^{4} + {\left(a^{8} - 4 \, a^{7} + 4 \, a^{6} + 4 \, a^{5} - 10 \, a^{4} + 4 \, a^{3} + 4 \, a^{2} - 4 \, a + 1\right)} x^{3}\right)}}\right]"," ",0,"[-1/12*(3*((6*a^2 + 18*a + 11)*b^4*x^4 + (6*a^3 + 12*a^2 - 7*a - 11)*b^3*x^3)*sqrt(-a^2 + 1)*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 2*(2*a^7 + (2*a^4 + 51*a^3 + 50*a^2 - 51*a - 52)*b^3*x^3 - 2*a^6 - 6*a^5 + (16*a^4 + 3*a^3 - 35*a^2 - 3*a + 19)*b^2*x^2 + 6*a^4 + 6*a^3 - 7*(a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*b*x - 6*a^2 - 2*a + 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 - 3*a^6 + a^5 + 5*a^4 - 5*a^3 - a^2 + 3*a - 1)*b*x^4 + (a^8 - 4*a^7 + 4*a^6 + 4*a^5 - 10*a^4 + 4*a^3 + 4*a^2 - 4*a + 1)*x^3), 1/6*(3*((6*a^2 + 18*a + 11)*b^4*x^4 + (6*a^3 + 12*a^2 - 7*a - 11)*b^3*x^3)*sqrt(a^2 - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) - (2*a^7 + (2*a^4 + 51*a^3 + 50*a^2 - 51*a - 52)*b^3*x^3 - 2*a^6 - 6*a^5 + (16*a^4 + 3*a^3 - 35*a^2 - 3*a + 19)*b^2*x^2 + 6*a^4 + 6*a^3 - 7*(a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*b*x - 6*a^2 - 2*a + 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 - 3*a^6 + a^5 + 5*a^4 - 5*a^3 - a^2 + 3*a - 1)*b*x^4 + (a^8 - 4*a^7 + 4*a^6 + 4*a^5 - 10*a^4 + 4*a^3 + 4*a^2 - 4*a + 1)*x^3)]","A",0
843,1,143,0,0.678375," ","integrate(x^3/(b*x+a+1)*(1-(b*x+a)^2)^(1/2),x, algorithm=""fricas"")","\frac{3 \, {\left(8 \, a^{3} + 12 \, a^{2} + 12 \, a + 3\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + {\left(6 \, b^{3} x^{3} - 2 \, {\left(3 \, a + 4\right)} b^{2} x^{2} - 6 \, a^{3} + {\left(6 \, a^{2} + 20 \, a + 9\right)} b x - 44 \, a^{2} - 39 \, a - 16\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{24 \, b^{4}}"," ",0,"1/24*(3*(8*a^3 + 12*a^2 + 12*a + 3)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (6*b^3*x^3 - 2*(3*a + 4)*b^2*x^2 - 6*a^3 + (6*a^2 + 20*a + 9)*b*x - 44*a^2 - 39*a - 16)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/b^4","A",0
844,1,117,0,0.484464," ","integrate(x^2/(b*x+a+1)*(1-(b*x+a)^2)^(1/2),x, algorithm=""fricas"")","-\frac{3 \, {\left(2 \, a^{2} + 2 \, a + 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) - {\left(2 \, b^{2} x^{2} - {\left(2 \, a + 3\right)} b x + 2 \, a^{2} + 9 \, a + 4\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, b^{3}}"," ",0,"-1/6*(3*(2*a^2 + 2*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - (2*b^2*x^2 - (2*a + 3)*b*x + 2*a^2 + 9*a + 4)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/b^3","A",0
845,1,91,0,0.646052," ","integrate(x/(b*x+a+1)*(1-(b*x+a)^2)^(1/2),x, algorithm=""fricas"")","\frac{{\left(2 \, a + 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x - a - 2\right)}}{2 \, b^{2}}"," ",0,"1/2*((2*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x - a - 2))/b^2","A",0
846,1,77,0,0.587166," ","integrate(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} - \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right)}{b}"," ",0,"(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1) - arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)))/b","B",0
847,1,303,0,0.610360," ","integrate(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{1}{2} \, \sqrt{-\frac{a - 1}{a + 1}} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 4 \, a^{2} + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{3} + {\left(a^{2} + a\right)} b x + a^{2} - a - 1\right)} \sqrt{-\frac{a - 1}{a + 1}} + 2}{x^{2}}\right) + \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right), -\sqrt{\frac{a - 1}{a + 1}} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{\frac{a - 1}{a + 1}}}{{\left(a - 1\right)} b^{2} x^{2} + a^{3} + 2 \, {\left(a^{2} - a\right)} b x - a^{2} - a + 1}\right) + \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right)\right]"," ",0,"[1/2*sqrt(-(a - 1)/(a + 1))*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2 + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^2 + a)*b*x + a^2 - a - 1)*sqrt(-(a - 1)/(a + 1)) + 2)/x^2) + arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)), -sqrt((a - 1)/(a + 1))*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt((a - 1)/(a + 1))/((a - 1)*b^2*x^2 + a^3 + 2*(a^2 - a)*b*x - a^2 - a + 1)) + arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1))]","B",0
848,1,277,0,0.583488," ","integrate(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x^2,x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + 1} b x \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{2} - 1\right)}}{2 \, {\left(a^{3} + a^{2} - a - 1\right)} x}, -\frac{\sqrt{a^{2} - 1} b x \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) + \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{2} - 1\right)}}{{\left(a^{3} + a^{2} - a - 1\right)} x}\right]"," ",0,"[-1/2*(sqrt(-a^2 + 1)*b*x*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1))/((a^3 + a^2 - a - 1)*x), -(sqrt(a^2 - 1)*b*x*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1))/((a^3 + a^2 - a - 1)*x)]","A",0
849,1,356,0,0.659496," ","integrate(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x^3,x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + 1} {\left(2 \, a - 1\right)} b^{2} x^{2} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) + 2 \, {\left(a^{4} - {\left(a^{3} - 2 \, a^{2} - a + 2\right)} b x - 2 \, a^{2} + 1\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left(a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right)} x^{2}}, \frac{\sqrt{a^{2} - 1} {\left(2 \, a - 1\right)} b^{2} x^{2} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) - {\left(a^{4} - {\left(a^{3} - 2 \, a^{2} - a + 2\right)} b x - 2 \, a^{2} + 1\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left(a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right)} x^{2}}\right]"," ",0,"[-1/4*(sqrt(-a^2 + 1)*(2*a - 1)*b^2*x^2*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 2*(a^4 - (a^3 - 2*a^2 - a + 2)*b*x - 2*a^2 + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 + a^4 - 2*a^3 - 2*a^2 + a + 1)*x^2), 1/2*(sqrt(a^2 - 1)*(2*a - 1)*b^2*x^2*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) - (a^4 - (a^3 - 2*a^2 - a + 2)*b*x - 2*a^2 + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 + a^4 - 2*a^3 - 2*a^2 + a + 1)*x^2)]","A",0
850,1,483,0,0.716374," ","integrate(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x^4,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(2 \, a^{2} - 2 \, a + 1\right)} \sqrt{-a^{2} + 1} b^{3} x^{3} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) + 2 \, {\left(2 \, a^{6} + {\left(2 \, a^{4} - 9 \, a^{3} + 2 \, a^{2} + 9 \, a - 4\right)} b^{2} x^{2} - 6 \, a^{4} - {\left(2 \, a^{5} - 3 \, a^{4} - 4 \, a^{3} + 6 \, a^{2} + 2 \, a - 3\right)} b x + 6 \, a^{2} - 2\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{12 \, {\left(a^{7} + a^{6} - 3 \, a^{5} - 3 \, a^{4} + 3 \, a^{3} + 3 \, a^{2} - a - 1\right)} x^{3}}, -\frac{3 \, {\left(2 \, a^{2} - 2 \, a + 1\right)} \sqrt{a^{2} - 1} b^{3} x^{3} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) + {\left(2 \, a^{6} + {\left(2 \, a^{4} - 9 \, a^{3} + 2 \, a^{2} + 9 \, a - 4\right)} b^{2} x^{2} - 6 \, a^{4} - {\left(2 \, a^{5} - 3 \, a^{4} - 4 \, a^{3} + 6 \, a^{2} + 2 \, a - 3\right)} b x + 6 \, a^{2} - 2\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left(a^{7} + a^{6} - 3 \, a^{5} - 3 \, a^{4} + 3 \, a^{3} + 3 \, a^{2} - a - 1\right)} x^{3}}\right]"," ",0,"[-1/12*(3*(2*a^2 - 2*a + 1)*sqrt(-a^2 + 1)*b^3*x^3*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 2*(2*a^6 + (2*a^4 - 9*a^3 + 2*a^2 + 9*a - 4)*b^2*x^2 - 6*a^4 - (2*a^5 - 3*a^4 - 4*a^3 + 6*a^2 + 2*a - 3)*b*x + 6*a^2 - 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 + a^6 - 3*a^5 - 3*a^4 + 3*a^3 + 3*a^2 - a - 1)*x^3), -1/6*(3*(2*a^2 - 2*a + 1)*sqrt(a^2 - 1)*b^3*x^3*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + (2*a^6 + (2*a^4 - 9*a^3 + 2*a^2 + 9*a - 4)*b^2*x^2 - 6*a^4 - (2*a^5 - 3*a^4 - 4*a^3 + 6*a^2 + 2*a - 3)*b*x + 6*a^2 - 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 + a^6 - 3*a^5 - 3*a^4 + 3*a^3 + 3*a^2 - a - 1)*x^3)]","A",0
851,1,93,0,0.509980," ","integrate(x^4/(b*x+a+1)^2*(1-(b*x+a)^2),x, algorithm=""fricas"")","-\frac{6 \, b^{5} x^{5} - 15 \, b^{4} x^{4} + 20 \, {\left(a + 1\right)} b^{3} x^{3} - 30 \, {\left(a^{2} + 2 \, a + 1\right)} b^{2} x^{2} + 60 \, {\left(a^{3} + 3 \, a^{2} + 3 \, a + 1\right)} b x - 60 \, {\left(a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right)} \log\left(b x + a + 1\right)}{30 \, b^{5}}"," ",0,"-1/30*(6*b^5*x^5 - 15*b^4*x^4 + 20*(a + 1)*b^3*x^3 - 30*(a^2 + 2*a + 1)*b^2*x^2 + 60*(a^3 + 3*a^2 + 3*a + 1)*b*x - 60*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1))/b^5","A",0
852,1,67,0,2.260425," ","integrate(x^3/(b*x+a+1)^2*(1-(b*x+a)^2),x, algorithm=""fricas"")","-\frac{3 \, b^{4} x^{4} - 8 \, b^{3} x^{3} + 12 \, {\left(a + 1\right)} b^{2} x^{2} - 24 \, {\left(a^{2} + 2 \, a + 1\right)} b x + 24 \, {\left(a^{3} + 3 \, a^{2} + 3 \, a + 1\right)} \log\left(b x + a + 1\right)}{12 \, b^{4}}"," ",0,"-1/12*(3*b^4*x^4 - 8*b^3*x^3 + 12*(a + 1)*b^2*x^2 - 24*(a^2 + 2*a + 1)*b*x + 24*(a^3 + 3*a^2 + 3*a + 1)*log(b*x + a + 1))/b^4","A",0
853,1,45,0,0.673235," ","integrate(x^2/(b*x+a+1)^2*(1-(b*x+a)^2),x, algorithm=""fricas"")","-\frac{b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, {\left(a + 1\right)} b x - 6 \, {\left(a^{2} + 2 \, a + 1\right)} \log\left(b x + a + 1\right)}{3 \, b^{3}}"," ",0,"-1/3*(b^3*x^3 - 3*b^2*x^2 + 6*(a + 1)*b*x - 6*(a^2 + 2*a + 1)*log(b*x + a + 1))/b^3","A",0
854,1,29,0,0.481244," ","integrate(x/(b*x+a+1)^2*(1-(b*x+a)^2),x, algorithm=""fricas"")","-\frac{b^{2} x^{2} - 4 \, b x + 4 \, {\left(a + 1\right)} \log\left(b x + a + 1\right)}{2 \, b^{2}}"," ",0,"-1/2*(b^2*x^2 - 4*b*x + 4*(a + 1)*log(b*x + a + 1))/b^2","A",0
855,1,18,0,1.307586," ","integrate(1/(b*x+a+1)^2*(1-(b*x+a)^2),x, algorithm=""fricas"")","-\frac{b x - 2 \, \log\left(b x + a + 1\right)}{b}"," ",0,"-(b*x - 2*log(b*x + a + 1))/b","A",0
856,1,23,0,0.588712," ","integrate(1/(b*x+a+1)^2*(1-(b*x+a)^2)/x,x, algorithm=""fricas"")","-\frac{{\left(a - 1\right)} \log\left(x\right) + 2 \, \log\left(b x + a + 1\right)}{a + 1}"," ",0,"-((a - 1)*log(x) + 2*log(b*x + a + 1))/(a + 1)","A",0
857,1,36,0,2.384471," ","integrate(1/(b*x+a+1)^2*(1-(b*x+a)^2)/x^2,x, algorithm=""fricas"")","\frac{2 \, b x \log\left(b x + a + 1\right) - 2 \, b x \log\left(x\right) + a^{2} - 1}{{\left(a^{2} + 2 \, a + 1\right)} x}"," ",0,"(2*b*x*log(b*x + a + 1) - 2*b*x*log(x) + a^2 - 1)/((a^2 + 2*a + 1)*x)","A",0
858,1,65,0,0.602141," ","integrate(1/(b*x+a+1)^2*(1-(b*x+a)^2)/x^3,x, algorithm=""fricas"")","-\frac{4 \, b^{2} x^{2} \log\left(b x + a + 1\right) - 4 \, b^{2} x^{2} \log\left(x\right) - a^{3} - 4 \, {\left(a + 1\right)} b x - a^{2} + a + 1}{2 \, {\left(a^{3} + 3 \, a^{2} + 3 \, a + 1\right)} x^{2}}"," ",0,"-1/2*(4*b^2*x^2*log(b*x + a + 1) - 4*b^2*x^2*log(x) - a^3 - 4*(a + 1)*b*x - a^2 + a + 1)/((a^3 + 3*a^2 + 3*a + 1)*x^2)","A",0
859,1,86,0,0.577725," ","integrate(1/(b*x+a+1)^2*(1-(b*x+a)^2)/x^4,x, algorithm=""fricas"")","\frac{6 \, b^{3} x^{3} \log\left(b x + a + 1\right) - 6 \, b^{3} x^{3} \log\left(x\right) - 6 \, {\left(a + 1\right)} b^{2} x^{2} + a^{4} + 2 \, a^{3} + 3 \, {\left(a^{2} + 2 \, a + 1\right)} b x - 2 \, a - 1}{3 \, {\left(a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right)} x^{3}}"," ",0,"1/3*(6*b^3*x^3*log(b*x + a + 1) - 6*b^3*x^3*log(x) - 6*(a + 1)*b^2*x^2 + a^4 + 2*a^3 + 3*(a^2 + 2*a + 1)*b*x - 2*a - 1)/((a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*x^3)","A",0
860,1,191,0,0.818262," ","integrate(x^3/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2),x, algorithm=""fricas"")","-\frac{3 \, {\left(8 \, a^{4} + 44 \, a^{3} + {\left(8 \, a^{3} + 36 \, a^{2} + 44 \, a + 17\right)} b x + 80 \, a^{2} + 61 \, a + 17\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + {\left(2 \, b^{4} x^{4} - 6 \, b^{3} x^{3} + {\left(10 \, a + 11\right)} b^{2} x^{2} - 2 \, a^{4} - 78 \, a^{3} - {\left(22 \, a^{2} + 54 \, a + 29\right)} b x - 233 \, a^{2} - 237 \, a - 80\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{8 \, {\left(b^{5} x + {\left(a + 1\right)} b^{4}\right)}}"," ",0,"-1/8*(3*(8*a^4 + 44*a^3 + (8*a^3 + 36*a^2 + 44*a + 17)*b*x + 80*a^2 + 61*a + 17)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (2*b^4*x^4 - 6*b^3*x^3 + (10*a + 11)*b^2*x^2 - 2*a^4 - 78*a^3 - (22*a^2 + 54*a + 29)*b*x - 233*a^2 - 237*a - 80)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^5*x + (a + 1)*b^4)","A",0
861,1,159,0,0.664699," ","integrate(x^2/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2),x, algorithm=""fricas"")","\frac{3 \, {\left(6 \, a^{3} + {\left(6 \, a^{2} + 18 \, a + 11\right)} b x + 24 \, a^{2} + 29 \, a + 11\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) - {\left(2 \, b^{3} x^{3} - 7 \, b^{2} x^{2} + 2 \, a^{3} + {\left(16 \, a + 19\right)} b x + 53 \, a^{2} + 103 \, a + 52\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left(b^{4} x + {\left(a + 1\right)} b^{3}\right)}}"," ",0,"1/6*(3*(6*a^3 + (6*a^2 + 18*a + 11)*b*x + 24*a^2 + 29*a + 11)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - (2*b^3*x^3 - 7*b^2*x^2 + 2*a^3 + (16*a + 19)*b*x + 53*a^2 + 103*a + 52)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^4*x + (a + 1)*b^3)","A",0
862,1,130,0,1.263570," ","integrate(x/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2),x, algorithm=""fricas"")","-\frac{3 \, {\left({\left(2 \, a + 3\right)} b x + 2 \, a^{2} + 5 \, a + 3\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + {\left(b^{2} x^{2} - a^{2} - 5 \, b x - 15 \, a - 14\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left(b^{3} x + {\left(a + 1\right)} b^{2}\right)}}"," ",0,"-1/2*(3*((2*a + 3)*b*x + 2*a^2 + 5*a + 3)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (b^2*x^2 - a^2 - 5*b*x - 15*a - 14)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^3*x + (a + 1)*b^2)","A",0
863,1,101,0,0.627699," ","integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2),x, algorithm=""fricas"")","\frac{3 \, {\left(b x + a + 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a + 5\right)}}{b^{2} x + {\left(a + 1\right)} b}"," ",0,"(3*(b*x + a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a + 5))/(b^2*x + (a + 1)*b)","A",0
864,1,438,0,0.456507," ","integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left({\left(a - 1\right)} b x + a^{2} - 1\right)} \sqrt{-\frac{a - 1}{a + 1}} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 4 \, a^{2} - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{3} + {\left(a^{2} + a\right)} b x + a^{2} - a - 1\right)} \sqrt{-\frac{a - 1}{a + 1}} + 2}{x^{2}}\right) - 2 \, {\left({\left(a + 1\right)} b x + a^{2} + 2 \, a + 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + 8 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left({\left(a + 1\right)} b x + a^{2} + 2 \, a + 1\right)}}, \frac{{\left({\left(a - 1\right)} b x + a^{2} - 1\right)} \sqrt{\frac{a - 1}{a + 1}} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{\frac{a - 1}{a + 1}}}{{\left(a - 1\right)} b^{2} x^{2} + a^{3} + 2 \, {\left(a^{2} - a\right)} b x - a^{2} - a + 1}\right) - {\left({\left(a + 1\right)} b x + a^{2} + 2 \, a + 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + 4 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{{\left(a + 1\right)} b x + a^{2} + 2 \, a + 1}\right]"," ",0,"[1/2*(((a - 1)*b*x + a^2 - 1)*sqrt(-(a - 1)/(a + 1))*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2 - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^2 + a)*b*x + a^2 - a - 1)*sqrt(-(a - 1)/(a + 1)) + 2)/x^2) - 2*((a + 1)*b*x + a^2 + 2*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + 8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a + 1)*b*x + a^2 + 2*a + 1), (((a - 1)*b*x + a^2 - 1)*sqrt((a - 1)/(a + 1))*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt((a - 1)/(a + 1))/((a - 1)*b^2*x^2 + a^3 + 2*(a^2 - a)*b*x - a^2 - a + 1)) - ((a + 1)*b*x + a^2 + 2*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + 4*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a + 1)*b*x + a^2 + 2*a + 1)]","B",0
865,1,369,0,0.701550," ","integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^2,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(b^{2} x^{2} + {\left(a + 1\right)} b x\right)} \sqrt{-\frac{a - 1}{a + 1}} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 4 \, a^{2} - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{3} + {\left(a^{2} + a\right)} b x + a^{2} - a - 1\right)} \sqrt{-\frac{a - 1}{a + 1}} + 2}{x^{2}}\right) + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left({\left(a - 5\right)} b x + a^{2} - 1\right)}}{2 \, {\left({\left(a^{2} + 2 \, a + 1\right)} b x^{2} + {\left(a^{3} + 3 \, a^{2} + 3 \, a + 1\right)} x\right)}}, \frac{3 \, {\left(b^{2} x^{2} + {\left(a + 1\right)} b x\right)} \sqrt{\frac{a - 1}{a + 1}} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{\frac{a - 1}{a + 1}}}{{\left(a - 1\right)} b^{2} x^{2} + a^{3} + 2 \, {\left(a^{2} - a\right)} b x - a^{2} - a + 1}\right) + \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left({\left(a - 5\right)} b x + a^{2} - 1\right)}}{{\left(a^{2} + 2 \, a + 1\right)} b x^{2} + {\left(a^{3} + 3 \, a^{2} + 3 \, a + 1\right)} x}\right]"," ",0,"[1/2*(3*(b^2*x^2 + (a + 1)*b*x)*sqrt(-(a - 1)/(a + 1))*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2 - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^2 + a)*b*x + a^2 - a - 1)*sqrt(-(a - 1)/(a + 1)) + 2)/x^2) + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*((a - 5)*b*x + a^2 - 1))/((a^2 + 2*a + 1)*b*x^2 + (a^3 + 3*a^2 + 3*a + 1)*x), (3*(b^2*x^2 + (a + 1)*b*x)*sqrt((a - 1)/(a + 1))*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt((a - 1)/(a + 1))/((a - 1)*b^2*x^2 + a^3 + 2*(a^2 - a)*b*x - a^2 - a + 1)) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*((a - 5)*b*x + a^2 - 1))/((a^2 + 2*a + 1)*b*x^2 + (a^3 + 3*a^2 + 3*a + 1)*x)]","A",0
866,1,520,0,0.664256," ","integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^3,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(2 \, a - 3\right)} b^{3} x^{3} + {\left(2 \, a^{2} - a - 3\right)} b^{2} x^{2}\right)} \sqrt{-a^{2} + 1} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) - 2 \, {\left(a^{5} - {\left(a^{3} - 14 \, a^{2} - a + 14\right)} b^{2} x^{2} + a^{4} - 2 \, a^{3} + 5 \, {\left(a^{3} + a^{2} - a - 1\right)} b x - 2 \, a^{2} + a + 1\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left({\left(a^{5} + 3 \, a^{4} + 2 \, a^{3} - 2 \, a^{2} - 3 \, a - 1\right)} b x^{3} + {\left(a^{6} + 4 \, a^{5} + 5 \, a^{4} - 5 \, a^{2} - 4 \, a - 1\right)} x^{2}\right)}}, -\frac{3 \, {\left({\left(2 \, a - 3\right)} b^{3} x^{3} + {\left(2 \, a^{2} - a - 3\right)} b^{2} x^{2}\right)} \sqrt{a^{2} - 1} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) - {\left(a^{5} - {\left(a^{3} - 14 \, a^{2} - a + 14\right)} b^{2} x^{2} + a^{4} - 2 \, a^{3} + 5 \, {\left(a^{3} + a^{2} - a - 1\right)} b x - 2 \, a^{2} + a + 1\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left({\left(a^{5} + 3 \, a^{4} + 2 \, a^{3} - 2 \, a^{2} - 3 \, a - 1\right)} b x^{3} + {\left(a^{6} + 4 \, a^{5} + 5 \, a^{4} - 5 \, a^{2} - 4 \, a - 1\right)} x^{2}\right)}}\right]"," ",0,"[-1/4*(3*((2*a - 3)*b^3*x^3 + (2*a^2 - a - 3)*b^2*x^2)*sqrt(-a^2 + 1)*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) - 2*(a^5 - (a^3 - 14*a^2 - a + 14)*b^2*x^2 + a^4 - 2*a^3 + 5*(a^3 + a^2 - a - 1)*b*x - 2*a^2 + a + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 + 3*a^4 + 2*a^3 - 2*a^2 - 3*a - 1)*b*x^3 + (a^6 + 4*a^5 + 5*a^4 - 5*a^2 - 4*a - 1)*x^2), -1/2*(3*((2*a - 3)*b^3*x^3 + (2*a^2 - a - 3)*b^2*x^2)*sqrt(a^2 - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) - (a^5 - (a^3 - 14*a^2 - a + 14)*b^2*x^2 + a^4 - 2*a^3 + 5*(a^3 + a^2 - a - 1)*b*x - 2*a^2 + a + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 + 3*a^4 + 2*a^3 - 2*a^2 - 3*a - 1)*b*x^3 + (a^6 + 4*a^5 + 5*a^4 - 5*a^2 - 4*a - 1)*x^2)]","A",0
867,1,697,0,0.761739," ","integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^4,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(6 \, a^{2} - 18 \, a + 11\right)} b^{4} x^{4} + {\left(6 \, a^{3} - 12 \, a^{2} - 7 \, a + 11\right)} b^{3} x^{3}\right)} \sqrt{-a^{2} + 1} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) - 2 \, {\left(2 \, a^{7} + {\left(2 \, a^{4} - 51 \, a^{3} + 50 \, a^{2} + 51 \, a - 52\right)} b^{3} x^{3} + 2 \, a^{6} - 6 \, a^{5} - {\left(16 \, a^{4} - 3 \, a^{3} - 35 \, a^{2} + 3 \, a + 19\right)} b^{2} x^{2} - 6 \, a^{4} + 6 \, a^{3} + 7 \, {\left(a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right)} b x + 6 \, a^{2} - 2 \, a - 2\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{12 \, {\left({\left(a^{7} + 3 \, a^{6} + a^{5} - 5 \, a^{4} - 5 \, a^{3} + a^{2} + 3 \, a + 1\right)} b x^{4} + {\left(a^{8} + 4 \, a^{7} + 4 \, a^{6} - 4 \, a^{5} - 10 \, a^{4} - 4 \, a^{3} + 4 \, a^{2} + 4 \, a + 1\right)} x^{3}\right)}}, \frac{3 \, {\left({\left(6 \, a^{2} - 18 \, a + 11\right)} b^{4} x^{4} + {\left(6 \, a^{3} - 12 \, a^{2} - 7 \, a + 11\right)} b^{3} x^{3}\right)} \sqrt{a^{2} - 1} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) + {\left(2 \, a^{7} + {\left(2 \, a^{4} - 51 \, a^{3} + 50 \, a^{2} + 51 \, a - 52\right)} b^{3} x^{3} + 2 \, a^{6} - 6 \, a^{5} - {\left(16 \, a^{4} - 3 \, a^{3} - 35 \, a^{2} + 3 \, a + 19\right)} b^{2} x^{2} - 6 \, a^{4} + 6 \, a^{3} + 7 \, {\left(a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right)} b x + 6 \, a^{2} - 2 \, a - 2\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left({\left(a^{7} + 3 \, a^{6} + a^{5} - 5 \, a^{4} - 5 \, a^{3} + a^{2} + 3 \, a + 1\right)} b x^{4} + {\left(a^{8} + 4 \, a^{7} + 4 \, a^{6} - 4 \, a^{5} - 10 \, a^{4} - 4 \, a^{3} + 4 \, a^{2} + 4 \, a + 1\right)} x^{3}\right)}}\right]"," ",0,"[-1/12*(3*((6*a^2 - 18*a + 11)*b^4*x^4 + (6*a^3 - 12*a^2 - 7*a + 11)*b^3*x^3)*sqrt(-a^2 + 1)*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) - 2*(2*a^7 + (2*a^4 - 51*a^3 + 50*a^2 + 51*a - 52)*b^3*x^3 + 2*a^6 - 6*a^5 - (16*a^4 - 3*a^3 - 35*a^2 + 3*a + 19)*b^2*x^2 - 6*a^4 + 6*a^3 + 7*(a^5 + a^4 - 2*a^3 - 2*a^2 + a + 1)*b*x + 6*a^2 - 2*a - 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 + 3*a^6 + a^5 - 5*a^4 - 5*a^3 + a^2 + 3*a + 1)*b*x^4 + (a^8 + 4*a^7 + 4*a^6 - 4*a^5 - 10*a^4 - 4*a^3 + 4*a^2 + 4*a + 1)*x^3), 1/6*(3*((6*a^2 - 18*a + 11)*b^4*x^4 + (6*a^3 - 12*a^2 - 7*a + 11)*b^3*x^3)*sqrt(a^2 - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + (2*a^7 + (2*a^4 - 51*a^3 + 50*a^2 + 51*a - 52)*b^3*x^3 + 2*a^6 - 6*a^5 - (16*a^4 - 3*a^3 - 35*a^2 + 3*a + 19)*b^2*x^2 - 6*a^4 + 6*a^3 + 7*(a^5 + a^4 - 2*a^3 - 2*a^2 + a + 1)*b*x + 6*a^2 - 2*a - 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 + 3*a^6 + a^5 - 5*a^4 - 5*a^3 + a^2 + 3*a + 1)*b*x^4 + (a^8 + 4*a^7 + 4*a^6 - 4*a^5 - 10*a^4 - 4*a^3 + 4*a^2 + 4*a + 1)*x^3)]","A",0
868,1,28,0,0.636383," ","integrate(1/(1-(b*x+1)^2)^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, b x}}{b x}\right)}{b}"," ",0,"-2*arctan(sqrt(-b^2*x^2 - 2*b*x)/(b*x))/b","B",0
869,1,150,0,0.667435," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x^3/(-b^2*x^2-2*a*b*x-a^2+1),x, algorithm=""fricas"")","\frac{3 \, {\left(2 \, a^{3} + {\left(2 \, a^{2} - 2 \, a + 1\right)} b x - 4 \, a^{2} + 3 \, a - 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + {\left(b^{2} x^{2} + 2 \, a^{3} - {\left(4 \, a - 1\right)} b x - 11 \, a^{2} + 13 \, a - 4\right)} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left(b^{5} x + {\left(a - 1\right)} b^{4}\right)}}"," ",0,"1/2*(3*(2*a^3 + (2*a^2 - 2*a + 1)*b*x - 4*a^2 + 3*a - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (b^2*x^2 + 2*a^3 - (4*a - 1)*b*x - 11*a^2 + 13*a - 4)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^5*x + (a - 1)*b^4)","A",0
870,1,120,0,0.828023," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x^2/(-b^2*x^2-2*a*b*x-a^2+1),x, algorithm=""fricas"")","-\frac{{\left({\left(2 \, a - 1\right)} b x + 2 \, a^{2} - 3 \, a + 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{2} - b x - 3 \, a + 2\right)}}{b^{4} x + {\left(a - 1\right)} b^{3}}"," ",0,"-(((2*a - 1)*b*x + 2*a^2 - 3*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - b*x - 3*a + 2))/(b^4*x + (a - 1)*b^3)","A",0
871,1,98,0,0.565917," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)*x/(-b^2*x^2-2*a*b*x-a^2+1),x, algorithm=""fricas"")","\frac{{\left(b x + a - 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(b x + a\right)}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right) + \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a - 1\right)}}{b^{3} x + {\left(a - 1\right)} b^{2}}"," ",0,"((b*x + a - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a - 1))/(b^3*x + (a - 1)*b^2)","B",0
872,1,37,0,0.529735," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/(-b^2*x^2-2*a*b*x-a^2+1),x, algorithm=""fricas"")","-\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{2} x + {\left(a - 1\right)} b}"," ",0,"-sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(b^2*x + (a - 1)*b)","A",0
873,1,316,0,0.621536," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x/(-b^2*x^2-2*a*b*x-a^2+1),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + 1} {\left(b x + a - 1\right)} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{2} - 1\right)}}{2 \, {\left(a^{4} - 2 \, a^{3} + {\left(a^{3} - a^{2} - a + 1\right)} b x + 2 \, a - 1\right)}}, -\frac{\sqrt{a^{2} - 1} {\left(b x + a - 1\right)} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{2} - 1\right)}}{a^{4} - 2 \, a^{3} + {\left(a^{3} - a^{2} - a + 1\right)} b x + 2 \, a - 1}\right]"," ",0,"[-1/2*(sqrt(-a^2 + 1)*(b*x + a - 1)*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1))/(a^4 - 2*a^3 + (a^3 - a^2 - a + 1)*b*x + 2*a - 1), -(sqrt(a^2 - 1)*(b*x + a - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1))/(a^4 - 2*a^3 + (a^3 - a^2 - a + 1)*b*x + 2*a - 1)]","A",0
874,1,459,0,0.716952," ","integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^2/(-b^2*x^2-2*a*b*x-a^2+1),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(2 \, a + 1\right)} b^{2} x^{2} + {\left(2 \, a^{2} - a - 1\right)} b x\right)} \sqrt{-a^{2} + 1} \log\left(\frac{{\left(2 \, a^{2} - 1\right)} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left(a^{3} - a\right)} b x + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right) + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{3} + {\left(a^{3} + 2 \, a^{2} - a - 2\right)} b x - a^{2} - a + 1\right)}}{2 \, {\left({\left(a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right)} b x^{2} + {\left(a^{6} - 2 \, a^{5} - a^{4} + 4 \, a^{3} - a^{2} - 2 \, a + 1\right)} x\right)}}, \frac{{\left({\left(2 \, a + 1\right)} b^{2} x^{2} + {\left(2 \, a^{2} - a - 1\right)} b x\right)} \sqrt{a^{2} - 1} \arctan\left(\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a b x + a^{2} - 1\right)} \sqrt{a^{2} - 1}}{{\left(a^{2} - 1\right)} b^{2} x^{2} + a^{4} + 2 \, {\left(a^{3} - a\right)} b x - 2 \, a^{2} + 1}\right) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left(a^{3} + {\left(a^{3} + 2 \, a^{2} - a - 2\right)} b x - a^{2} - a + 1\right)}}{{\left(a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right)} b x^{2} + {\left(a^{6} - 2 \, a^{5} - a^{4} + 4 \, a^{3} - a^{2} - 2 \, a + 1\right)} x}\right]"," ",0,"[-1/2*(((2*a + 1)*b^2*x^2 + (2*a^2 - a - 1)*b*x)*sqrt(-a^2 + 1)*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^3 + 2*a^2 - a - 2)*b*x - a^2 - a + 1))/((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*b*x^2 + (a^6 - 2*a^5 - a^4 + 4*a^3 - a^2 - 2*a + 1)*x), (((2*a + 1)*b^2*x^2 + (2*a^2 - a - 1)*b*x)*sqrt(a^2 - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^3 + 2*a^2 - a - 2)*b*x - a^2 - a + 1))/((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*b*x^2 + (a^6 - 2*a^5 - a^4 + 4*a^3 - a^2 - 2*a + 1)*x)]","A",0
875,0,0,0,0.633281," ","integrate(exp(n*arctanh(b*x+a))*x^m,x, algorithm=""fricas"")","{\rm integral}\left(x^{m} \left(\frac{b x + a + 1}{b x + a - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(x^m*((b*x + a + 1)/(b*x + a - 1))^(1/2*n), x)","F",0
876,0,0,0,0.702304," ","integrate(exp(n*arctanh(b*x+a))*x^3,x, algorithm=""fricas"")","{\rm integral}\left(x^{3} \left(\frac{b x + a + 1}{b x + a - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(x^3*((b*x + a + 1)/(b*x + a - 1))^(1/2*n), x)","F",0
877,0,0,0,0.739782," ","integrate(exp(n*arctanh(b*x+a))*x^2,x, algorithm=""fricas"")","{\rm integral}\left(x^{2} \left(\frac{b x + a + 1}{b x + a - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(x^2*((b*x + a + 1)/(b*x + a - 1))^(1/2*n), x)","F",0
878,0,0,0,0.676243," ","integrate(exp(n*arctanh(b*x+a))*x,x, algorithm=""fricas"")","{\rm integral}\left(x \left(\frac{b x + a + 1}{b x + a - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(x*((b*x + a + 1)/(b*x + a - 1))^(1/2*n), x)","F",0
879,0,0,0,0.585112," ","integrate(exp(n*arctanh(b*x+a)),x, algorithm=""fricas"")","{\rm integral}\left(\left(\frac{b x + a + 1}{b x + a - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(((b*x + a + 1)/(b*x + a - 1))^(1/2*n), x)","F",0
880,0,0,0,0.676071," ","integrate(exp(n*arctanh(b*x+a))/x,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{b x + a + 1}{b x + a - 1}\right)^{\frac{1}{2} \, n}}{x}, x\right)"," ",0,"integral(((b*x + a + 1)/(b*x + a - 1))^(1/2*n)/x, x)","F",0
881,0,0,0,0.599465," ","integrate(exp(n*arctanh(b*x+a))/x^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{b x + a + 1}{b x + a - 1}\right)^{\frac{1}{2} \, n}}{x^{2}}, x\right)"," ",0,"integral(((b*x + a + 1)/(b*x + a - 1))^(1/2*n)/x^2, x)","F",0
882,0,0,0,0.590043," ","integrate(exp(n*arctanh(b*x+a))/x^3,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{b x + a + 1}{b x + a - 1}\right)^{\frac{1}{2} \, n}}{x^{3}}, x\right)"," ",0,"integral(((b*x + a + 1)/(b*x + a - 1))^(1/2*n)/x^3, x)","F",0
883,1,136,0,0.673089," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","-\frac{630 \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(128 \, a^{8} c^{4} x^{8} + 144 \, a^{7} c^{4} x^{7} - 512 \, a^{6} c^{4} x^{6} - 600 \, a^{5} c^{4} x^{5} + 768 \, a^{4} c^{4} x^{4} + 978 \, a^{3} c^{4} x^{3} - 512 \, a^{2} c^{4} x^{2} - 837 \, a c^{4} x + 128 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{1152 \, a}"," ",0,"-1/1152*(630*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (128*a^8*c^4*x^8 + 144*a^7*c^4*x^7 - 512*a^6*c^4*x^6 - 600*a^5*c^4*x^5 + 768*a^4*c^4*x^4 + 978*a^3*c^4*x^3 - 512*a^2*c^4*x^2 - 837*a*c^4*x + 128*c^4)*sqrt(-a^2*x^2 + 1))/a","A",0
884,1,115,0,0.645971," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{210 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(48 \, a^{6} c^{3} x^{6} + 56 \, a^{5} c^{3} x^{5} - 144 \, a^{4} c^{3} x^{4} - 182 \, a^{3} c^{3} x^{3} + 144 \, a^{2} c^{3} x^{2} + 231 \, a c^{3} x - 48 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{336 \, a}"," ",0,"-1/336*(210*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (48*a^6*c^3*x^6 + 56*a^5*c^3*x^5 - 144*a^4*c^3*x^4 - 182*a^3*c^3*x^3 + 144*a^2*c^3*x^2 + 231*a*c^3*x - 48*c^3)*sqrt(-a^2*x^2 + 1))/a","A",0
885,1,92,0,0.475157," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{30 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(8 \, a^{4} c^{2} x^{4} + 10 \, a^{3} c^{2} x^{3} - 16 \, a^{2} c^{2} x^{2} - 25 \, a c^{2} x + 8 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{40 \, a}"," ",0,"-1/40*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (8*a^4*c^2*x^4 + 10*a^3*c^2*x^3 - 16*a^2*c^2*x^2 - 25*a*c^2*x + 8*c^2)*sqrt(-a^2*x^2 + 1))/a","A",0
886,1,63,0,0.543117," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{6 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(2 \, a^{2} c x^{2} + 3 \, a c x - 2 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a}"," ",0,"-1/6*(6*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^2*c*x^2 + 3*a*c*x - 2*c)*sqrt(-a^2*x^2 + 1))/a","A",0
887,1,86,0,0.614113," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{16 \, a x + 18 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{3} x^{3} + a^{2} x^{2} + 7 \, a x - 16\right)} \sqrt{-a^{2} x^{2} + 1} - 16}{6 \, {\left(a^{6} c x - a^{5} c\right)}}"," ",0,"1/6*(16*a*x + 18*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*x^3 + a^2*x^2 + 7*a*x - 16)*sqrt(-a^2*x^2 + 1) - 16)/(a^6*c*x - a^5*c)","A",0
888,1,77,0,0.500359," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{4 \, a x + 6 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(a^{2} x^{2} + a x - 4\right)} \sqrt{-a^{2} x^{2} + 1} - 4}{2 \, {\left(a^{5} c x - a^{4} c\right)}}"," ",0,"1/2*(4*a*x + 6*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a^2*x^2 + a*x - 4)*sqrt(-a^2*x^2 + 1) - 4)/(a^5*c*x - a^4*c)","A",0
889,1,69,0,0.571762," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{2 \, a x + 2 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(a x - 2\right)} - 2}{a^{4} c x - a^{3} c}"," ",0,"(2*a*x + 2*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*(a*x - 2) - 2)/(a^4*c*x - a^3*c)","A",0
890,1,64,0,0.745975," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{a x + 2 \, {\left(a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - \sqrt{-a^{2} x^{2} + 1} - 1}{a^{3} c x - a^{2} c}"," ",0,"(a*x + 2*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1) - 1)/(a^3*c*x - a^2*c)","A",0
891,1,33,0,0.489074," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{a x - \sqrt{-a^{2} x^{2} + 1} - 1}{a^{2} c x - a c}"," ",0,"(a*x - sqrt(-a^2*x^2 + 1) - 1)/(a^2*c*x - a*c)","A",0
892,1,55,0,0.700355," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{a x + {\left(a x - 1\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \sqrt{-a^{2} x^{2} + 1} - 1}{a c x - c}"," ",0,"(a*x + (a*x - 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - sqrt(-a^2*x^2 + 1) - 1)/(a*c*x - c)","A",0
893,1,78,0,0.596457," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{a^{2} x^{2} - a x + {\left(a^{2} x^{2} - a x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \sqrt{-a^{2} x^{2} + 1} {\left(2 \, a x - 1\right)}}{a c x^{2} - c x}"," ",0,"(a^2*x^2 - a*x + (a^2*x^2 - a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - sqrt(-a^2*x^2 + 1)*(2*a*x - 1))/(a*c*x^2 - c*x)","A",0
894,1,99,0,0.571216," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(4 \, a^{2} x^{2} - a x - 1\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a c x^{3} - c x^{2}\right)}}"," ",0,"1/2*(2*a^3*x^3 - 2*a^2*x^2 + 3*(a^3*x^3 - a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (4*a^2*x^2 - a*x - 1)*sqrt(-a^2*x^2 + 1))/(a*c*x^3 - c*x^2)","A",0
895,1,107,0,0.638444," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{6 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 9 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(16 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - a x - 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a c x^{4} - c x^{3}\right)}}"," ",0,"1/6*(6*a^4*x^4 - 6*a^3*x^3 + 9*(a^4*x^4 - a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (16*a^3*x^3 - 7*a^2*x^2 - a*x - 2)*sqrt(-a^2*x^2 + 1))/(a*c*x^4 - c*x^3)","A",0
896,1,159,0,0.562368," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{32 \, a^{3} x^{3} - 32 \, a^{2} x^{2} - 32 \, a x + 30 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{5} x^{5} + a^{4} x^{4} + 11 \, a^{3} x^{3} - 31 \, a^{2} x^{2} - 17 \, a x + 32\right)} \sqrt{-a^{2} x^{2} + 1} + 32}{6 \, {\left(a^{10} c^{2} x^{3} - a^{9} c^{2} x^{2} - a^{8} c^{2} x + a^{7} c^{2}\right)}}"," ",0,"-1/6*(32*a^3*x^3 - 32*a^2*x^2 - 32*a*x + 30*(a^3*x^3 - a^2*x^2 - a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^5*x^5 + a^4*x^4 + 11*a^3*x^3 - 31*a^2*x^2 - 17*a*x + 32)*sqrt(-a^2*x^2 + 1) + 32)/(a^10*c^2*x^3 - a^9*c^2*x^2 - a^8*c^2*x + a^7*c^2)","A",0
897,1,152,0,0.725155," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{16 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 16 \, a x + 30 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 23 \, a^{2} x^{2} - a x + 16\right)} \sqrt{-a^{2} x^{2} + 1} + 16}{6 \, {\left(a^{9} c^{2} x^{3} - a^{8} c^{2} x^{2} - a^{7} c^{2} x + a^{6} c^{2}\right)}}"," ",0,"-1/6*(16*a^3*x^3 - 16*a^2*x^2 - 16*a*x + 30*(a^3*x^3 - a^2*x^2 - a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^4*x^4 + 3*a^3*x^3 - 23*a^2*x^2 - a*x + 16)*sqrt(-a^2*x^2 + 1) + 16)/(a^9*c^2*x^3 - a^8*c^2*x^2 - a^7*c^2*x + a^6*c^2)","A",0
898,1,144,0,0.678093," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{8 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 8 \, a x + 6 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} + 8}{3 \, {\left(a^{8} c^{2} x^{3} - a^{7} c^{2} x^{2} - a^{6} c^{2} x + a^{5} c^{2}\right)}}"," ",0,"-1/3*(8*a^3*x^3 - 8*a^2*x^2 - 8*a*x + 6*(a^3*x^3 - a^2*x^2 - a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^3*x^3 - 7*a^2*x^2 - 5*a*x + 8)*sqrt(-a^2*x^2 + 1) + 8)/(a^8*c^2*x^3 - a^7*c^2*x^2 - a^6*c^2*x + a^5*c^2)","A",0
899,1,137,0,0.630481," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 2 \, a x + 6 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(4 \, a^{2} x^{2} - a x - 2\right)} \sqrt{-a^{2} x^{2} + 1} + 2}{3 \, {\left(a^{7} c^{2} x^{3} - a^{6} c^{2} x^{2} - a^{5} c^{2} x + a^{4} c^{2}\right)}}"," ",0,"-1/3*(2*a^3*x^3 - 2*a^2*x^2 - 2*a*x + 6*(a^3*x^3 - a^2*x^2 - a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (4*a^2*x^2 - a*x - 2)*sqrt(-a^2*x^2 + 1) + 2)/(a^7*c^2*x^3 - a^6*c^2*x^2 - a^5*c^2*x + a^4*c^2)","B",0
900,1,91,0,0.495276," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 2 \, a x - {\left(a^{2} x^{2} + 2 \, a x - 2\right)} \sqrt{-a^{2} x^{2} + 1} + 2}{3 \, {\left(a^{6} c^{2} x^{3} - a^{5} c^{2} x^{2} - a^{4} c^{2} x + a^{3} c^{2}\right)}}"," ",0,"-1/3*(2*a^3*x^3 - 2*a^2*x^2 - 2*a*x - (a^2*x^2 + 2*a*x - 2)*sqrt(-a^2*x^2 + 1) + 2)/(a^6*c^2*x^3 - a^5*c^2*x^2 - a^4*c^2*x + a^3*c^2)","A",0
901,1,89,0,0.524099," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{a^{3} x^{3} - a^{2} x^{2} - a x + {\left(a^{2} x^{2} - a x + 1\right)} \sqrt{-a^{2} x^{2} + 1} + 1}{3 \, {\left(a^{5} c^{2} x^{3} - a^{4} c^{2} x^{2} - a^{3} c^{2} x + a^{2} c^{2}\right)}}"," ",0,"1/3*(a^3*x^3 - a^2*x^2 - a*x + (a^2*x^2 - a*x + 1)*sqrt(-a^2*x^2 + 1) + 1)/(a^5*c^2*x^3 - a^4*c^2*x^2 - a^3*c^2*x + a^2*c^2)","A",0
902,1,89,0,0.547699," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{a^{3} x^{3} - a^{2} x^{2} - a x - {\left(2 \, a^{2} x^{2} - 2 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1} + 1}{3 \, {\left(a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} - a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"1/3*(a^3*x^3 - a^2*x^2 - a*x - (2*a^2*x^2 - 2*a*x - 1)*sqrt(-a^2*x^2 + 1) + 1)/(a^4*c^2*x^3 - a^3*c^2*x^2 - a^2*c^2*x + a*c^2)","B",0
903,1,127,0,0.739944," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{4 \, a^{3} x^{3} - 4 \, a^{2} x^{2} - 4 \, a x + 3 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(2 \, a^{2} x^{2} + a x - 4\right)} \sqrt{-a^{2} x^{2} + 1} + 4}{3 \, {\left(a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - a c^{2} x + c^{2}\right)}}"," ",0,"1/3*(4*a^3*x^3 - 4*a^2*x^2 - 4*a*x + 3*(a^3*x^3 - a^2*x^2 - a*x + 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (2*a^2*x^2 + a*x - 4)*sqrt(-a^2*x^2 + 1) + 4)/(a^3*c^2*x^3 - a^2*c^2*x^2 - a*c^2*x + c^2)","A",0
904,1,153,0,0.636488," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{4 \, a^{4} x^{4} - 4 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + 4 \, a x + 3 \, {\left(a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(8 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 7 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1}}{3 \, {\left(a^{3} c^{2} x^{4} - a^{2} c^{2} x^{3} - a c^{2} x^{2} + c^{2} x\right)}}"," ",0,"1/3*(4*a^4*x^4 - 4*a^3*x^3 - 4*a^2*x^2 + 4*a*x + 3*(a^4*x^4 - a^3*x^3 - a^2*x^2 + a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (8*a^3*x^3 - 5*a^2*x^2 - 7*a*x + 3)*sqrt(-a^2*x^2 + 1))/(a^3*c^2*x^4 - a^2*c^2*x^3 - a*c^2*x^2 + c^2*x)","A",0
905,1,171,0,0.735137," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{14 \, a^{5} x^{5} - 14 \, a^{4} x^{4} - 14 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 15 \, {\left(a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(16 \, a^{4} x^{4} - a^{3} x^{3} - 23 \, a^{2} x^{2} + 3 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{3} c^{2} x^{5} - a^{2} c^{2} x^{4} - a c^{2} x^{3} + c^{2} x^{2}\right)}}"," ",0,"1/6*(14*a^5*x^5 - 14*a^4*x^4 - 14*a^3*x^3 + 14*a^2*x^2 + 15*(a^5*x^5 - a^4*x^4 - a^3*x^3 + a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (16*a^4*x^4 - a^3*x^3 - 23*a^2*x^2 + 3*a*x + 3)*sqrt(-a^2*x^2 + 1))/(a^3*c^2*x^5 - a^2*c^2*x^4 - a*c^2*x^3 + c^2*x^2)","A",0
906,1,178,0,0.842890," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{14 \, a^{6} x^{6} - 14 \, a^{5} x^{5} - 14 \, a^{4} x^{4} + 14 \, a^{3} x^{3} + 15 \, {\left(a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(32 \, a^{5} x^{5} - 17 \, a^{4} x^{4} - 31 \, a^{3} x^{3} + 11 \, a^{2} x^{2} + a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{3} c^{2} x^{6} - a^{2} c^{2} x^{5} - a c^{2} x^{4} + c^{2} x^{3}\right)}}"," ",0,"1/6*(14*a^6*x^6 - 14*a^5*x^5 - 14*a^4*x^4 + 14*a^3*x^3 + 15*(a^6*x^6 - a^5*x^5 - a^4*x^4 + a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (32*a^5*x^5 - 17*a^4*x^4 - 31*a^3*x^3 + 11*a^2*x^2 + a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^3*c^2*x^6 - a^2*c^2*x^5 - a*c^2*x^4 + c^2*x^3)","A",0
907,1,221,0,0.739880," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^7/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{96 \, a^{5} x^{5} - 96 \, a^{4} x^{4} - 192 \, a^{3} x^{3} + 192 \, a^{2} x^{2} + 96 \, a x + 210 \, {\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(15 \, a^{6} x^{6} + 15 \, a^{5} x^{5} - 176 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 249 \, a^{2} x^{2} - 9 \, a x - 96\right)} \sqrt{-a^{2} x^{2} + 1} - 96}{30 \, {\left(a^{13} c^{3} x^{5} - a^{12} c^{3} x^{4} - 2 \, a^{11} c^{3} x^{3} + 2 \, a^{10} c^{3} x^{2} + a^{9} c^{3} x - a^{8} c^{3}\right)}}"," ",0,"1/30*(96*a^5*x^5 - 96*a^4*x^4 - 192*a^3*x^3 + 192*a^2*x^2 + 96*a*x + 210*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^6*x^6 + 15*a^5*x^5 - 176*a^4*x^4 - 4*a^3*x^3 + 249*a^2*x^2 - 9*a*x - 96)*sqrt(-a^2*x^2 + 1) - 96)/(a^13*c^3*x^5 - a^12*c^3*x^4 - 2*a^11*c^3*x^3 + 2*a^10*c^3*x^2 + a^9*c^3*x - a^8*c^3)","A",0
908,1,213,0,0.605928," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{48 \, a^{5} x^{5} - 48 \, a^{4} x^{4} - 96 \, a^{3} x^{3} + 96 \, a^{2} x^{2} + 48 \, a x + 30 \, {\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(15 \, a^{5} x^{5} - 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} + 87 \, a^{2} x^{2} + 33 \, a x - 48\right)} \sqrt{-a^{2} x^{2} + 1} - 48}{15 \, {\left(a^{12} c^{3} x^{5} - a^{11} c^{3} x^{4} - 2 \, a^{10} c^{3} x^{3} + 2 \, a^{9} c^{3} x^{2} + a^{8} c^{3} x - a^{7} c^{3}\right)}}"," ",0,"1/15*(48*a^5*x^5 - 48*a^4*x^4 - 96*a^3*x^3 + 96*a^2*x^2 + 48*a*x + 30*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^5*x^5 - 38*a^4*x^4 - 52*a^3*x^3 + 87*a^2*x^2 + 33*a*x - 48)*sqrt(-a^2*x^2 + 1) - 48)/(a^12*c^3*x^5 - a^11*c^3*x^4 - 2*a^10*c^3*x^3 + 2*a^9*c^3*x^2 + a^8*c^3*x - a^7*c^3)","A",0
909,1,206,0,0.721473," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{8 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 8 \, a x + 30 \, {\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(23 \, a^{4} x^{4} - 8 \, a^{3} x^{3} - 27 \, a^{2} x^{2} + 7 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} - 8}{15 \, {\left(a^{11} c^{3} x^{5} - a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{8} c^{3} x^{2} + a^{7} c^{3} x - a^{6} c^{3}\right)}}"," ",0,"1/15*(8*a^5*x^5 - 8*a^4*x^4 - 16*a^3*x^3 + 16*a^2*x^2 + 8*a*x + 30*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (23*a^4*x^4 - 8*a^3*x^3 - 27*a^2*x^2 + 7*a*x + 8)*sqrt(-a^2*x^2 + 1) - 8)/(a^11*c^3*x^5 - a^10*c^3*x^4 - 2*a^9*c^3*x^3 + 2*a^8*c^3*x^2 + a^7*c^3*x - a^6*c^3)","B",0
910,1,146,0,0.747332," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{8 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 8 \, a x - {\left(3 \, a^{4} x^{4} + 12 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 8 \, a x + 8\right)} \sqrt{-a^{2} x^{2} + 1} - 8}{15 \, {\left(a^{10} c^{3} x^{5} - a^{9} c^{3} x^{4} - 2 \, a^{8} c^{3} x^{3} + 2 \, a^{7} c^{3} x^{2} + a^{6} c^{3} x - a^{5} c^{3}\right)}}"," ",0,"1/15*(8*a^5*x^5 - 8*a^4*x^4 - 16*a^3*x^3 + 16*a^2*x^2 + 8*a*x - (3*a^4*x^4 + 12*a^3*x^3 - 12*a^2*x^2 - 8*a*x + 8)*sqrt(-a^2*x^2 + 1) - 8)/(a^10*c^3*x^5 - a^9*c^3*x^4 - 2*a^8*c^3*x^3 + 2*a^7*c^3*x^2 + a^6*c^3*x - a^5*c^3)","B",0
911,1,145,0,0.553661," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{2 \, a^{5} x^{5} - 2 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + 2 \, a x + {\left(3 \, a^{4} x^{4} - 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 \, a x - 2\right)} \sqrt{-a^{2} x^{2} + 1} - 2}{15 \, {\left(a^{9} c^{3} x^{5} - a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{6} c^{3} x^{2} + a^{5} c^{3} x - a^{4} c^{3}\right)}}"," ",0,"-1/15*(2*a^5*x^5 - 2*a^4*x^4 - 4*a^3*x^3 + 4*a^2*x^2 + 2*a*x + (3*a^4*x^4 - 3*a^3*x^3 + 3*a^2*x^2 + 2*a*x - 2)*sqrt(-a^2*x^2 + 1) - 2)/(a^9*c^3*x^5 - a^8*c^3*x^4 - 2*a^7*c^3*x^3 + 2*a^6*c^3*x^2 + a^5*c^3*x - a^4*c^3)","B",0
912,1,146,0,0.804513," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{2 \, a^{5} x^{5} - 2 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + 2 \, a x - {\left(2 \, a^{4} x^{4} - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} - 2}{15 \, {\left(a^{8} c^{3} x^{5} - a^{7} c^{3} x^{4} - 2 \, a^{6} c^{3} x^{3} + 2 \, a^{5} c^{3} x^{2} + a^{4} c^{3} x - a^{3} c^{3}\right)}}"," ",0,"-1/15*(2*a^5*x^5 - 2*a^4*x^4 - 4*a^3*x^3 + 4*a^2*x^2 + 2*a*x - (2*a^4*x^4 - 2*a^3*x^3 - 3*a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1) - 2)/(a^8*c^3*x^5 - a^7*c^3*x^4 - 2*a^6*c^3*x^3 + 2*a^5*c^3*x^2 + a^4*c^3*x - a^3*c^3)","A",0
913,1,145,0,0.682785," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 3 \, a x + {\left(2 \, a^{4} x^{4} - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 3\right)} \sqrt{-a^{2} x^{2} + 1} - 3}{15 \, {\left(a^{7} c^{3} x^{5} - a^{6} c^{3} x^{4} - 2 \, a^{5} c^{3} x^{3} + 2 \, a^{4} c^{3} x^{2} + a^{3} c^{3} x - a^{2} c^{3}\right)}}"," ",0,"1/15*(3*a^5*x^5 - 3*a^4*x^4 - 6*a^3*x^3 + 6*a^2*x^2 + 3*a*x + (2*a^4*x^4 - 2*a^3*x^3 - 3*a^2*x^2 + 3*a*x - 3)*sqrt(-a^2*x^2 + 1) - 3)/(a^7*c^3*x^5 - a^6*c^3*x^4 - 2*a^5*c^3*x^3 + 2*a^4*c^3*x^2 + a^3*c^3*x - a^2*c^3)","B",0
914,1,144,0,0.555702," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 3 \, a x - {\left(8 \, a^{4} x^{4} - 8 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 12 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1} - 3}{15 \, {\left(a^{6} c^{3} x^{5} - a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"1/15*(3*a^5*x^5 - 3*a^4*x^4 - 6*a^3*x^3 + 6*a^2*x^2 + 3*a*x - (8*a^4*x^4 - 8*a^3*x^3 - 12*a^2*x^2 + 12*a*x + 3)*sqrt(-a^2*x^2 + 1) - 3)/(a^6*c^3*x^5 - a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 + a^2*c^3*x - a*c^3)","B",0
915,1,198,0,0.605717," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{23 \, a^{5} x^{5} - 23 \, a^{4} x^{4} - 46 \, a^{3} x^{3} + 46 \, a^{2} x^{2} + 23 \, a x + 15 \, {\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(8 \, a^{4} x^{4} + 7 \, a^{3} x^{3} - 27 \, a^{2} x^{2} - 8 \, a x + 23\right)} \sqrt{-a^{2} x^{2} + 1} - 23}{15 \, {\left(a^{5} c^{3} x^{5} - a^{4} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} + a c^{3} x - c^{3}\right)}}"," ",0,"1/15*(23*a^5*x^5 - 23*a^4*x^4 - 46*a^3*x^3 + 46*a^2*x^2 + 23*a*x + 15*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (8*a^4*x^4 + 7*a^3*x^3 - 27*a^2*x^2 - 8*a*x + 23)*sqrt(-a^2*x^2 + 1) - 23)/(a^5*c^3*x^5 - a^4*c^3*x^4 - 2*a^3*c^3*x^3 + 2*a^2*c^3*x^2 + a*c^3*x - c^3)","B",0
916,1,223,0,0.651520," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{23 \, a^{6} x^{6} - 23 \, a^{5} x^{5} - 46 \, a^{4} x^{4} + 46 \, a^{3} x^{3} + 23 \, a^{2} x^{2} - 23 \, a x + 15 \, {\left(a^{6} x^{6} - a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + a^{2} x^{2} - a x\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(48 \, a^{5} x^{5} - 33 \, a^{4} x^{4} - 87 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 38 \, a x - 15\right)} \sqrt{-a^{2} x^{2} + 1}}{15 \, {\left(a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{3} + a c^{3} x^{2} - c^{3} x\right)}}"," ",0,"1/15*(23*a^6*x^6 - 23*a^5*x^5 - 46*a^4*x^4 + 46*a^3*x^3 + 23*a^2*x^2 - 23*a*x + 15*(a^6*x^6 - a^5*x^5 - 2*a^4*x^4 + 2*a^3*x^3 + a^2*x^2 - a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (48*a^5*x^5 - 33*a^4*x^4 - 87*a^3*x^3 + 52*a^2*x^2 + 38*a*x - 15)*sqrt(-a^2*x^2 + 1))/(a^5*c^3*x^6 - a^4*c^3*x^5 - 2*a^3*c^3*x^4 + 2*a^2*c^3*x^3 + a*c^3*x^2 - c^3*x)","A",0
917,1,241,0,0.748801," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{116 \, a^{7} x^{7} - 116 \, a^{6} x^{6} - 232 \, a^{5} x^{5} + 232 \, a^{4} x^{4} + 116 \, a^{3} x^{3} - 116 \, a^{2} x^{2} + 105 \, {\left(a^{7} x^{7} - a^{6} x^{6} - 2 \, a^{5} x^{5} + 2 \, a^{4} x^{4} + a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(96 \, a^{6} x^{6} + 9 \, a^{5} x^{5} - 249 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 176 \, a^{2} x^{2} - 15 \, a x - 15\right)} \sqrt{-a^{2} x^{2} + 1}}{30 \, {\left(a^{5} c^{3} x^{7} - a^{4} c^{3} x^{6} - 2 \, a^{3} c^{3} x^{5} + 2 \, a^{2} c^{3} x^{4} + a c^{3} x^{3} - c^{3} x^{2}\right)}}"," ",0,"1/30*(116*a^7*x^7 - 116*a^6*x^6 - 232*a^5*x^5 + 232*a^4*x^4 + 116*a^3*x^3 - 116*a^2*x^2 + 105*(a^7*x^7 - a^6*x^6 - 2*a^5*x^5 + 2*a^4*x^4 + a^3*x^3 - a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (96*a^6*x^6 + 9*a^5*x^5 - 249*a^4*x^4 + 4*a^3*x^3 + 176*a^2*x^2 - 15*a*x - 15)*sqrt(-a^2*x^2 + 1))/(a^5*c^3*x^7 - a^4*c^3*x^6 - 2*a^3*c^3*x^5 + 2*a^2*c^3*x^4 + a*c^3*x^3 - c^3*x^2)","A",0
918,1,198,0,0.735829," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","\frac{5 \, a^{7} x^{7} - 5 \, a^{6} x^{6} - 15 \, a^{5} x^{5} + 15 \, a^{4} x^{4} + 15 \, a^{3} x^{3} - 15 \, a^{2} x^{2} - 5 \, a x - {\left(16 \, a^{6} x^{6} - 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} + 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} - 30 \, a x - 5\right)} \sqrt{-a^{2} x^{2} + 1} + 5}{35 \, {\left(a^{8} c^{4} x^{7} - a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"1/35*(5*a^7*x^7 - 5*a^6*x^6 - 15*a^5*x^5 + 15*a^4*x^4 + 15*a^3*x^3 - 15*a^2*x^2 - 5*a*x - (16*a^6*x^6 - 16*a^5*x^5 - 40*a^4*x^4 + 40*a^3*x^3 + 30*a^2*x^2 - 30*a*x - 5)*sqrt(-a^2*x^2 + 1) + 5)/(a^8*c^4*x^7 - a^7*c^4*x^6 - 3*a^6*c^4*x^5 + 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 - 3*a^3*c^4*x^2 - a^2*c^4*x + a*c^4)","B",0
919,1,252,0,0.789512," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x, algorithm=""fricas"")","\frac{35 \, a^{9} x^{9} - 35 \, a^{8} x^{8} - 140 \, a^{7} x^{7} + 140 \, a^{6} x^{6} + 210 \, a^{5} x^{5} - 210 \, a^{4} x^{4} - 140 \, a^{3} x^{3} + 140 \, a^{2} x^{2} + 35 \, a x - {\left(128 \, a^{8} x^{8} - 128 \, a^{7} x^{7} - 448 \, a^{6} x^{6} + 448 \, a^{5} x^{5} + 560 \, a^{4} x^{4} - 560 \, a^{3} x^{3} - 280 \, a^{2} x^{2} + 280 \, a x + 35\right)} \sqrt{-a^{2} x^{2} + 1} - 35}{315 \, {\left(a^{10} c^{5} x^{9} - a^{9} c^{5} x^{8} - 4 \, a^{8} c^{5} x^{7} + 4 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 4 \, a^{3} c^{5} x^{2} + a^{2} c^{5} x - a c^{5}\right)}}"," ",0,"1/315*(35*a^9*x^9 - 35*a^8*x^8 - 140*a^7*x^7 + 140*a^6*x^6 + 210*a^5*x^5 - 210*a^4*x^4 - 140*a^3*x^3 + 140*a^2*x^2 + 35*a*x - (128*a^8*x^8 - 128*a^7*x^7 - 448*a^6*x^6 + 448*a^5*x^5 + 560*a^4*x^4 - 560*a^3*x^3 - 280*a^2*x^2 + 280*a*x + 35)*sqrt(-a^2*x^2 + 1) - 35)/(a^10*c^5*x^9 - a^9*c^5*x^8 - 4*a^8*c^5*x^7 + 4*a^7*c^5*x^6 + 6*a^6*c^5*x^5 - 6*a^5*c^5*x^4 - 4*a^4*c^5*x^3 + 4*a^3*c^5*x^2 + a^2*c^5*x - a*c^5)","B",0
920,1,42,0,0.486786," ","integrate((a*x+1)/(-a^2*x^2+1)*x^4,x, algorithm=""fricas"")","-\frac{3 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 12 \, a x + 12 \, \log\left(a x - 1\right)}{12 \, a^{5}}"," ",0,"-1/12*(3*a^4*x^4 + 4*a^3*x^3 + 6*a^2*x^2 + 12*a*x + 12*log(a*x - 1))/a^5","A",0
921,1,34,0,0.594940," ","integrate((a*x+1)/(-a^2*x^2+1)*x^3,x, algorithm=""fricas"")","-\frac{2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x + 6 \, \log\left(a x - 1\right)}{6 \, a^{4}}"," ",0,"-1/6*(2*a^3*x^3 + 3*a^2*x^2 + 6*a*x + 6*log(a*x - 1))/a^4","A",0
922,1,25,0,0.540346," ","integrate((a*x+1)/(-a^2*x^2+1)*x^2,x, algorithm=""fricas"")","-\frac{a^{2} x^{2} + 2 \, a x + 2 \, \log\left(a x - 1\right)}{2 \, a^{3}}"," ",0,"-1/2*(a^2*x^2 + 2*a*x + 2*log(a*x - 1))/a^3","A",0
923,1,15,0,0.535280," ","integrate((a*x+1)/(-a^2*x^2+1)*x,x, algorithm=""fricas"")","-\frac{a x + \log\left(a x - 1\right)}{a^{2}}"," ",0,"-(a*x + log(a*x - 1))/a^2","A",0
924,1,11,0,0.592647," ","integrate((a*x+1)/(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{\log\left(a x - 1\right)}{a}"," ",0,"-log(a*x - 1)/a","A",0
925,1,11,0,0.541552," ","integrate((a*x+1)/(-a^2*x^2+1)/x,x, algorithm=""fricas"")","-\log\left(a x - 1\right) + \log\left(x\right)"," ",0,"-log(a*x - 1) + log(x)","A",0
926,1,22,0,0.582717," ","integrate((a*x+1)/(-a^2*x^2+1)/x^2,x, algorithm=""fricas"")","-\frac{a x \log\left(a x - 1\right) - a x \log\left(x\right) + 1}{x}"," ",0,"-(a*x*log(a*x - 1) - a*x*log(x) + 1)/x","A",0
927,1,35,0,0.440535," ","integrate((a*x+1)/(-a^2*x^2+1)/x^3,x, algorithm=""fricas"")","-\frac{2 \, a^{2} x^{2} \log\left(a x - 1\right) - 2 \, a^{2} x^{2} \log\left(x\right) + 2 \, a x + 1}{2 \, x^{2}}"," ",0,"-1/2*(2*a^2*x^2*log(a*x - 1) - 2*a^2*x^2*log(x) + 2*a*x + 1)/x^2","A",0
928,1,43,0,0.758105," ","integrate((a*x+1)/(-a^2*x^2+1)/x^4,x, algorithm=""fricas"")","-\frac{6 \, a^{3} x^{3} \log\left(a x - 1\right) - 6 \, a^{3} x^{3} \log\left(x\right) + 6 \, a^{2} x^{2} + 3 \, a x + 2}{6 \, x^{3}}"," ",0,"-1/6*(6*a^3*x^3*log(a*x - 1) - 6*a^3*x^3*log(x) + 6*a^2*x^2 + 3*a*x + 2)/x^3","A",0
929,1,62,0,0.611386," ","integrate((a*x+1)/(-a^2*x^2+1)^2*x^4,x, algorithm=""fricas"")","\frac{2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 4 \, a x + {\left(a x - 1\right)} \log\left(a x + 1\right) + 7 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 2}{4 \, {\left(a^{6} x - a^{5}\right)}}"," ",0,"1/4*(2*a^3*x^3 + 2*a^2*x^2 - 4*a*x + (a*x - 1)*log(a*x + 1) + 7*(a*x - 1)*log(a*x - 1) - 2)/(a^6*x - a^5)","A",0
930,1,55,0,0.567685," ","integrate((a*x+1)/(-a^2*x^2+1)^2*x^3,x, algorithm=""fricas"")","\frac{4 \, a^{2} x^{2} - 4 \, a x - {\left(a x - 1\right)} \log\left(a x + 1\right) + 5 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 2}{4 \, {\left(a^{5} x - a^{4}\right)}}"," ",0,"1/4*(4*a^2*x^2 - 4*a*x - (a*x - 1)*log(a*x + 1) + 5*(a*x - 1)*log(a*x - 1) - 2)/(a^5*x - a^4)","A",0
931,1,42,0,0.754636," ","integrate((a*x+1)/(-a^2*x^2+1)^2*x^2,x, algorithm=""fricas"")","\frac{{\left(a x - 1\right)} \log\left(a x + 1\right) + 3 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 2}{4 \, {\left(a^{4} x - a^{3}\right)}}"," ",0,"1/4*((a*x - 1)*log(a*x + 1) + 3*(a*x - 1)*log(a*x - 1) - 2)/(a^4*x - a^3)","A",0
932,1,42,0,0.507000," ","integrate((a*x+1)/(-a^2*x^2+1)^2*x,x, algorithm=""fricas"")","-\frac{{\left(a x - 1\right)} \log\left(a x + 1\right) - {\left(a x - 1\right)} \log\left(a x - 1\right) + 2}{4 \, {\left(a^{3} x - a^{2}\right)}}"," ",0,"-1/4*((a*x - 1)*log(a*x + 1) - (a*x - 1)*log(a*x - 1) + 2)/(a^3*x - a^2)","A",0
933,1,40,0,0.584665," ","integrate((a*x+1)/(-a^2*x^2+1)^2,x, algorithm=""fricas"")","\frac{{\left(a x - 1\right)} \log\left(a x + 1\right) - {\left(a x - 1\right)} \log\left(a x - 1\right) - 2}{4 \, {\left(a^{2} x - a\right)}}"," ",0,"1/4*((a*x - 1)*log(a*x + 1) - (a*x - 1)*log(a*x - 1) - 2)/(a^2*x - a)","A",0
934,1,45,0,0.667381," ","integrate((a*x+1)/(-a^2*x^2+1)^2/x,x, algorithm=""fricas"")","-\frac{{\left(a x - 1\right)} \log\left(a x + 1\right) + 3 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 4 \, {\left(a x - 1\right)} \log\left(x\right) + 2}{4 \, {\left(a x - 1\right)}}"," ",0,"-1/4*((a*x - 1)*log(a*x + 1) + 3*(a*x - 1)*log(a*x - 1) - 4*(a*x - 1)*log(x) + 2)/(a*x - 1)","A",0
935,1,75,0,0.644385," ","integrate((a*x+1)/(-a^2*x^2+1)^2/x^2,x, algorithm=""fricas"")","-\frac{6 \, a x - {\left(a^{2} x^{2} - a x\right)} \log\left(a x + 1\right) + 5 \, {\left(a^{2} x^{2} - a x\right)} \log\left(a x - 1\right) - 4 \, {\left(a^{2} x^{2} - a x\right)} \log\left(x\right) - 4}{4 \, {\left(a x^{2} - x\right)}}"," ",0,"-1/4*(6*a*x - (a^2*x^2 - a*x)*log(a*x + 1) + 5*(a^2*x^2 - a*x)*log(a*x - 1) - 4*(a^2*x^2 - a*x)*log(x) - 4)/(a*x^2 - x)","A",0
936,1,96,0,0.623044," ","integrate((a*x+1)/(-a^2*x^2+1)^2/x^3,x, algorithm=""fricas"")","-\frac{6 \, a^{2} x^{2} - 2 \, a x + {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(a x + 1\right) + 7 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(a x - 1\right) - 8 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(x\right) - 2}{4 \, {\left(a x^{3} - x^{2}\right)}}"," ",0,"-1/4*(6*a^2*x^2 - 2*a*x + (a^3*x^3 - a^2*x^2)*log(a*x + 1) + 7*(a^3*x^3 - a^2*x^2)*log(a*x - 1) - 8*(a^3*x^3 - a^2*x^2)*log(x) - 2)/(a*x^3 - x^2)","A",0
937,1,105,0,0.592892," ","integrate((a*x+1)/(-a^2*x^2+1)^2/x^4,x, algorithm=""fricas"")","-\frac{30 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 2 \, a x - 3 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(a x + 1\right) + 27 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(a x - 1\right) - 24 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(x\right) - 4}{12 \, {\left(a x^{4} - x^{3}\right)}}"," ",0,"-1/12*(30*a^3*x^3 - 18*a^2*x^2 - 2*a*x - 3*(a^4*x^4 - a^3*x^3)*log(a*x + 1) + 27*(a^4*x^4 - a^3*x^3)*log(a*x - 1) - 24*(a^4*x^4 - a^3*x^3)*log(x) - 4)/(a*x^4 - x^3)","A",0
938,1,125,0,0.656232," ","integrate((a*x+1)/(-a^2*x^2+1)^3*x^6,x, algorithm=""fricas"")","-\frac{8 \, a^{5} x^{5} + 8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 10 \, a x + 9 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x + 1\right) + 39 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x - 1\right) + 20}{16 \, {\left(a^{10} x^{3} - a^{9} x^{2} - a^{8} x + a^{7}\right)}}"," ",0,"-1/16*(8*a^5*x^5 + 8*a^4*x^4 - 24*a^3*x^3 - 26*a^2*x^2 + 10*a*x + 9*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) + 39*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) + 20)/(a^10*x^3 - a^9*x^2 - a^8*x + a^7)","A",0
939,1,117,0,0.539536," ","integrate((a*x+1)/(-a^2*x^2+1)^3*x^5,x, algorithm=""fricas"")","-\frac{16 \, a^{4} x^{4} - 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} + 18 \, a x - 7 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x + 1\right) + 23 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x - 1\right) + 12}{16 \, {\left(a^{9} x^{3} - a^{8} x^{2} - a^{7} x + a^{6}\right)}}"," ",0,"-1/16*(16*a^4*x^4 - 16*a^3*x^3 - 34*a^2*x^2 + 18*a*x - 7*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) + 23*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) + 12)/(a^9*x^3 - a^8*x^2 - a^7*x + a^6)","A",0
940,1,101,0,0.499960," ","integrate((a*x+1)/(-a^2*x^2+1)^3*x^4,x, algorithm=""fricas"")","\frac{10 \, a^{2} x^{2} + 6 \, a x - 5 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x + 1\right) - 11 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x - 1\right) - 12}{16 \, {\left(a^{8} x^{3} - a^{7} x^{2} - a^{6} x + a^{5}\right)}}"," ",0,"1/16*(10*a^2*x^2 + 6*a*x - 5*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) - 11*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) - 12)/(a^8*x^3 - a^7*x^2 - a^6*x + a^5)","A",0
941,1,101,0,0.735845," ","integrate((a*x+1)/(-a^2*x^2+1)^3*x^3,x, algorithm=""fricas"")","\frac{10 \, a^{2} x^{2} - 2 \, a x + 3 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x + 1\right) - 3 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x - 1\right) - 4}{16 \, {\left(a^{7} x^{3} - a^{6} x^{2} - a^{5} x + a^{4}\right)}}"," ",0,"1/16*(10*a^2*x^2 - 2*a*x + 3*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) - 3*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) - 4)/(a^7*x^3 - a^6*x^2 - a^5*x + a^4)","B",0
942,1,100,0,0.719438," ","integrate((a*x+1)/(-a^2*x^2+1)^3*x^2,x, algorithm=""fricas"")","\frac{2 \, a^{2} x^{2} + 6 \, a x - {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x + 1\right) + {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x - 1\right) - 4}{16 \, {\left(a^{6} x^{3} - a^{5} x^{2} - a^{4} x + a^{3}\right)}}"," ",0,"1/16*(2*a^2*x^2 + 6*a*x - (a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) + (a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) - 4)/(a^6*x^3 - a^5*x^2 - a^4*x + a^3)","B",0
943,1,100,0,0.527346," ","integrate((a*x+1)/(-a^2*x^2+1)^3*x,x, algorithm=""fricas"")","\frac{2 \, a^{2} x^{2} - 2 \, a x - {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x + 1\right) + {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x - 1\right) + 4}{16 \, {\left(a^{5} x^{3} - a^{4} x^{2} - a^{3} x + a^{2}\right)}}"," ",0,"1/16*(2*a^2*x^2 - 2*a*x - (a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) + (a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) + 4)/(a^5*x^3 - a^4*x^2 - a^3*x + a^2)","B",0
944,1,99,0,0.601591," ","integrate((a*x+1)/(-a^2*x^2+1)^3,x, algorithm=""fricas"")","-\frac{6 \, a^{2} x^{2} - 6 \, a x - 3 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x + 1\right) + 3 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x - 1\right) - 4}{16 \, {\left(a^{4} x^{3} - a^{3} x^{2} - a^{2} x + a\right)}}"," ",0,"-1/16*(6*a^2*x^2 - 6*a*x - 3*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) + 3*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) - 4)/(a^4*x^3 - a^3*x^2 - a^2*x + a)","B",0
945,1,122,0,0.591650," ","integrate((a*x+1)/(-a^2*x^2+1)^3/x,x, algorithm=""fricas"")","-\frac{6 \, a^{2} x^{2} + 2 \, a x + 5 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x + 1\right) + 11 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(a x - 1\right) - 16 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \log\left(x\right) - 12}{16 \, {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)}}"," ",0,"-1/16*(6*a^2*x^2 + 2*a*x + 5*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) + 11*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) - 16*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(x) - 12)/(a^3*x^3 - a^2*x^2 - a*x + 1)","B",0
946,1,150,0,0.653476," ","integrate((a*x+1)/(-a^2*x^2+1)^3/x^2,x, algorithm=""fricas"")","-\frac{30 \, a^{3} x^{3} - 22 \, a^{2} x^{2} - 28 \, a x - 7 \, {\left(a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right)} \log\left(a x + 1\right) + 23 \, {\left(a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right)} \log\left(a x - 1\right) - 16 \, {\left(a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right)} \log\left(x\right) + 16}{16 \, {\left(a^{3} x^{4} - a^{2} x^{3} - a x^{2} + x\right)}}"," ",0,"-1/16*(30*a^3*x^3 - 22*a^2*x^2 - 28*a*x - 7*(a^4*x^4 - a^3*x^3 - a^2*x^2 + a*x)*log(a*x + 1) + 23*(a^4*x^4 - a^3*x^3 - a^2*x^2 + a*x)*log(a*x - 1) - 16*(a^4*x^4 - a^3*x^3 - a^2*x^2 + a*x)*log(x) + 16)/(a^3*x^4 - a^2*x^3 - a*x^2 + x)","B",0
947,1,172,0,0.688524," ","integrate((a*x+1)/(-a^2*x^2+1)^3/x^3,x, algorithm=""fricas"")","-\frac{30 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - 44 \, a^{2} x^{2} + 8 \, a x + 9 \, {\left(a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(a x + 1\right) + 39 \, {\left(a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(a x - 1\right) - 48 \, {\left(a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(x\right) + 8}{16 \, {\left(a^{3} x^{5} - a^{2} x^{4} - a x^{3} + x^{2}\right)}}"," ",0,"-1/16*(30*a^4*x^4 - 6*a^3*x^3 - 44*a^2*x^2 + 8*a*x + 9*(a^5*x^5 - a^4*x^4 - a^3*x^3 + a^2*x^2)*log(a*x + 1) + 39*(a^5*x^5 - a^4*x^4 - a^3*x^3 + a^2*x^2)*log(a*x - 1) - 48*(a^5*x^5 - a^4*x^4 - a^3*x^3 + a^2*x^2)*log(x) + 8)/(a^3*x^5 - a^2*x^4 - a*x^3 + x^2)","B",0
948,1,180,0,0.688967," ","integrate((a*x+1)/(-a^2*x^2+1)^3/x^4,x, algorithm=""fricas"")","-\frac{210 \, a^{5} x^{5} - 138 \, a^{4} x^{4} - 212 \, a^{3} x^{3} + 104 \, a^{2} x^{2} + 8 \, a x - 33 \, {\left(a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(a x + 1\right) + 177 \, {\left(a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(a x - 1\right) - 144 \, {\left(a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(x\right) + 16}{48 \, {\left(a^{3} x^{6} - a^{2} x^{5} - a x^{4} + x^{3}\right)}}"," ",0,"-1/48*(210*a^5*x^5 - 138*a^4*x^4 - 212*a^3*x^3 + 104*a^2*x^2 + 8*a*x - 33*(a^6*x^6 - a^5*x^5 - a^4*x^4 + a^3*x^3)*log(a*x + 1) + 177*(a^6*x^6 - a^5*x^5 - a^4*x^4 + a^3*x^3)*log(a*x - 1) - 144*(a^6*x^6 - a^5*x^5 - a^4*x^4 + a^3*x^3)*log(x) + 16)/(a^3*x^6 - a^2*x^5 - a*x^4 + x^3)","B",0
949,1,50,0,0.672031," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} c x^{2} + c} {\left(3 \, a x^{4} + 4 \, x^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{12 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/12*sqrt(-a^2*c*x^2 + c)*(3*a*x^4 + 4*x^3)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
950,1,50,0,0.556226," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(2 \, a x^{3} + 3 \, x^{2}\right)}}{6 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/6*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(2*a*x^3 + 3*x^2)/(a^2*x^2 - 1)","A",0
951,1,47,0,0.890407," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x^{2} + 2 \, x\right)}}{2 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x^2 + 2*x)/(a^2*x^2 - 1)","A",0
952,1,262,0,0.868942," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x^{4} - 1\right)} \sqrt{c} - c}{a^{2} x^{4} - x^{2}}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x - a\right)}}{2 \, {\left(a^{2} x^{2} - 1\right)}}, \frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x^{2} + 1\right)} \sqrt{-c}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x - a\right)}}{a^{2} x^{2} - 1}\right]"," ",0,"[1/2*((a^2*x^2 - 1)*sqrt(c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x^4 - 1)*sqrt(c) - c)/(a^2*x^4 - x^2)) - 2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x - a))/(a^2*x^2 - 1), ((a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x^2 + 1)*sqrt(-c)/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x - a))/(a^2*x^2 - 1)]","B",0
953,1,264,0,0.745204," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^(1/2)/x^2,x, algorithm=""fricas"")","\left[\frac{{\left(a^{3} x^{3} - a x\right)} \sqrt{c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x^{4} - 1\right)} \sqrt{c} - c}{a^{2} x^{4} - x^{2}}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x - 1\right)}}{2 \, {\left(a^{2} x^{3} - x\right)}}, \frac{{\left(a^{3} x^{3} - a x\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x^{2} + 1\right)} \sqrt{-c}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) - \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x - 1\right)}}{a^{2} x^{3} - x}\right]"," ",0,"[1/2*((a^3*x^3 - a*x)*sqrt(c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x^4 - 1)*sqrt(c) - c)/(a^2*x^4 - x^2)) - 2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x - 1))/(a^2*x^3 - x), ((a^3*x^3 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x^2 + 1)*sqrt(-c)/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x - 1))/(a^2*x^3 - x)]","B",0
954,1,68,0,0.807148," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\frac{{\left(3 \, a^{3} c x^{4} + 4 \, a^{2} c x^{3} - 6 \, a c x^{2} - 12 \, c x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{12 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/12*(3*a^3*c*x^4 + 4*a^2*c*x^3 - 6*a*c*x^2 - 12*c*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
955,1,98,0,0.873085," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(5 \, a^{5} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{5} - 15 \, a^{3} c^{2} x^{4} - 20 \, a^{2} c^{2} x^{3} + 15 \, a c^{2} x^{2} + 30 \, c^{2} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{30 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/30*(5*a^5*c^2*x^6 + 6*a^4*c^2*x^5 - 15*a^3*c^2*x^4 - 20*a^2*c^2*x^3 + 15*a*c^2*x^2 + 30*c^2*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
956,1,120,0,0.688033," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","\frac{{\left(35 \, a^{7} c^{3} x^{8} + 40 \, a^{6} c^{3} x^{7} - 140 \, a^{5} c^{3} x^{6} - 168 \, a^{4} c^{3} x^{5} + 210 \, a^{3} c^{3} x^{4} + 280 \, a^{2} c^{3} x^{3} - 140 \, a c^{3} x^{2} - 280 \, c^{3} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{280 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/280*(35*a^7*c^3*x^8 + 40*a^6*c^3*x^7 - 140*a^5*c^3*x^6 - 168*a^4*c^3*x^5 + 210*a^3*c^3*x^4 + 280*a^2*c^3*x^3 - 140*a*c^3*x^2 - 280*c^3*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
957,1,387,0,0.681864," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) + {\left(3 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 12 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{12 \, {\left(a^{7} c x^{2} - a^{5} c\right)}}, -\frac{12 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right) - {\left(3 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 12 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{12 \, {\left(a^{7} c x^{2} - a^{5} c\right)}}\right]"," ",0,"[1/12*(6*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) + (3*a^4*x^4 + 4*a^3*x^3 + 6*a^2*x^2 + 12*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^7*c*x^2 - a^5*c), -1/12*(12*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x)) - (3*a^4*x^4 + 4*a^3*x^3 + 6*a^2*x^2 + 12*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^7*c*x^2 - a^5*c)]","A",0
958,1,371,0,0.810431," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) + {\left(2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{6} c x^{2} - a^{4} c\right)}}, -\frac{6 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right) - {\left(2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{6} c x^{2} - a^{4} c\right)}}\right]"," ",0,"[1/6*(3*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) + (2*a^3*x^3 + 3*a^2*x^2 + 6*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*c*x^2 - a^4*c), -1/6*(6*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x)) - (2*a^3*x^3 + 3*a^2*x^2 + 6*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*c*x^2 - a^4*c)]","A",0
959,1,352,0,0.898856," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 2 \, a x\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a^{5} c x^{2} - a^{3} c\right)}}, -\frac{2 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 2 \, a x\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a^{5} c x^{2} - a^{3} c\right)}}\right]"," ",0,"[1/2*((a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) + sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 2*a*x)*sqrt(-a^2*x^2 + 1))/(a^5*c*x^2 - a^3*c), -1/2*(2*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x)) - sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 2*a*x)*sqrt(-a^2*x^2 + 1))/(a^5*c*x^2 - a^3*c)]","A",0
960,1,331,0,0.668518," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x + {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right)}{2 \, {\left(a^{4} c x^{2} - a^{2} c\right)}}, \frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x - {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right)}{a^{4} c x^{2} - a^{2} c}\right]"," ",0,"[1/2*(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x + (a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)))/(a^4*c*x^2 - a^2*c), (sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x - (a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x)))/(a^4*c*x^2 - a^2*c)]","B",0
961,1,227,0,0.573694," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right)}{2 \, a \sqrt{c}}, -\frac{\sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right)}{a c}\right]"," ",0,"[1/2*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1))/(a*sqrt(c)), -sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x))/(a*c)]","B",0
962,1,302,0,0.815210," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{3} x^{3} - {\left(4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right)}{2 \, \sqrt{c}}, -\frac{\sqrt{-c} \arctan\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a^{2} - 2 \, a + 1\right)} x^{2} - 2 \, a x + 1\right)} \sqrt{-c}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right)}{c}\right]"," ",0,"[1/2*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 - 6*a^2 + 4*a - 1)*x^4 - 6*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2))/sqrt(c), -sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 - 2*a + 1)*x^2 - 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c))/c]","A",0
963,1,416,0,0.828244," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{3} x^{3} - a x\right)} \sqrt{c} \log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{3} x^{3} - {\left(4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x - 1\right)}}{2 \, {\left(a^{2} c x^{3} - c x\right)}}, -\frac{{\left(a^{3} x^{3} - a x\right)} \sqrt{-c} \arctan\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a^{2} - 2 \, a + 1\right)} x^{2} - 2 \, a x + 1\right)} \sqrt{-c}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right) + \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x - 1\right)}}{a^{2} c x^{3} - c x}\right]"," ",0,"[1/2*((a^3*x^3 - a*x)*sqrt(c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 - 6*a^2 + 4*a - 1)*x^4 - 6*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) - 2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x - 1))/(a^2*c*x^3 - c*x), -((a^3*x^3 - a*x)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 - 2*a + 1)*x^2 - 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) + sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x - 1))/(a^2*c*x^3 - c*x)]","A",0
964,1,453,0,0.819075," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{c} \log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{3} x^{3} - {\left(4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right) - \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a + 1\right)} x^{2} - 2 \, a x - 1\right)}}{2 \, {\left(a^{2} c x^{4} - c x^{2}\right)}}, -\frac{2 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{-c} \arctan\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a^{2} - 2 \, a + 1\right)} x^{2} - 2 \, a x + 1\right)} \sqrt{-c}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right) + \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a + 1\right)} x^{2} - 2 \, a x - 1\right)}}{2 \, {\left(a^{2} c x^{4} - c x^{2}\right)}}\right]"," ",0,"[1/2*((a^4*x^4 - a^2*x^2)*sqrt(c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 - 6*a^2 + 4*a - 1)*x^4 - 6*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) - sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a + 1)*x^2 - 2*a*x - 1))/(a^2*c*x^4 - c*x^2), -1/2*(2*(a^4*x^4 - a^2*x^2)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 - 2*a + 1)*x^2 - 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) + sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a + 1)*x^2 - 2*a*x - 1))/(a^2*c*x^4 - c*x^2)]","A",0
965,1,482,0,0.892646," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{c} \log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{3} x^{3} - {\left(4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(6 \, a^{2} x^{2} - {\left(6 \, a^{2} + 3 \, a + 2\right)} x^{3} + 3 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{2} c x^{5} - c x^{3}\right)}}, -\frac{6 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{-c} \arctan\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a^{2} - 2 \, a + 1\right)} x^{2} - 2 \, a x + 1\right)} \sqrt{-c}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(6 \, a^{2} x^{2} - {\left(6 \, a^{2} + 3 \, a + 2\right)} x^{3} + 3 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{2} c x^{5} - c x^{3}\right)}}\right]"," ",0,"[1/6*(3*(a^5*x^5 - a^3*x^3)*sqrt(c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 - 6*a^2 + 4*a - 1)*x^4 - 6*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) + sqrt(-a^2*c*x^2 + c)*(6*a^2*x^2 - (6*a^2 + 3*a + 2)*x^3 + 3*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*c*x^5 - c*x^3), -1/6*(6*(a^5*x^5 - a^3*x^3)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 - 2*a + 1)*x^2 - 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) - sqrt(-a^2*c*x^2 + c)*(6*a^2*x^2 - (6*a^2 + 3*a + 2)*x^3 + 3*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*c*x^5 - c*x^3)]","A",0
966,0,0,0,1.083091," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{5}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^5/(a^5*c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)","F",0
967,0,0,0,0.775948," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{4}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^4/(a^5*c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)","F",0
968,0,0,0,0.894325," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{3}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^3/(a^5*c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)","F",0
969,0,0,0,0.724307," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^2/(a^5*c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)","F",0
970,1,348,0,1.265302," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x + {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} + 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right)}{8 \, {\left(a^{5} c^{2} x^{3} - a^{4} c^{2} x^{2} - a^{3} c^{2} x + a^{2} c^{2}\right)}}, \frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x - {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right)}{4 \, {\left(a^{5} c^{2} x^{3} - a^{4} c^{2} x^{2} - a^{3} c^{2} x + a^{2} c^{2}\right)}}\right]"," ",0,"[1/8*(4*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x + (a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 + 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)))/(a^5*c^2*x^3 - a^4*c^2*x^2 - a^3*c^2*x + a^2*c^2), 1/4*(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x - (a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)))/(a^5*c^2*x^3 - a^4*c^2*x^2 - a^3*c^2*x + a^2*c^2)]","A",0
971,1,343,0,0.756104," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x + {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right)}{8 \, {\left(a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} - a^{2} c^{2} x + a c^{2}\right)}}, \frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x + {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right)}{4 \, {\left(a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} - a^{2} c^{2} x + a c^{2}\right)}}\right]"," ",0,"[1/8*(4*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x + (a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)))/(a^4*c^2*x^3 - a^3*c^2*x^2 - a^2*c^2*x + a*c^2), 1/4*(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x + (a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)))/(a^4*c^2*x^3 - a^3*c^2*x^2 - a^2*c^2*x + a*c^2)]","A",0
972,0,0,0,0.872143," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{a^{5} c^{2} x^{6} - a^{4} c^{2} x^{5} - 2 \, a^{3} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{3} + a c^{2} x^{2} - c^{2} x}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^5*c^2*x^6 - a^4*c^2*x^5 - 2*a^3*c^2*x^4 + 2*a^2*c^2*x^3 + a*c^2*x^2 - c^2*x), x)","F",0
973,0,0,0,0.769238," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{a^{5} c^{2} x^{7} - a^{4} c^{2} x^{6} - 2 \, a^{3} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{4} + a c^{2} x^{3} - c^{2} x^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^5*c^2*x^7 - a^4*c^2*x^6 - 2*a^3*c^2*x^5 + 2*a^2*c^2*x^4 + a*c^2*x^3 - c^2*x^2), x)","F",0
974,0,0,0,0.859407," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{a^{5} c^{2} x^{8} - a^{4} c^{2} x^{7} - 2 \, a^{3} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{5} + a c^{2} x^{4} - c^{2} x^{3}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^5*c^2*x^8 - a^4*c^2*x^7 - 2*a^3*c^2*x^6 + 2*a^2*c^2*x^5 + a*c^2*x^4 - c^2*x^3), x)","F",0
975,0,0,0,0.767130," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{a^{5} c^{2} x^{9} - a^{4} c^{2} x^{8} - 2 \, a^{3} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{6} + a c^{2} x^{5} - c^{2} x^{4}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^5*c^2*x^9 - a^4*c^2*x^8 - 2*a^3*c^2*x^7 + 2*a^2*c^2*x^6 + a*c^2*x^5 - c^2*x^4), x)","F",0
976,0,0,0,0.848137," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{6}}{a^{7} c^{3} x^{7} - a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} - a c^{3} x + c^{3}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^6/(a^7*c^3*x^7 - a^6*c^3*x^6 - 3*a^5*c^3*x^5 + 3*a^4*c^3*x^4 + 3*a^3*c^3*x^3 - 3*a^2*c^3*x^2 - a*c^3*x + c^3), x)","F",0
977,0,0,0,0.634994," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{5}}{a^{7} c^{3} x^{7} - a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} - a c^{3} x + c^{3}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^5/(a^7*c^3*x^7 - a^6*c^3*x^6 - 3*a^5*c^3*x^5 + 3*a^4*c^3*x^4 + 3*a^3*c^3*x^3 - 3*a^2*c^3*x^2 - a*c^3*x + c^3), x)","F",0
978,0,0,0,0.744287," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{4}}{a^{7} c^{3} x^{7} - a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} - a c^{3} x + c^{3}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^4/(a^7*c^3*x^7 - a^6*c^3*x^6 - 3*a^5*c^3*x^5 + 3*a^4*c^3*x^4 + 3*a^3*c^3*x^3 - 3*a^2*c^3*x^2 - a*c^3*x + c^3), x)","F",0
979,1,461,0,0.687052," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) - 4 \, {\left(2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - 3 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{32 \, {\left(a^{9} c^{3} x^{5} - a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{6} c^{3} x^{2} + a^{5} c^{3} x - a^{4} c^{3}\right)}}, \frac{3 \, {\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) - 2 \, {\left(2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - 3 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{16 \, {\left(a^{9} c^{3} x^{5} - a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{6} c^{3} x^{2} + a^{5} c^{3} x - a^{4} c^{3}\right)}}\right]"," ",0,"[1/32*(3*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) - 4*(2*a^3*x^3 + 3*a^2*x^2 - 3*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^9*c^3*x^5 - a^8*c^3*x^4 - 2*a^7*c^3*x^3 + 2*a^6*c^3*x^2 + a^5*c^3*x - a^4*c^3), 1/16*(3*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) - 2*(2*a^3*x^3 + 3*a^2*x^2 - 3*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^9*c^3*x^5 - a^8*c^3*x^4 - 2*a^7*c^3*x^3 + 2*a^6*c^3*x^2 + a^5*c^3*x - a^4*c^3)]","A",0
980,1,457,0,0.542798," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} + 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) - 4 \, {\left(2 \, a^{3} x^{3} - a^{2} x^{2} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{32 \, {\left(a^{8} c^{3} x^{5} - a^{7} c^{3} x^{4} - 2 \, a^{6} c^{3} x^{3} + 2 \, a^{5} c^{3} x^{2} + a^{4} c^{3} x - a^{3} c^{3}\right)}}, -\frac{{\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) + 2 \, {\left(2 \, a^{3} x^{3} - a^{2} x^{2} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{16 \, {\left(a^{8} c^{3} x^{5} - a^{7} c^{3} x^{4} - 2 \, a^{6} c^{3} x^{3} + 2 \, a^{5} c^{3} x^{2} + a^{4} c^{3} x - a^{3} c^{3}\right)}}\right]"," ",0,"[1/32*((a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 + 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) - 4*(2*a^3*x^3 - a^2*x^2 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^3*x^5 - a^7*c^3*x^4 - 2*a^6*c^3*x^3 + 2*a^5*c^3*x^2 + a^4*c^3*x - a^3*c^3), -1/16*((a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) + 2*(2*a^3*x^3 - a^2*x^2 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^3*x^5 - a^7*c^3*x^4 - 2*a^6*c^3*x^3 + 2*a^5*c^3*x^2 + a^4*c^3*x - a^3*c^3)]","A",0
981,1,459,0,0.674905," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} + 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) + 4 \, {\left(2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{32 \, {\left(a^{7} c^{3} x^{5} - a^{6} c^{3} x^{4} - 2 \, a^{5} c^{3} x^{3} + 2 \, a^{4} c^{3} x^{2} + a^{3} c^{3} x - a^{2} c^{3}\right)}}, -\frac{{\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) - 2 \, {\left(2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{16 \, {\left(a^{7} c^{3} x^{5} - a^{6} c^{3} x^{4} - 2 \, a^{5} c^{3} x^{3} + 2 \, a^{4} c^{3} x^{2} + a^{3} c^{3} x - a^{2} c^{3}\right)}}\right]"," ",0,"[1/32*((a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 + 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) + 4*(2*a^3*x^3 - 3*a^2*x^2 - a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^7*c^3*x^5 - a^6*c^3*x^4 - 2*a^5*c^3*x^3 + 2*a^4*c^3*x^2 + a^3*c^3*x - a^2*c^3), -1/16*((a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) - 2*(2*a^3*x^3 - 3*a^2*x^2 - a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^7*c^3*x^5 - a^6*c^3*x^4 - 2*a^5*c^3*x^3 + 2*a^4*c^3*x^2 + a^3*c^3*x - a^2*c^3)]","A",0
982,1,455,0,0.709441," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) + 4 \, {\left(2 \, a^{3} x^{3} + a^{2} x^{2} - 5 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{32 \, {\left(a^{6} c^{3} x^{5} - a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x - a c^{3}\right)}}, \frac{3 \, {\left(a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) + 2 \, {\left(2 \, a^{3} x^{3} + a^{2} x^{2} - 5 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{16 \, {\left(a^{6} c^{3} x^{5} - a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x - a c^{3}\right)}}\right]"," ",0,"[1/32*(3*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) + 4*(2*a^3*x^3 + a^2*x^2 - 5*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*c^3*x^5 - a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 + a^2*c^3*x - a*c^3), 1/16*(3*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) + 2*(2*a^3*x^3 + a^2*x^2 - 5*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*c^3*x^5 - a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 + a^2*c^3*x - a*c^3)]","A",0
983,0,0,0,0.799057," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{a^{7} c^{3} x^{8} - a^{6} c^{3} x^{7} - 3 \, a^{5} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{3} c^{3} x^{4} - 3 \, a^{2} c^{3} x^{3} - a c^{3} x^{2} + c^{3} x}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^7*c^3*x^8 - a^6*c^3*x^7 - 3*a^5*c^3*x^6 + 3*a^4*c^3*x^5 + 3*a^3*c^3*x^4 - 3*a^2*c^3*x^3 - a*c^3*x^2 + c^3*x), x)","F",0
984,0,0,0,0.761887," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{a^{7} c^{3} x^{9} - a^{6} c^{3} x^{8} - 3 \, a^{5} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{3} c^{3} x^{5} - 3 \, a^{2} c^{3} x^{4} - a c^{3} x^{3} + c^{3} x^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^7*c^3*x^9 - a^6*c^3*x^8 - 3*a^5*c^3*x^7 + 3*a^4*c^3*x^6 + 3*a^3*c^3*x^5 - 3*a^2*c^3*x^4 - a*c^3*x^3 + c^3*x^2), x)","F",0
985,0,0,0,0.850216," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{a^{7} c^{3} x^{10} - a^{6} c^{3} x^{9} - 3 \, a^{5} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{3} c^{3} x^{6} - 3 \, a^{2} c^{3} x^{5} - a c^{3} x^{4} + c^{3} x^{3}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^7*c^3*x^10 - a^6*c^3*x^9 - 3*a^5*c^3*x^8 + 3*a^4*c^3*x^7 + 3*a^3*c^3*x^6 - 3*a^2*c^3*x^5 - a*c^3*x^4 + c^3*x^3), x)","F",0
986,1,567,0,1.201292," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{7} x^{7} - a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) + 4 \, {\left(8 \, a^{5} x^{5} + 7 \, a^{4} x^{4} - 31 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 33 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{192 \, {\left(a^{8} c^{4} x^{7} - a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right)}}, \frac{15 \, {\left(a^{7} x^{7} - a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x + 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) + 2 \, {\left(8 \, a^{5} x^{5} + 7 \, a^{4} x^{4} - 31 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 33 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{96 \, {\left(a^{8} c^{4} x^{7} - a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right)}}\right]"," ",0,"[1/192*(15*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2*x^2 - a*x + 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) + 4*(8*a^5*x^5 + 7*a^4*x^4 - 31*a^3*x^3 - 9*a^2*x^2 + 33*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 - a^7*c^4*x^6 - 3*a^6*c^4*x^5 + 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 - 3*a^3*c^4*x^2 - a^2*c^4*x + a*c^4), 1/96*(15*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2*x^2 - a*x + 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) + 2*(8*a^5*x^5 + 7*a^4*x^4 - 31*a^3*x^3 - 9*a^2*x^2 + 33*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 - a^7*c^4*x^6 - 3*a^6*c^4*x^5 + 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 - 3*a^3*c^4*x^2 - a^2*c^4*x + a*c^4)]","A",0
987,0,0,0,0.578382," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{3} c^{2} x^{3} + a^{2} c^{2} x^{2} - a c^{2} x - c^{2}\right)} \sqrt{-a^{2} x^{2} + 1} x^{m}, x\right)"," ",0,"integral(-(a^3*c^2*x^3 + a^2*c^2*x^2 - a*c^2*x - c^2)*sqrt(-a^2*x^2 + 1)*x^m, x)","F",0
988,0,0,0,0.756551," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m*(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-a^{2} x^{2} + 1} {\left(a c x + c\right)} x^{m}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*c*x + c)*x^m, x)","F",0
989,0,0,0,0.720890," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m/(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} x^{m}}{a^{3} c x^{3} - a^{2} c x^{2} - a c x + c}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*x^m/(a^3*c*x^3 - a^2*c*x^2 - a*c*x + c), x)","F",0
990,0,0,0,0.720145," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} x^{m}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*x^m/(a^5*c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)","F",0
991,0,0,0,0.741165," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} x^{m}}{a^{7} c^{3} x^{7} - a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} - a c^{3} x + c^{3}}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*x^m/(a^7*c^3*x^7 - a^6*c^3*x^6 - 3*a^5*c^3*x^5 + 3*a^4*c^3*x^4 + 3*a^3*c^3*x^3 - 3*a^2*c^3*x^2 - a*c^3*x + c^3), x)","F",0
992,1,285,0,0.670459," ","integrate((a*x+1)*(-a^2*x^2+1)^2*x^m,x, algorithm=""fricas"")","\frac{{\left({\left(a^{5} m^{5} + 15 \, a^{5} m^{4} + 85 \, a^{5} m^{3} + 225 \, a^{5} m^{2} + 274 \, a^{5} m + 120 \, a^{5}\right)} x^{6} + {\left(a^{4} m^{5} + 16 \, a^{4} m^{4} + 95 \, a^{4} m^{3} + 260 \, a^{4} m^{2} + 324 \, a^{4} m + 144 \, a^{4}\right)} x^{5} - 2 \, {\left(a^{3} m^{5} + 17 \, a^{3} m^{4} + 107 \, a^{3} m^{3} + 307 \, a^{3} m^{2} + 396 \, a^{3} m + 180 \, a^{3}\right)} x^{4} - 2 \, {\left(a^{2} m^{5} + 18 \, a^{2} m^{4} + 121 \, a^{2} m^{3} + 372 \, a^{2} m^{2} + 508 \, a^{2} m + 240 \, a^{2}\right)} x^{3} + {\left(a m^{5} + 19 \, a m^{4} + 137 \, a m^{3} + 461 \, a m^{2} + 702 \, a m + 360 \, a\right)} x^{2} + {\left(m^{5} + 20 \, m^{4} + 155 \, m^{3} + 580 \, m^{2} + 1044 \, m + 720\right)} x\right)} x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720}"," ",0,"((a^5*m^5 + 15*a^5*m^4 + 85*a^5*m^3 + 225*a^5*m^2 + 274*a^5*m + 120*a^5)*x^6 + (a^4*m^5 + 16*a^4*m^4 + 95*a^4*m^3 + 260*a^4*m^2 + 324*a^4*m + 144*a^4)*x^5 - 2*(a^3*m^5 + 17*a^3*m^4 + 107*a^3*m^3 + 307*a^3*m^2 + 396*a^3*m + 180*a^3)*x^4 - 2*(a^2*m^5 + 18*a^2*m^4 + 121*a^2*m^3 + 372*a^2*m^2 + 508*a^2*m + 240*a^2)*x^3 + (a*m^5 + 19*a*m^4 + 137*a*m^3 + 461*a*m^2 + 702*a*m + 360*a)*x^2 + (m^5 + 20*m^4 + 155*m^3 + 580*m^2 + 1044*m + 720)*x)*x^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)","B",0
993,1,128,0,0.504717," ","integrate((a*x+1)*(-a^2*x^2+1)*x^m,x, algorithm=""fricas"")","-\frac{{\left({\left(a^{3} m^{3} + 6 \, a^{3} m^{2} + 11 \, a^{3} m + 6 \, a^{3}\right)} x^{4} + {\left(a^{2} m^{3} + 7 \, a^{2} m^{2} + 14 \, a^{2} m + 8 \, a^{2}\right)} x^{3} - {\left(a m^{3} + 8 \, a m^{2} + 19 \, a m + 12 \, a\right)} x^{2} - {\left(m^{3} + 9 \, m^{2} + 26 \, m + 24\right)} x\right)} x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24}"," ",0,"-((a^3*m^3 + 6*a^3*m^2 + 11*a^3*m + 6*a^3)*x^4 + (a^2*m^3 + 7*a^2*m^2 + 14*a^2*m + 8*a^2)*x^3 - (a*m^3 + 8*a*m^2 + 19*a*m + 12*a)*x^2 - (m^3 + 9*m^2 + 26*m + 24)*x)*x^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)","B",0
994,1,29,0,0.607789," ","integrate((a*x+1)*x^m,x, algorithm=""fricas"")","\frac{{\left({\left(a m + a\right)} x^{2} + {\left(m + 2\right)} x\right)} x^{m}}{m^{2} + 3 \, m + 2}"," ",0,"((a*m + a)*x^2 + (m + 2)*x)*x^m/(m^2 + 3*m + 2)","A",0
995,0,0,0,0.641286," ","integrate((a*x+1)/(-a^2*x^2+1)*x^m,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{x^{m}}{a x - 1}, x\right)"," ",0,"integral(-x^m/(a*x - 1), x)","F",0
996,0,0,0,0.666814," ","integrate((a*x+1)/(-a^2*x^2+1)^2*x^m,x, algorithm=""fricas"")","{\rm integral}\left(\frac{x^{m}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}, x\right)"," ",0,"integral(x^m/(a^3*x^3 - a^2*x^2 - a*x + 1), x)","F",0
997,0,0,0,0.708322," ","integrate((a*x+1)/(-a^2*x^2+1)^3*x^m,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{x^{m}}{a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1}, x\right)"," ",0,"integral(-x^m/(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1), x)","F",0
998,1,476,0,0.516938," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m*(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\frac{{\left({\left(a^{5} c^{2} m^{5} + 15 \, a^{5} c^{2} m^{4} + 85 \, a^{5} c^{2} m^{3} + 225 \, a^{5} c^{2} m^{2} + 274 \, a^{5} c^{2} m + 120 \, a^{5} c^{2}\right)} x^{6} + {\left(a^{4} c^{2} m^{5} + 16 \, a^{4} c^{2} m^{4} + 95 \, a^{4} c^{2} m^{3} + 260 \, a^{4} c^{2} m^{2} + 324 \, a^{4} c^{2} m + 144 \, a^{4} c^{2}\right)} x^{5} - 2 \, {\left(a^{3} c^{2} m^{5} + 17 \, a^{3} c^{2} m^{4} + 107 \, a^{3} c^{2} m^{3} + 307 \, a^{3} c^{2} m^{2} + 396 \, a^{3} c^{2} m + 180 \, a^{3} c^{2}\right)} x^{4} - 2 \, {\left(a^{2} c^{2} m^{5} + 18 \, a^{2} c^{2} m^{4} + 121 \, a^{2} c^{2} m^{3} + 372 \, a^{2} c^{2} m^{2} + 508 \, a^{2} c^{2} m + 240 \, a^{2} c^{2}\right)} x^{3} + {\left(a c^{2} m^{5} + 19 \, a c^{2} m^{4} + 137 \, a c^{2} m^{3} + 461 \, a c^{2} m^{2} + 702 \, a c^{2} m + 360 \, a c^{2}\right)} x^{2} + {\left(c^{2} m^{5} + 20 \, c^{2} m^{4} + 155 \, c^{2} m^{3} + 580 \, c^{2} m^{2} + 1044 \, c^{2} m + 720 \, c^{2}\right)} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} - {\left(a^{2} m^{6} + 21 \, a^{2} m^{5} + 175 \, a^{2} m^{4} + 735 \, a^{2} m^{3} + 1624 \, a^{2} m^{2} + 1764 \, a^{2} m + 720 \, a^{2}\right)} x^{2} + 1624 \, m^{2} + 1764 \, m + 720}"," ",0,"((a^5*c^2*m^5 + 15*a^5*c^2*m^4 + 85*a^5*c^2*m^3 + 225*a^5*c^2*m^2 + 274*a^5*c^2*m + 120*a^5*c^2)*x^6 + (a^4*c^2*m^5 + 16*a^4*c^2*m^4 + 95*a^4*c^2*m^3 + 260*a^4*c^2*m^2 + 324*a^4*c^2*m + 144*a^4*c^2)*x^5 - 2*(a^3*c^2*m^5 + 17*a^3*c^2*m^4 + 107*a^3*c^2*m^3 + 307*a^3*c^2*m^2 + 396*a^3*c^2*m + 180*a^3*c^2)*x^4 - 2*(a^2*c^2*m^5 + 18*a^2*c^2*m^4 + 121*a^2*c^2*m^3 + 372*a^2*c^2*m^2 + 508*a^2*c^2*m + 240*a^2*c^2)*x^3 + (a*c^2*m^5 + 19*a*c^2*m^4 + 137*a*c^2*m^3 + 461*a*c^2*m^2 + 702*a*c^2*m + 360*a*c^2)*x^2 + (c^2*m^5 + 20*c^2*m^4 + 155*c^2*m^3 + 580*c^2*m^2 + 1044*c^2*m + 720*c^2)*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 - (a^2*m^6 + 21*a^2*m^5 + 175*a^2*m^4 + 735*a^2*m^3 + 1624*a^2*m^2 + 1764*a^2*m + 720*a^2)*x^2 + 1624*m^2 + 1764*m + 720)","A",0
999,1,211,0,0.741915," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m*(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} c x^{2} + c} {\left({\left(a^{3} c m^{3} + 6 \, a^{3} c m^{2} + 11 \, a^{3} c m + 6 \, a^{3} c\right)} x^{4} + {\left(a^{2} c m^{3} + 7 \, a^{2} c m^{2} + 14 \, a^{2} c m + 8 \, a^{2} c\right)} x^{3} - {\left(a c m^{3} + 8 \, a c m^{2} + 19 \, a c m + 12 \, a c\right)} x^{2} - {\left(c m^{3} + 9 \, c m^{2} + 26 \, c m + 24 \, c\right)} x\right)} \sqrt{-a^{2} x^{2} + 1} x^{m}}{m^{4} + 10 \, m^{3} - {\left(a^{2} m^{4} + 10 \, a^{2} m^{3} + 35 \, a^{2} m^{2} + 50 \, a^{2} m + 24 \, a^{2}\right)} x^{2} + 35 \, m^{2} + 50 \, m + 24}"," ",0,"-sqrt(-a^2*c*x^2 + c)*((a^3*c*m^3 + 6*a^3*c*m^2 + 11*a^3*c*m + 6*a^3*c)*x^4 + (a^2*c*m^3 + 7*a^2*c*m^2 + 14*a^2*c*m + 8*a^2*c)*x^3 - (a*c*m^3 + 8*a*c*m^2 + 19*a*c*m + 12*a*c)*x^2 - (c*m^3 + 9*c*m^2 + 26*c*m + 24*c)*x)*sqrt(-a^2*x^2 + 1)*x^m/(m^4 + 10*m^3 - (a^2*m^4 + 10*a^2*m^3 + 35*a^2*m^2 + 50*a^2*m + 24*a^2)*x^2 + 35*m^2 + 50*m + 24)","A",0
1000,1,80,0,0.731438," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(a m + a\right)} x^{2} + {\left(m + 2\right)} x\right)} x^{m}}{{\left(a^{2} m^{2} + 3 \, a^{2} m + 2 \, a^{2}\right)} x^{2} - m^{2} - 3 \, m - 2}"," ",0,"-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((a*m + a)*x^2 + (m + 2)*x)*x^m/((a^2*m^2 + 3*a^2*m + 2*a^2)*x^2 - m^2 - 3*m - 2)","A",0
1001,0,0,0,0.796909," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{m}}{a^{3} c x^{3} - a^{2} c x^{2} - a c x + c}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^m/(a^3*c*x^3 - a^2*c*x^2 - a*c*x + c), x)","F",0
1002,0,0,0,0.617786," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{m}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^m/(a^5*c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)","F",0
1003,0,0,0,0.549356," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{m}}{a^{7} c^{3} x^{7} - a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} - a c^{3} x + c^{3}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^m/(a^7*c^3*x^7 - a^6*c^3*x^6 - 3*a^5*c^3*x^5 + 3*a^4*c^3*x^4 + 3*a^3*c^3*x^3 - 3*a^2*c^3*x^2 - a*c^3*x + c^3), x)","F",0
1004,0,0,0,0.571286," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^m*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p} x^{m}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p*x^m/(a*x - 1), x)","F",0
1005,0,0,0,0.594562," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a^2*x^2+1)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p} x^{3}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p*x^3/(a*x - 1), x)","F",0
1006,0,0,0,0.744771," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a^2*x^2+1)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p} x^{2}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p*x^2/(a*x - 1), x)","F",0
1007,0,0,0,0.578912," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a^2*x^2+1)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p} x}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p*x/(a*x - 1), x)","F",0
1008,0,0,0,0.633256," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p/(a*x - 1), x)","F",0
1009,0,0,0,0.592006," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p/x,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p}}{a x^{2} - x}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p/(a*x^2 - x), x)","F",0
1010,0,0,0,0.659003," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p/x^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p}}{a x^{3} - x^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p/(a*x^3 - x^2), x)","F",0
1011,0,0,0,0.821795," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p/x^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p}}{a x^{4} - x^{3}}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p/(a*x^4 - x^3), x)","F",0
1012,0,0,0,0.564514," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p} x^{3}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p*x^3/(a*x - 1), x)","F",0
1013,0,0,0,0.734622," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p} x^{2}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p*x^2/(a*x - 1), x)","F",0
1014,0,0,0,0.766018," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p} x}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p*x/(a*x - 1), x)","F",0
1015,0,0,0,0.731880," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p/(a*x - 1), x)","F",0
1016,0,0,0,0.676989," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p/x,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a x^{2} - x}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p/(a*x^2 - x), x)","F",0
1017,0,0,0,0.625315," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p/x^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a x^{3} - x^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p/(a*x^3 - x^2), x)","F",0
1018,0,0,0,0.610712," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p/x^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a x^{4} - x^{3}}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p/(a*x^4 - x^3), x)","F",0
1019,1,23,0,0.525990," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^4*(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{1}{7} \, a^{2} c x^{7} + \frac{1}{3} \, a c x^{6} + \frac{1}{5} \, c x^{5}"," ",0,"1/7*a^2*c*x^7 + 1/3*a*c*x^6 + 1/5*c*x^5","A",0
1020,1,23,0,0.627154," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{1}{6} \, a^{2} c x^{6} + \frac{2}{5} \, a c x^{5} + \frac{1}{4} \, c x^{4}"," ",0,"1/6*a^2*c*x^6 + 2/5*a*c*x^5 + 1/4*c*x^4","A",0
1021,1,23,0,0.518964," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{1}{5} \, a^{2} c x^{5} + \frac{1}{2} \, a c x^{4} + \frac{1}{3} \, c x^{3}"," ",0,"1/5*a^2*c*x^5 + 1/2*a*c*x^4 + 1/3*c*x^3","A",0
1022,1,23,0,0.566347," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x*(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{1}{4} \, a^{2} c x^{4} + \frac{2}{3} \, a c x^{3} + \frac{1}{2} \, c x^{2}"," ",0,"1/4*a^2*c*x^4 + 2/3*a*c*x^3 + 1/2*c*x^2","A",0
1023,1,19,0,0.667158," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{1}{3} \, a^{2} c x^{3} + a c x^{2} + c x"," ",0,"1/3*a^2*c*x^3 + a*c*x^2 + c*x","A",0
1024,1,19,0,1.022949," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)/x,x, algorithm=""fricas"")","\frac{1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log\left(x\right)"," ",0,"1/2*a^2*c*x^2 + 2*a*c*x + c*log(x)","A",0
1025,1,23,0,0.612478," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)/x^2,x, algorithm=""fricas"")","\frac{a^{2} c x^{2} + 2 \, a c x \log\left(x\right) - c}{x}"," ",0,"(a^2*c*x^2 + 2*a*c*x*log(x) - c)/x","A",0
1026,1,25,0,1.005454," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)/x^3,x, algorithm=""fricas"")","\frac{2 \, a^{2} c x^{2} \log\left(x\right) - 4 \, a c x - c}{2 \, x^{2}}"," ",0,"1/2*(2*a^2*c*x^2*log(x) - 4*a*c*x - c)/x^2","A",0
1027,1,21,0,0.476033," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)/x^4,x, algorithm=""fricas"")","-\frac{3 \, a^{2} c x^{2} + 3 \, a c x + c}{3 \, x^{3}}"," ",0,"-1/3*(3*a^2*c*x^2 + 3*a*c*x + c)/x^3","A",0
1028,1,40,0,0.522643," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^4*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{1}{9} \, a^{4} c^{2} x^{9} - \frac{1}{4} \, a^{3} c^{2} x^{8} + \frac{1}{3} \, a c^{2} x^{6} + \frac{1}{5} \, c^{2} x^{5}"," ",0,"-1/9*a^4*c^2*x^9 - 1/4*a^3*c^2*x^8 + 1/3*a*c^2*x^6 + 1/5*c^2*x^5","A",0
1029,1,40,0,0.538110," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{1}{8} \, a^{4} c^{2} x^{8} - \frac{2}{7} \, a^{3} c^{2} x^{7} + \frac{2}{5} \, a c^{2} x^{5} + \frac{1}{4} \, c^{2} x^{4}"," ",0,"-1/8*a^4*c^2*x^8 - 2/7*a^3*c^2*x^7 + 2/5*a*c^2*x^5 + 1/4*c^2*x^4","A",0
1030,1,40,0,0.660365," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{1}{7} \, a^{4} c^{2} x^{7} - \frac{1}{3} \, a^{3} c^{2} x^{6} + \frac{1}{2} \, a c^{2} x^{4} + \frac{1}{3} \, c^{2} x^{3}"," ",0,"-1/7*a^4*c^2*x^7 - 1/3*a^3*c^2*x^6 + 1/2*a*c^2*x^4 + 1/3*c^2*x^3","A",0
1031,1,40,0,0.548420," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{1}{6} \, a^{4} c^{2} x^{6} - \frac{2}{5} \, a^{3} c^{2} x^{5} + \frac{2}{3} \, a c^{2} x^{3} + \frac{1}{2} \, c^{2} x^{2}"," ",0,"-1/6*a^4*c^2*x^6 - 2/5*a^3*c^2*x^5 + 2/3*a*c^2*x^3 + 1/2*c^2*x^2","A",0
1032,1,36,0,0.709583," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{1}{5} \, a^{4} c^{2} x^{5} - \frac{1}{2} \, a^{3} c^{2} x^{4} + a c^{2} x^{2} + c^{2} x"," ",0,"-1/5*a^4*c^2*x^5 - 1/2*a^3*c^2*x^4 + a*c^2*x^2 + c^2*x","A",0
1033,1,36,0,0.620705," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2/x,x, algorithm=""fricas"")","-\frac{1}{4} \, a^{4} c^{2} x^{4} - \frac{2}{3} \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x + c^{2} \log\left(x\right)"," ",0,"-1/4*a^4*c^2*x^4 - 2/3*a^3*c^2*x^3 + 2*a*c^2*x + c^2*log(x)","A",0
1034,1,41,0,0.663886," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2/x^2,x, algorithm=""fricas"")","-\frac{a^{4} c^{2} x^{4} + 3 \, a^{3} c^{2} x^{3} - 6 \, a c^{2} x \log\left(x\right) + 3 \, c^{2}}{3 \, x}"," ",0,"-1/3*(a^4*c^2*x^4 + 3*a^3*c^2*x^3 - 6*a*c^2*x*log(x) + 3*c^2)/x","A",0
1035,1,37,0,0.505781," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2/x^3,x, algorithm=""fricas"")","-\frac{a^{4} c^{2} x^{4} + 4 \, a^{3} c^{2} x^{3} + 4 \, a c^{2} x + c^{2}}{2 \, x^{2}}"," ",0,"-1/2*(a^4*c^2*x^4 + 4*a^3*c^2*x^3 + 4*a*c^2*x + c^2)/x^2","B",0
1036,1,40,0,0.879669," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2/x^4,x, algorithm=""fricas"")","-\frac{3 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} \log\left(x\right) + 3 \, a c^{2} x + c^{2}}{3 \, x^{3}}"," ",0,"-1/3*(3*a^4*c^2*x^4 + 6*a^3*c^2*x^3*log(x) + 3*a*c^2*x + c^2)/x^3","A",0
1037,1,42,0,0.666658," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2/x^5,x, algorithm=""fricas"")","-\frac{12 \, a^{4} c^{2} x^{4} \log\left(x\right) - 24 \, a^{3} c^{2} x^{3} + 8 \, a c^{2} x + 3 \, c^{2}}{12 \, x^{4}}"," ",0,"-1/12*(12*a^4*c^2*x^4*log(x) - 24*a^3*c^2*x^3 + 8*a*c^2*x + 3*c^2)/x^4","A",0
1038,1,40,0,0.542303," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^2/x^6,x, algorithm=""fricas"")","\frac{10 \, a^{4} c^{2} x^{4} + 10 \, a^{3} c^{2} x^{3} - 5 \, a c^{2} x - 2 \, c^{2}}{10 \, x^{5}}"," ",0,"1/10*(10*a^4*c^2*x^4 + 10*a^3*c^2*x^3 - 5*a*c^2*x - 2*c^2)/x^5","A",0
1039,1,73,0,0.722457," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^4*(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{1}{11} \, a^{6} c^{3} x^{11} + \frac{1}{5} \, a^{5} c^{3} x^{10} - \frac{1}{9} \, a^{4} c^{3} x^{9} - \frac{1}{2} \, a^{3} c^{3} x^{8} - \frac{1}{7} \, a^{2} c^{3} x^{7} + \frac{1}{3} \, a c^{3} x^{6} + \frac{1}{5} \, c^{3} x^{5}"," ",0,"1/11*a^6*c^3*x^11 + 1/5*a^5*c^3*x^10 - 1/9*a^4*c^3*x^9 - 1/2*a^3*c^3*x^8 - 1/7*a^2*c^3*x^7 + 1/3*a*c^3*x^6 + 1/5*c^3*x^5","A",0
1040,1,73,0,0.585344," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{1}{10} \, a^{6} c^{3} x^{10} + \frac{2}{9} \, a^{5} c^{3} x^{9} - \frac{1}{8} \, a^{4} c^{3} x^{8} - \frac{4}{7} \, a^{3} c^{3} x^{7} - \frac{1}{6} \, a^{2} c^{3} x^{6} + \frac{2}{5} \, a c^{3} x^{5} + \frac{1}{4} \, c^{3} x^{4}"," ",0,"1/10*a^6*c^3*x^10 + 2/9*a^5*c^3*x^9 - 1/8*a^4*c^3*x^8 - 4/7*a^3*c^3*x^7 - 1/6*a^2*c^3*x^6 + 2/5*a*c^3*x^5 + 1/4*c^3*x^4","A",0
1041,1,73,0,0.625730," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{1}{9} \, a^{6} c^{3} x^{9} + \frac{1}{4} \, a^{5} c^{3} x^{8} - \frac{1}{7} \, a^{4} c^{3} x^{7} - \frac{2}{3} \, a^{3} c^{3} x^{6} - \frac{1}{5} \, a^{2} c^{3} x^{5} + \frac{1}{2} \, a c^{3} x^{4} + \frac{1}{3} \, c^{3} x^{3}"," ",0,"1/9*a^6*c^3*x^9 + 1/4*a^5*c^3*x^8 - 1/7*a^4*c^3*x^7 - 2/3*a^3*c^3*x^6 - 1/5*a^2*c^3*x^5 + 1/2*a*c^3*x^4 + 1/3*c^3*x^3","A",0
1042,1,73,0,0.485410," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x*(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{1}{8} \, a^{6} c^{3} x^{8} + \frac{2}{7} \, a^{5} c^{3} x^{7} - \frac{1}{6} \, a^{4} c^{3} x^{6} - \frac{4}{5} \, a^{3} c^{3} x^{5} - \frac{1}{4} \, a^{2} c^{3} x^{4} + \frac{2}{3} \, a c^{3} x^{3} + \frac{1}{2} \, c^{3} x^{2}"," ",0,"1/8*a^6*c^3*x^8 + 2/7*a^5*c^3*x^7 - 1/6*a^4*c^3*x^6 - 4/5*a^3*c^3*x^5 - 1/4*a^2*c^3*x^4 + 2/3*a*c^3*x^3 + 1/2*c^3*x^2","A",0
1043,1,69,0,0.731516," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{1}{7} \, a^{6} c^{3} x^{7} + \frac{1}{3} \, a^{5} c^{3} x^{6} - \frac{1}{5} \, a^{4} c^{3} x^{5} - a^{3} c^{3} x^{4} - \frac{1}{3} \, a^{2} c^{3} x^{3} + a c^{3} x^{2} + c^{3} x"," ",0,"1/7*a^6*c^3*x^7 + 1/3*a^5*c^3*x^6 - 1/5*a^4*c^3*x^5 - a^3*c^3*x^4 - 1/3*a^2*c^3*x^3 + a*c^3*x^2 + c^3*x","A",0
1044,1,69,0,1.358248," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x,x, algorithm=""fricas"")","\frac{1}{6} \, a^{6} c^{3} x^{6} + \frac{2}{5} \, a^{5} c^{3} x^{5} - \frac{1}{4} \, a^{4} c^{3} x^{4} - \frac{4}{3} \, a^{3} c^{3} x^{3} - \frac{1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log\left(x\right)"," ",0,"1/6*a^6*c^3*x^6 + 2/5*a^5*c^3*x^5 - 1/4*a^4*c^3*x^4 - 4/3*a^3*c^3*x^3 - 1/2*a^2*c^3*x^2 + 2*a*c^3*x + c^3*log(x)","A",0
1045,1,75,0,0.677891," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x^2,x, algorithm=""fricas"")","\frac{6 \, a^{6} c^{3} x^{6} + 15 \, a^{5} c^{3} x^{5} - 10 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} - 30 \, a^{2} c^{3} x^{2} + 60 \, a c^{3} x \log\left(x\right) - 30 \, c^{3}}{30 \, x}"," ",0,"1/30*(6*a^6*c^3*x^6 + 15*a^5*c^3*x^5 - 10*a^4*c^3*x^4 - 60*a^3*c^3*x^3 - 30*a^2*c^3*x^2 + 60*a*c^3*x*log(x) - 30*c^3)/x","A",0
1046,1,75,0,0.670843," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x^3,x, algorithm=""fricas"")","\frac{3 \, a^{6} c^{3} x^{6} + 8 \, a^{5} c^{3} x^{5} - 6 \, a^{4} c^{3} x^{4} - 48 \, a^{3} c^{3} x^{3} - 12 \, a^{2} c^{3} x^{2} \log\left(x\right) - 24 \, a c^{3} x - 6 \, c^{3}}{12 \, x^{2}}"," ",0,"1/12*(3*a^6*c^3*x^6 + 8*a^5*c^3*x^5 - 6*a^4*c^3*x^4 - 48*a^3*c^3*x^3 - 12*a^2*c^3*x^2*log(x) - 24*a*c^3*x - 6*c^3)/x^2","A",0
1047,1,74,0,0.635098," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x^4,x, algorithm=""fricas"")","\frac{a^{6} c^{3} x^{6} + 3 \, a^{5} c^{3} x^{5} - 3 \, a^{4} c^{3} x^{4} - 12 \, a^{3} c^{3} x^{3} \log\left(x\right) + 3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}}"," ",0,"1/3*(a^6*c^3*x^6 + 3*a^5*c^3*x^5 - 3*a^4*c^3*x^4 - 12*a^3*c^3*x^3*log(x) + 3*a^2*c^3*x^2 - 3*a*c^3*x - c^3)/x^3","A",0
1048,1,79,0,0.584658," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","-\frac{1}{9} \, a^{8} c^{4} x^{9} - \frac{1}{4} \, a^{7} c^{4} x^{8} + \frac{2}{7} \, a^{6} c^{4} x^{7} + a^{5} c^{4} x^{6} - \frac{3}{2} \, a^{3} c^{4} x^{4} - \frac{2}{3} \, a^{2} c^{4} x^{3} + a c^{4} x^{2} + c^{4} x"," ",0,"-1/9*a^8*c^4*x^9 - 1/4*a^7*c^4*x^8 + 2/7*a^6*c^4*x^7 + a^5*c^4*x^6 - 3/2*a^3*c^4*x^4 - 2/3*a^2*c^4*x^3 + a*c^4*x^2 + c^4*x","A",0
1049,1,59,0,0.537452," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{a^{4} x^{4} + 2 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 9 \, a x + 12 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 3}{3 \, {\left(a^{6} c x - a^{5} c\right)}}"," ",0,"1/3*(a^4*x^4 + 2*a^3*x^3 + 6*a^2*x^2 - 9*a*x + 12*(a*x - 1)*log(a*x - 1) - 3)/(a^6*c*x - a^5*c)","A",0
1050,1,51,0,0.582565," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{a^{3} x^{3} + 3 \, a^{2} x^{2} - 4 \, a x + 6 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 2}{2 \, {\left(a^{5} c x - a^{4} c\right)}}"," ",0,"1/2*(a^3*x^3 + 3*a^2*x^2 - 4*a*x + 6*(a*x - 1)*log(a*x - 1) - 2)/(a^5*c*x - a^4*c)","A",0
1051,1,42,0,0.625614," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{a^{2} x^{2} - a x + 2 \, {\left(a x - 1\right)} \log\left(a x - 1\right) - 1}{a^{4} c x - a^{3} c}"," ",0,"(a^2*x^2 - a*x + 2*(a*x - 1)*log(a*x - 1) - 1)/(a^4*c*x - a^3*c)","A",0
1052,1,30,0,0.639967," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{{\left(a x - 1\right)} \log\left(a x - 1\right) - 1}{a^{3} c x - a^{2} c}"," ",0,"((a*x - 1)*log(a*x - 1) - 1)/(a^3*c*x - a^2*c)","A",0
1053,1,15,0,0.449735," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{1}{a^{2} c x - a c}"," ",0,"-1/(a^2*c*x - a*c)","A",0
1054,1,35,0,0.645458," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x/(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{{\left(a x - 1\right)} \log\left(a x - 1\right) - {\left(a x - 1\right)} \log\left(x\right) + 1}{a c x - c}"," ",0,"-((a*x - 1)*log(a*x - 1) - (a*x - 1)*log(x) + 1)/(a*c*x - c)","A",0
1055,1,57,0,0.707928," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^2/(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{2 \, a x + 2 \, {\left(a^{2} x^{2} - a x\right)} \log\left(a x - 1\right) - 2 \, {\left(a^{2} x^{2} - a x\right)} \log\left(x\right) - 1}{a c x^{2} - c x}"," ",0,"-(2*a*x + 2*(a^2*x^2 - a*x)*log(a*x - 1) - 2*(a^2*x^2 - a*x)*log(x) - 1)/(a*c*x^2 - c*x)","A",0
1056,1,75,0,0.542696," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^3/(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{6 \, a^{2} x^{2} - 3 \, a x + 6 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(a x - 1\right) - 6 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(x\right) - 1}{2 \, {\left(a c x^{3} - c x^{2}\right)}}"," ",0,"-1/2*(6*a^2*x^2 - 3*a*x + 6*(a^3*x^3 - a^2*x^2)*log(a*x - 1) - 6*(a^3*x^3 - a^2*x^2)*log(x) - 1)/(a*c*x^3 - c*x^2)","A",0
1057,1,83,0,0.574151," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^4/(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{12 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 2 \, a x + 12 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(a x - 1\right) - 12 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \log\left(x\right) - 1}{3 \, {\left(a c x^{4} - c x^{3}\right)}}"," ",0,"-1/3*(12*a^3*x^3 - 6*a^2*x^2 - 2*a*x + 12*(a^4*x^4 - a^3*x^3)*log(a*x - 1) - 12*(a^4*x^4 - a^3*x^3)*log(x) - 1)/(a*c*x^4 - c*x^3)","A",0
1058,1,95,0,0.613167," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{8 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 6 \, a x - {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) + 17 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) + 12}{8 \, {\left(a^{7} c^{2} x^{2} - 2 \, a^{6} c^{2} x + a^{5} c^{2}\right)}}"," ",0,"-1/8*(8*a^3*x^3 - 16*a^2*x^2 - 6*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 17*(a^2*x^2 - 2*a*x + 1)*log(a*x - 1) + 12)/(a^7*c^2*x^2 - 2*a^6*c^2*x + a^5*c^2)","A",0
1059,1,79,0,0.504302," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{10 \, a x - {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) - 7 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) - 8}{8 \, {\left(a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right)}}"," ",0,"1/8*(10*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) - 7*(a^2*x^2 - 2*a*x + 1)*log(a*x - 1) - 8)/(a^6*c^2*x^2 - 2*a^5*c^2*x + a^4*c^2)","A",0
1060,1,78,0,0.582614," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{6 \, a x + {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) - {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) - 4}{8 \, {\left(a^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} c^{2}\right)}}"," ",0,"1/8*(6*a*x + (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) - (a^2*x^2 - 2*a*x + 1)*log(a*x - 1) - 4)/(a^5*c^2*x^2 - 2*a^4*c^2*x + a^3*c^2)","A",0
1061,1,77,0,0.535486," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{2 \, a x - {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) + {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right)}{8 \, {\left(a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right)}}"," ",0,"1/8*(2*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + (a^2*x^2 - 2*a*x + 1)*log(a*x - 1))/(a^4*c^2*x^2 - 2*a^3*c^2*x + a^2*c^2)","A",0
1062,1,76,0,0.513258," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{2 \, a x - {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) + {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) - 4}{8 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"-1/8*(2*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + (a^2*x^2 - 2*a*x + 1)*log(a*x - 1) - 4)/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)","A",0
1063,1,89,0,0.555171," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{6 \, a x + {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) + 7 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) - 8 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(x\right) - 8}{8 \, {\left(a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right)}}"," ",0,"-1/8*(6*a*x + (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 7*(a^2*x^2 - 2*a*x + 1)*log(a*x - 1) - 8*(a^2*x^2 - 2*a*x + 1)*log(x) - 8)/(a^2*c^2*x^2 - 2*a*c^2*x + c^2)","A",0
1064,1,120,0,0.649670," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^2/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{18 \, a^{2} x^{2} - 28 \, a x - {\left(a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right)} \log\left(a x + 1\right) + 17 \, {\left(a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right)} \log\left(a x - 1\right) - 16 \, {\left(a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right)} \log\left(x\right) + 8}{8 \, {\left(a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right)}}"," ",0,"-1/8*(18*a^2*x^2 - 28*a*x - (a^3*x^3 - 2*a^2*x^2 + a*x)*log(a*x + 1) + 17*(a^3*x^3 - 2*a^2*x^2 + a*x)*log(a*x - 1) - 16*(a^3*x^3 - 2*a^2*x^2 + a*x)*log(x) + 8)/(a^2*c^2*x^3 - 2*a*c^2*x^2 + c^2*x)","A",0
1065,1,141,0,0.669092," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^3/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{30 \, a^{3} x^{3} - 44 \, a^{2} x^{2} + 8 \, a x + {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(a x + 1\right) + 31 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(a x - 1\right) - 32 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right)} \log\left(x\right) + 4}{8 \, {\left(a^{2} c^{2} x^{4} - 2 \, a c^{2} x^{3} + c^{2} x^{2}\right)}}"," ",0,"-1/8*(30*a^3*x^3 - 44*a^2*x^2 + 8*a*x + (a^4*x^4 - 2*a^3*x^3 + a^2*x^2)*log(a*x + 1) + 31*(a^4*x^4 - 2*a^3*x^3 + a^2*x^2)*log(a*x - 1) - 32*(a^4*x^4 - 2*a^3*x^3 + a^2*x^2)*log(x) + 4)/(a^2*c^2*x^4 - 2*a*c^2*x^3 + c^2*x^2)","A",0
1066,1,150,0,0.628375," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^4/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{150 \, a^{4} x^{4} - 228 \, a^{3} x^{3} + 56 \, a^{2} x^{2} + 8 \, a x - 3 \, {\left(a^{5} x^{5} - 2 \, a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(a x + 1\right) + 147 \, {\left(a^{5} x^{5} - 2 \, a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(a x - 1\right) - 144 \, {\left(a^{5} x^{5} - 2 \, a^{4} x^{4} + a^{3} x^{3}\right)} \log\left(x\right) + 8}{24 \, {\left(a^{2} c^{2} x^{5} - 2 \, a c^{2} x^{4} + c^{2} x^{3}\right)}}"," ",0,"-1/24*(150*a^4*x^4 - 228*a^3*x^3 + 56*a^2*x^2 + 8*a*x - 3*(a^5*x^5 - 2*a^4*x^4 + a^3*x^3)*log(a*x + 1) + 147*(a^5*x^5 - 2*a^4*x^4 + a^3*x^3)*log(a*x - 1) - 144*(a^5*x^5 - 2*a^4*x^4 + a^3*x^3)*log(x) + 8)/(a^2*c^2*x^5 - 2*a*c^2*x^4 + c^2*x^3)","A",0
1067,1,123,0,0.647705," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^5/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{66 \, a^{3} x^{3} - 36 \, a^{2} x^{2} - 74 \, a x - 9 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x + 1\right) - 39 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x - 1\right) + 52}{48 \, {\left(a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{7} c^{3} x - a^{6} c^{3}\right)}}"," ",0,"-1/48*(66*a^3*x^3 - 36*a^2*x^2 - 74*a*x - 9*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x + 1) - 39*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x - 1) + 52)/(a^10*c^3*x^4 - 2*a^9*c^3*x^3 + 2*a^7*c^3*x - a^6*c^3)","A",0
1068,1,123,0,0.546389," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^4/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{18 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 10 \, a x + 3 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x + 1\right) - 3 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x - 1\right) + 8}{24 \, {\left(a^{9} c^{3} x^{4} - 2 \, a^{8} c^{3} x^{3} + 2 \, a^{6} c^{3} x - a^{5} c^{3}\right)}}"," ",0,"-1/24*(18*a^3*x^3 - 12*a^2*x^2 - 10*a*x + 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x + 1) - 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x - 1) + 8)/(a^9*c^3*x^4 - 2*a^8*c^3*x^3 + 2*a^6*c^3*x - a^5*c^3)","A",0
1069,1,123,0,0.609337," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 14 \, a x - 3 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x + 1\right) + 3 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x - 1\right) + 4}{48 \, {\left(a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{5} c^{3} x - a^{4} c^{3}\right)}}"," ",0,"-1/48*(6*a^3*x^3 + 12*a^2*x^2 - 14*a*x - 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x - 1) + 4)/(a^8*c^3*x^4 - 2*a^7*c^3*x^3 + 2*a^5*c^3*x - a^4*c^3)","A",0
1070,1,49,0,0.563887," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{2 \, a x - 1}{6 \, {\left(a^{7} c^{3} x^{4} - 2 \, a^{6} c^{3} x^{3} + 2 \, a^{4} c^{3} x - a^{3} c^{3}\right)}}"," ",0,"-1/6*(2*a*x - 1)/(a^7*c^3*x^4 - 2*a^6*c^3*x^3 + 2*a^4*c^3*x - a^3*c^3)","A",0
1071,1,123,0,0.603493," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{6 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 2 \, a x - 3 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x + 1\right) + 3 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x - 1\right) - 4}{48 \, {\left(a^{6} c^{3} x^{4} - 2 \, a^{5} c^{3} x^{3} + 2 \, a^{3} c^{3} x - a^{2} c^{3}\right)}}"," ",0,"1/48*(6*a^3*x^3 - 12*a^2*x^2 + 2*a*x - 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x - 1) - 4)/(a^6*c^3*x^4 - 2*a^5*c^3*x^3 + 2*a^3*c^3*x - a^2*c^3)","B",0
1072,1,121,0,0.653297," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{6 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 2 \, a x - 3 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x + 1\right) + 3 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x - 1\right) + 8}{24 \, {\left(a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/24*(6*a^3*x^3 - 12*a^2*x^2 + 2*a*x - 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x - 1) + 8)/(a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^2*c^3*x - a*c^3)","A",0
1073,1,143,0,0.794463," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{30 \, a^{3} x^{3} - 36 \, a^{2} x^{2} - 38 \, a x + 9 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x + 1\right) + 39 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(a x - 1\right) - 48 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right)} \log\left(x\right) + 52}{48 \, {\left(a^{4} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{3} + 2 \, a c^{3} x - c^{3}\right)}}"," ",0,"-1/48*(30*a^3*x^3 - 36*a^2*x^2 - 38*a*x + 9*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x + 1) + 39*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(a*x - 1) - 48*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(x) + 52)/(a^4*c^3*x^4 - 2*a^3*c^3*x^3 + 2*a*c^3*x - c^3)","A",0
1074,1,175,0,1.089652," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^2/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{30 \, a^{4} x^{4} - 48 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 46 \, a x - 3 \, {\left(a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right)} \log\left(a x + 1\right) + 27 \, {\left(a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right)} \log\left(a x - 1\right) - 24 \, {\left(a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right)} \log\left(x\right) - 12}{12 \, {\left(a^{4} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{4} + 2 \, a c^{3} x^{2} - c^{3} x\right)}}"," ",0,"-1/12*(30*a^4*x^4 - 48*a^3*x^3 - 14*a^2*x^2 + 46*a*x - 3*(a^5*x^5 - 2*a^4*x^4 + 2*a^2*x^2 - a*x)*log(a*x + 1) + 27*(a^5*x^5 - 2*a^4*x^4 + 2*a^2*x^2 - a*x)*log(a*x - 1) - 24*(a^5*x^5 - 2*a^4*x^4 + 2*a^2*x^2 - a*x)*log(x) - 12)/(a^4*c^3*x^5 - 2*a^3*c^3*x^4 + 2*a*c^3*x^2 - c^3*x)","A",0
1075,1,197,0,0.593469," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^3/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{210 \, a^{5} x^{5} - 300 \, a^{4} x^{4} - 170 \, a^{3} x^{3} + 340 \, a^{2} x^{2} - 48 \, a x + 15 \, {\left(a^{6} x^{6} - 2 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(a x + 1\right) + 225 \, {\left(a^{6} x^{6} - 2 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(a x - 1\right) - 240 \, {\left(a^{6} x^{6} - 2 \, a^{5} x^{5} + 2 \, a^{3} x^{3} - a^{2} x^{2}\right)} \log\left(x\right) - 24}{48 \, {\left(a^{4} c^{3} x^{6} - 2 \, a^{3} c^{3} x^{5} + 2 \, a c^{3} x^{3} - c^{3} x^{2}\right)}}"," ",0,"-1/48*(210*a^5*x^5 - 300*a^4*x^4 - 170*a^3*x^3 + 340*a^2*x^2 - 48*a*x + 15*(a^6*x^6 - 2*a^5*x^5 + 2*a^3*x^3 - a^2*x^2)*log(a*x + 1) + 225*(a^6*x^6 - 2*a^5*x^5 + 2*a^3*x^3 - a^2*x^2)*log(a*x - 1) - 240*(a^6*x^6 - 2*a^5*x^5 + 2*a^3*x^3 - a^2*x^2)*log(x) - 24)/(a^4*c^3*x^6 - 2*a^3*c^3*x^5 + 2*a*c^3*x^3 - c^3*x^2)","A",0
1076,1,217,0,0.588589," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","-\frac{30 \, a^{5} x^{5} - 60 \, a^{4} x^{4} - 20 \, a^{3} x^{3} + 100 \, a^{2} x^{2} - 34 \, a x - 15 \, {\left(a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x + 1\right) + 15 \, {\left(a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right)} \log\left(a x - 1\right) - 32}{128 \, {\left(a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/128*(30*a^5*x^5 - 60*a^4*x^4 - 20*a^3*x^3 + 100*a^2*x^2 - 34*a*x - 15*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 15*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(a*x - 1) - 32)/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)","B",0
1077,1,184,0,0.688189," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(4 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 15 \, a x + 24\right)} \sqrt{-a^{2} c x^{2} + c} - 15 \, \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right)}{40 \, a^{4}}, -\frac{{\left(4 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 15 \, a x + 24\right)} \sqrt{-a^{2} c x^{2} + c} + 15 \, \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right)}{20 \, a^{4}}\right]"," ",0,"[-1/40*(2*(4*a^4*x^4 + 10*a^3*x^3 + 12*a^2*x^2 + 15*a*x + 24)*sqrt(-a^2*c*x^2 + c) - 15*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a^4, -1/20*((4*a^4*x^4 + 10*a^3*x^3 + 12*a^2*x^2 + 15*a*x + 24)*sqrt(-a^2*c*x^2 + c) + 15*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))/a^4]","A",0
1078,1,168,0,0.744614," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(6 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 21 \, a x + 32\right)} \sqrt{-a^{2} c x^{2} + c} - 21 \, \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right)}{48 \, a^{3}}, -\frac{{\left(6 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 21 \, a x + 32\right)} \sqrt{-a^{2} c x^{2} + c} + 21 \, \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right)}{24 \, a^{3}}\right]"," ",0,"[-1/48*(2*(6*a^3*x^3 + 16*a^2*x^2 + 21*a*x + 32)*sqrt(-a^2*c*x^2 + c) - 21*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a^3, -1/24*((6*a^3*x^3 + 16*a^2*x^2 + 21*a*x + 32)*sqrt(-a^2*c*x^2 + c) + 21*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))/a^3]","A",0
1079,1,150,0,0.836121," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, \sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 3 \, a x + 5\right)} - 3 \, \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right)}{6 \, a^{2}}, -\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 3 \, a x + 5\right)} + 3 \, \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right)}{3 \, a^{2}}\right]"," ",0,"[-1/6*(2*sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 3*a*x + 5) - 3*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a^2, -1/3*(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 3*a*x + 5) + 3*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))/a^2]","A",0
1080,1,134,0,0.698122," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, \sqrt{-a^{2} c x^{2} + c} {\left(a x + 4\right)} - 3 \, \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right)}{4 \, a}, -\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a x + 4\right)} + 3 \, \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right)}{2 \, a}\right]"," ",0,"[-1/4*(2*sqrt(-a^2*c*x^2 + c)*(a*x + 4) - 3*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a, -1/2*(sqrt(-a^2*c*x^2 + c)*(a*x + 4) + 3*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))/a]","A",0
1081,1,196,0,0.656030," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2)/x,x, algorithm=""fricas"")","\left[-2 \, \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + \frac{1}{2} \, \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - \sqrt{-a^{2} c x^{2} + c}, -\sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - \sqrt{-a^{2} c x^{2} + c}\right]"," ",0,"[-2*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 1/2*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - sqrt(-a^2*c*x^2 + c), -sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - sqrt(-a^2*c*x^2 + c)]","A",0
1082,1,212,0,1.010260," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2)/x^2,x, algorithm=""fricas"")","\left[-\frac{a \sqrt{c} x \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - a \sqrt{c} x \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + \sqrt{-a^{2} c x^{2} + c}}{x}, -\frac{4 \, a \sqrt{-c} x \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - a \sqrt{-c} x \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 2 \, \sqrt{-a^{2} c x^{2} + c}}{2 \, x}\right]"," ",0,"[-(a*sqrt(c)*x*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - a*sqrt(c)*x*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + sqrt(-a^2*c*x^2 + c))/x, -1/2*(4*a*sqrt(-c)*x*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) - a*sqrt(-c)*x*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*sqrt(-a^2*c*x^2 + c))/x]","A",0
1083,1,148,0,1.077569," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2)/x^3,x, algorithm=""fricas"")","\left[\frac{3 \, a^{2} \sqrt{c} x^{2} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} {\left(4 \, a x + 1\right)}}{4 \, x^{2}}, -\frac{3 \, a^{2} \sqrt{-c} x^{2} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(4 \, a x + 1\right)}}{2 \, x^{2}}\right]"," ",0,"[1/4*(3*a^2*sqrt(c)*x^2*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*sqrt(-a^2*c*x^2 + c)*(4*a*x + 1))/x^2, -1/2*(3*a^2*sqrt(-c)*x^2*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(4*a*x + 1))/x^2]","A",0
1084,1,164,0,0.568574," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2)/x^4,x, algorithm=""fricas"")","\left[\frac{3 \, a^{3} \sqrt{c} x^{3} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} {\left(5 \, a^{2} x^{2} + 3 \, a x + 1\right)}}{6 \, x^{3}}, -\frac{3 \, a^{3} \sqrt{-c} x^{3} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(5 \, a^{2} x^{2} + 3 \, a x + 1\right)}}{3 \, x^{3}}\right]"," ",0,"[1/6*(3*a^3*sqrt(c)*x^3*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*sqrt(-a^2*c*x^2 + c)*(5*a^2*x^2 + 3*a*x + 1))/x^3, -1/3*(3*a^3*sqrt(-c)*x^3*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(5*a^2*x^2 + 3*a*x + 1))/x^3]","A",0
1085,1,180,0,0.680975," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2)/x^5,x, algorithm=""fricas"")","\left[\frac{21 \, a^{4} \sqrt{c} x^{4} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 2 \, {\left(32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right)} \sqrt{-a^{2} c x^{2} + c}}{48 \, x^{4}}, -\frac{21 \, a^{4} \sqrt{-c} x^{4} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right)} \sqrt{-a^{2} c x^{2} + c}}{24 \, x^{4}}\right]"," ",0,"[1/48*(21*a^4*sqrt(c)*x^4*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*(32*a^3*x^3 + 21*a^2*x^2 + 16*a*x + 6)*sqrt(-a^2*c*x^2 + c))/x^4, -1/24*(21*a^4*sqrt(-c)*x^4*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (32*a^3*x^3 + 21*a^2*x^2 + 16*a*x + 6)*sqrt(-a^2*c*x^2 + c))/x^4]","A",0
1086,1,234,0,0.740375," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{105 \, \sqrt{-c} c \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 2 \, {\left(120 \, a^{6} c x^{6} + 280 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} - 70 \, a^{3} c x^{3} - 88 \, a^{2} c x^{2} - 105 \, a c x - 176 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{1680 \, a^{4}}, -\frac{105 \, c^{\frac{3}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - {\left(120 \, a^{6} c x^{6} + 280 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} - 70 \, a^{3} c x^{3} - 88 \, a^{2} c x^{2} - 105 \, a c x - 176 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{840 \, a^{4}}\right]"," ",0,"[1/1680*(105*sqrt(-c)*c*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(120*a^6*c*x^6 + 280*a^5*c*x^5 + 144*a^4*c*x^4 - 70*a^3*c*x^3 - 88*a^2*c*x^2 - 105*a*c*x - 176*c)*sqrt(-a^2*c*x^2 + c))/a^4, -1/840*(105*c^(3/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (120*a^6*c*x^6 + 280*a^5*c*x^5 + 144*a^4*c*x^4 - 70*a^3*c*x^3 - 88*a^2*c*x^2 - 105*a*c*x - 176*c)*sqrt(-a^2*c*x^2 + c))/a^4]","A",0
1087,1,216,0,0.639957," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{45 \, \sqrt{-c} c \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 2 \, {\left(40 \, a^{5} c x^{5} + 96 \, a^{4} c x^{4} + 50 \, a^{3} c x^{3} - 32 \, a^{2} c x^{2} - 45 \, a c x - 64 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{480 \, a^{3}}, -\frac{45 \, c^{\frac{3}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - {\left(40 \, a^{5} c x^{5} + 96 \, a^{4} c x^{4} + 50 \, a^{3} c x^{3} - 32 \, a^{2} c x^{2} - 45 \, a c x - 64 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{240 \, a^{3}}\right]"," ",0,"[1/480*(45*sqrt(-c)*c*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(40*a^5*c*x^5 + 96*a^4*c*x^4 + 50*a^3*c*x^3 - 32*a^2*c*x^2 - 45*a*c*x - 64*c)*sqrt(-a^2*c*x^2 + c))/a^3, -1/240*(45*c^(3/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (40*a^5*c*x^5 + 96*a^4*c*x^4 + 50*a^3*c*x^3 - 32*a^2*c*x^2 - 45*a*c*x - 64*c)*sqrt(-a^2*c*x^2 + c))/a^3]","A",0
1088,1,198,0,0.877189," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x*(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{-c} c \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 2 \, {\left(12 \, a^{4} c x^{4} + 30 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} - 15 \, a c x - 28 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{120 \, a^{2}}, -\frac{15 \, c^{\frac{3}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - {\left(12 \, a^{4} c x^{4} + 30 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} - 15 \, a c x - 28 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{60 \, a^{2}}\right]"," ",0,"[1/120*(15*sqrt(-c)*c*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(12*a^4*c*x^4 + 30*a^3*c*x^3 + 16*a^2*c*x^2 - 15*a*c*x - 28*c)*sqrt(-a^2*c*x^2 + c))/a^2, -1/60*(15*c^(3/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (12*a^4*c*x^4 + 30*a^3*c*x^3 + 16*a^2*c*x^2 - 15*a*c*x - 28*c)*sqrt(-a^2*c*x^2 + c))/a^2]","A",0
1089,1,180,0,0.675584," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{-c} c \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 2 \, {\left(6 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} + 9 \, a c x - 16 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{48 \, a}, -\frac{15 \, c^{\frac{3}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - {\left(6 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} + 9 \, a c x - 16 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{24 \, a}\right]"," ",0,"[1/48*(15*sqrt(-c)*c*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(6*a^3*c*x^3 + 16*a^2*c*x^2 + 9*a*c*x - 16*c)*sqrt(-a^2*c*x^2 + c))/a, -1/24*(15*c^(3/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (6*a^3*c*x^3 + 16*a^2*c*x^2 + 9*a*c*x - 16*c)*sqrt(-a^2*c*x^2 + c))/a]","A",0
1090,1,233,0,0.693980," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x,x, algorithm=""fricas"")","\left[-c^{\frac{3}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + \frac{1}{2} \, c^{\frac{3}{2}} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + \frac{1}{3} \, {\left(a^{2} c x^{2} + 3 \, a c x + 2 \, c\right)} \sqrt{-a^{2} c x^{2} + c}, -\sqrt{-c} c \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \frac{1}{2} \, \sqrt{-c} c \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + \frac{1}{3} \, {\left(a^{2} c x^{2} + 3 \, a c x + 2 \, c\right)} \sqrt{-a^{2} c x^{2} + c}\right]"," ",0,"[-c^(3/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 1/2*c^(3/2)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 1/3*(a^2*c*x^2 + 3*a*c*x + 2*c)*sqrt(-a^2*c*x^2 + c), -sqrt(-c)*c*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 1/2*sqrt(-c)*c*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 1/3*(a^2*c*x^2 + 3*a*c*x + 2*c)*sqrt(-a^2*c*x^2 + c)]","A",0
1091,1,249,0,0.702475," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^2,x, algorithm=""fricas"")","\left[\frac{a c^{\frac{3}{2}} x \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + 2 \, a c^{\frac{3}{2}} x \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + {\left(a^{2} c x^{2} + 4 \, a c x - 2 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{2 \, x}, -\frac{8 \, a \sqrt{-c} c x \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - a \sqrt{-c} c x \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, {\left(a^{2} c x^{2} + 4 \, a c x - 2 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{4 \, x}\right]"," ",0,"[1/2*(a*c^(3/2)*x*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 2*a*c^(3/2)*x*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + (a^2*c*x^2 + 4*a*c*x - 2*c)*sqrt(-a^2*c*x^2 + c))/x, -1/4*(8*a*sqrt(-c)*c*x*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) - a*sqrt(-c)*c*x*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(a^2*c*x^2 + 4*a*c*x - 2*c)*sqrt(-a^2*c*x^2 + c))/x]","A",0
1092,1,267,0,0.662840," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^3,x, algorithm=""fricas"")","\left[\frac{8 \, a^{2} c^{\frac{3}{2}} x^{2} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + a^{2} c^{\frac{3}{2}} x^{2} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + 2 \, {\left(2 \, a^{2} c x^{2} - 4 \, a c x - c\right)} \sqrt{-a^{2} c x^{2} + c}}{4 \, x^{2}}, -\frac{a^{2} \sqrt{-c} c x^{2} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - 2 \, a^{2} \sqrt{-c} c x^{2} \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - {\left(2 \, a^{2} c x^{2} - 4 \, a c x - c\right)} \sqrt{-a^{2} c x^{2} + c}}{2 \, x^{2}}\right]"," ",0,"[1/4*(8*a^2*c^(3/2)*x^2*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + a^2*c^(3/2)*x^2*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 2*(2*a^2*c*x^2 - 4*a*c*x - c)*sqrt(-a^2*c*x^2 + c))/x^2, -1/2*(a^2*sqrt(-c)*c*x^2*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 2*a^2*sqrt(-c)*c*x^2*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - (2*a^2*c*x^2 - 4*a*c*x - c)*sqrt(-a^2*c*x^2 + c))/x^2]","A",0
1093,1,265,0,0.689420," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^4,x, algorithm=""fricas"")","\left[\frac{6 \, a^{3} c^{\frac{3}{2}} x^{3} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + 3 \, a^{3} c^{\frac{3}{2}} x^{3} \log\left(-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 2 \, {\left(2 \, a^{2} c x^{2} + 3 \, a c x + c\right)} \sqrt{-a^{2} c x^{2} + c}}{6 \, x^{3}}, \frac{6 \, a^{3} \sqrt{-c} c x^{3} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 3 \, a^{3} \sqrt{-c} c x^{3} \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, {\left(2 \, a^{2} c x^{2} + 3 \, a c x + c\right)} \sqrt{-a^{2} c x^{2} + c}}{6 \, x^{3}}\right]"," ",0,"[1/6*(6*a^3*c^(3/2)*x^3*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 3*a^3*c^(3/2)*x^3*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*(2*a^2*c*x^2 + 3*a*c*x + c)*sqrt(-a^2*c*x^2 + c))/x^3, 1/6*(6*a^3*sqrt(-c)*c*x^3*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 3*a^3*sqrt(-c)*c*x^3*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(2*a^2*c*x^2 + 3*a*c*x + c)*sqrt(-a^2*c*x^2 + c))/x^3]","A",0
1094,1,191,0,0.598893," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^5,x, algorithm=""fricas"")","\left[\frac{15 \, a^{4} c^{\frac{3}{2}} x^{4} \log\left(-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + 2 \, {\left(16 \, a^{3} c x^{3} - 9 \, a^{2} c x^{2} - 16 \, a c x - 6 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{48 \, x^{4}}, \frac{15 \, a^{4} \sqrt{-c} c x^{4} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(16 \, a^{3} c x^{3} - 9 \, a^{2} c x^{2} - 16 \, a c x - 6 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{24 \, x^{4}}\right]"," ",0,"[1/48*(15*a^4*c^(3/2)*x^4*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 2*(16*a^3*c*x^3 - 9*a^2*c*x^2 - 16*a*c*x - 6*c)*sqrt(-a^2*c*x^2 + c))/x^4, 1/24*(15*a^4*sqrt(-c)*c*x^4*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (16*a^3*c*x^3 - 9*a^2*c*x^2 - 16*a*c*x - 6*c)*sqrt(-a^2*c*x^2 + c))/x^4]","A",0
1095,1,209,0,0.653727," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^6,x, algorithm=""fricas"")","\left[\frac{15 \, a^{5} c^{\frac{3}{2}} x^{5} \log\left(-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + 2 \, {\left(28 \, a^{4} c x^{4} + 15 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} - 30 \, a c x - 12 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{120 \, x^{5}}, \frac{15 \, a^{5} \sqrt{-c} c x^{5} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(28 \, a^{4} c x^{4} + 15 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} - 30 \, a c x - 12 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{60 \, x^{5}}\right]"," ",0,"[1/120*(15*a^5*c^(3/2)*x^5*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 2*(28*a^4*c*x^4 + 15*a^3*c*x^3 - 16*a^2*c*x^2 - 30*a*c*x - 12*c)*sqrt(-a^2*c*x^2 + c))/x^5, 1/60*(15*a^5*sqrt(-c)*c*x^5*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (28*a^4*c*x^4 + 15*a^3*c*x^3 - 16*a^2*c*x^2 - 30*a*c*x - 12*c)*sqrt(-a^2*c*x^2 + c))/x^5]","A",0
1096,1,227,0,0.793712," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^7,x, algorithm=""fricas"")","\left[\frac{45 \, a^{6} c^{\frac{3}{2}} x^{6} \log\left(-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + 2 \, {\left(64 \, a^{5} c x^{5} + 45 \, a^{4} c x^{4} + 32 \, a^{3} c x^{3} - 50 \, a^{2} c x^{2} - 96 \, a c x - 40 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{480 \, x^{6}}, \frac{45 \, a^{6} \sqrt{-c} c x^{6} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(64 \, a^{5} c x^{5} + 45 \, a^{4} c x^{4} + 32 \, a^{3} c x^{3} - 50 \, a^{2} c x^{2} - 96 \, a c x - 40 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{240 \, x^{6}}\right]"," ",0,"[1/480*(45*a^6*c^(3/2)*x^6*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 2*(64*a^5*c*x^5 + 45*a^4*c*x^4 + 32*a^3*c*x^3 - 50*a^2*c*x^2 - 96*a*c*x - 40*c)*sqrt(-a^2*c*x^2 + c))/x^6, 1/240*(45*a^6*sqrt(-c)*c*x^6*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (64*a^5*c*x^5 + 45*a^4*c*x^4 + 32*a^3*c*x^3 - 50*a^2*c*x^2 - 96*a*c*x - 40*c)*sqrt(-a^2*c*x^2 + c))/x^6]","A",0
1097,1,245,0,0.739410," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^8,x, algorithm=""fricas"")","\left[\frac{105 \, a^{7} c^{\frac{3}{2}} x^{7} \log\left(-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + 2 \, {\left(176 \, a^{6} c x^{6} + 105 \, a^{5} c x^{5} + 88 \, a^{4} c x^{4} + 70 \, a^{3} c x^{3} - 144 \, a^{2} c x^{2} - 280 \, a c x - 120 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{1680 \, x^{7}}, \frac{105 \, a^{7} \sqrt{-c} c x^{7} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(176 \, a^{6} c x^{6} + 105 \, a^{5} c x^{5} + 88 \, a^{4} c x^{4} + 70 \, a^{3} c x^{3} - 144 \, a^{2} c x^{2} - 280 \, a c x - 120 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{840 \, x^{7}}\right]"," ",0,"[1/1680*(105*a^7*c^(3/2)*x^7*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 2*(176*a^6*c*x^6 + 105*a^5*c*x^5 + 88*a^4*c*x^4 + 70*a^3*c*x^3 - 144*a^2*c*x^2 - 280*a*c*x - 120*c)*sqrt(-a^2*c*x^2 + c))/x^7, 1/840*(105*a^7*sqrt(-c)*c*x^7*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (176*a^6*c*x^6 + 105*a^5*c*x^5 + 88*a^4*c*x^4 + 70*a^3*c*x^3 - 144*a^2*c*x^2 - 280*a*c*x - 120*c)*sqrt(-a^2*c*x^2 + c))/x^7]","A",0
1098,1,307,0,0.765206," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{945 \, \sqrt{-c} c^{2} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, {\left(2240 \, a^{8} c^{2} x^{8} + 5040 \, a^{7} c^{2} x^{7} - 320 \, a^{6} c^{2} x^{6} - 7560 \, a^{5} c^{2} x^{5} - 4416 \, a^{4} c^{2} x^{4} + 630 \, a^{3} c^{2} x^{3} + 832 \, a^{2} c^{2} x^{2} + 945 \, a c^{2} x + 1664 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{40320 \, a^{4}}, -\frac{945 \, c^{\frac{5}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + {\left(2240 \, a^{8} c^{2} x^{8} + 5040 \, a^{7} c^{2} x^{7} - 320 \, a^{6} c^{2} x^{6} - 7560 \, a^{5} c^{2} x^{5} - 4416 \, a^{4} c^{2} x^{4} + 630 \, a^{3} c^{2} x^{3} + 832 \, a^{2} c^{2} x^{2} + 945 \, a c^{2} x + 1664 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{20160 \, a^{4}}\right]"," ",0,"[1/40320*(945*sqrt(-c)*c^2*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(2240*a^8*c^2*x^8 + 5040*a^7*c^2*x^7 - 320*a^6*c^2*x^6 - 7560*a^5*c^2*x^5 - 4416*a^4*c^2*x^4 + 630*a^3*c^2*x^3 + 832*a^2*c^2*x^2 + 945*a*c^2*x + 1664*c^2)*sqrt(-a^2*c*x^2 + c))/a^4, -1/20160*(945*c^(5/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (2240*a^8*c^2*x^8 + 5040*a^7*c^2*x^7 - 320*a^6*c^2*x^6 - 7560*a^5*c^2*x^5 - 4416*a^4*c^2*x^4 + 630*a^3*c^2*x^3 + 832*a^2*c^2*x^2 + 945*a*c^2*x + 1664*c^2)*sqrt(-a^2*c*x^2 + c))/a^4]","A",0
1099,1,285,0,0.880435," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{1155 \, \sqrt{-c} c^{2} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, {\left(1680 \, a^{7} c^{2} x^{7} + 3840 \, a^{6} c^{2} x^{6} - 280 \, a^{5} c^{2} x^{5} - 6144 \, a^{4} c^{2} x^{4} - 3710 \, a^{3} c^{2} x^{3} + 768 \, a^{2} c^{2} x^{2} + 1155 \, a c^{2} x + 1536 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{26880 \, a^{3}}, -\frac{1155 \, c^{\frac{5}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + {\left(1680 \, a^{7} c^{2} x^{7} + 3840 \, a^{6} c^{2} x^{6} - 280 \, a^{5} c^{2} x^{5} - 6144 \, a^{4} c^{2} x^{4} - 3710 \, a^{3} c^{2} x^{3} + 768 \, a^{2} c^{2} x^{2} + 1155 \, a c^{2} x + 1536 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{13440 \, a^{3}}\right]"," ",0,"[1/26880*(1155*sqrt(-c)*c^2*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(1680*a^7*c^2*x^7 + 3840*a^6*c^2*x^6 - 280*a^5*c^2*x^5 - 6144*a^4*c^2*x^4 - 3710*a^3*c^2*x^3 + 768*a^2*c^2*x^2 + 1155*a*c^2*x + 1536*c^2)*sqrt(-a^2*c*x^2 + c))/a^3, -1/13440*(1155*c^(5/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (1680*a^7*c^2*x^7 + 3840*a^6*c^2*x^6 - 280*a^5*c^2*x^5 - 6144*a^4*c^2*x^4 - 3710*a^3*c^2*x^3 + 768*a^2*c^2*x^2 + 1155*a*c^2*x + 1536*c^2)*sqrt(-a^2*c*x^2 + c))/a^3]","A",0
1100,1,263,0,0.671752," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x*(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{105 \, \sqrt{-c} c^{2} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, {\left(120 \, a^{6} c^{2} x^{6} + 280 \, a^{5} c^{2} x^{5} - 24 \, a^{4} c^{2} x^{4} - 490 \, a^{3} c^{2} x^{3} - 312 \, a^{2} c^{2} x^{2} + 105 \, a c^{2} x + 216 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{1680 \, a^{2}}, -\frac{105 \, c^{\frac{5}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + {\left(120 \, a^{6} c^{2} x^{6} + 280 \, a^{5} c^{2} x^{5} - 24 \, a^{4} c^{2} x^{4} - 490 \, a^{3} c^{2} x^{3} - 312 \, a^{2} c^{2} x^{2} + 105 \, a c^{2} x + 216 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{840 \, a^{2}}\right]"," ",0,"[1/1680*(105*sqrt(-c)*c^2*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(120*a^6*c^2*x^6 + 280*a^5*c^2*x^5 - 24*a^4*c^2*x^4 - 490*a^3*c^2*x^3 - 312*a^2*c^2*x^2 + 105*a*c^2*x + 216*c^2)*sqrt(-a^2*c*x^2 + c))/a^2, -1/840*(105*c^(5/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (120*a^6*c^2*x^6 + 280*a^5*c^2*x^5 - 24*a^4*c^2*x^4 - 490*a^3*c^2*x^3 - 312*a^2*c^2*x^2 + 105*a*c^2*x + 216*c^2)*sqrt(-a^2*c*x^2 + c))/a^2]","A",0
1101,1,241,0,0.701900," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{105 \, \sqrt{-c} c^{2} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, {\left(40 \, a^{5} c^{2} x^{5} + 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x + 96 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{480 \, a}, -\frac{105 \, c^{\frac{5}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + {\left(40 \, a^{5} c^{2} x^{5} + 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x + 96 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{240 \, a}\right]"," ",0,"[1/480*(105*sqrt(-c)*c^2*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(40*a^5*c^2*x^5 + 96*a^4*c^2*x^4 - 10*a^3*c^2*x^3 - 192*a^2*c^2*x^2 - 135*a*c^2*x + 96*c^2)*sqrt(-a^2*c*x^2 + c))/a, -1/240*(105*c^(5/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (40*a^5*c^2*x^5 + 96*a^4*c^2*x^4 - 10*a^3*c^2*x^3 - 192*a^2*c^2*x^2 - 135*a*c^2*x + 96*c^2)*sqrt(-a^2*c*x^2 + c))/a]","A",0
1102,1,295,0,0.543073," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x,x, algorithm=""fricas"")","\left[-\frac{3}{4} \, c^{\frac{5}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + \frac{1}{2} \, c^{\frac{5}{2}} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - \frac{1}{60} \, {\left(12 \, a^{4} c^{2} x^{4} + 30 \, a^{3} c^{2} x^{3} - 4 \, a^{2} c^{2} x^{2} - 75 \, a c^{2} x - 68 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}, -\sqrt{-c} c^{2} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \frac{3}{8} \, \sqrt{-c} c^{2} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - \frac{1}{60} \, {\left(12 \, a^{4} c^{2} x^{4} + 30 \, a^{3} c^{2} x^{3} - 4 \, a^{2} c^{2} x^{2} - 75 \, a c^{2} x - 68 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}\right]"," ",0,"[-3/4*c^(5/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 1/2*c^(5/2)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 1/60*(12*a^4*c^2*x^4 + 30*a^3*c^2*x^3 - 4*a^2*c^2*x^2 - 75*a*c^2*x - 68*c^2)*sqrt(-a^2*c*x^2 + c), -sqrt(-c)*c^2*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 3/8*sqrt(-c)*c^2*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 1/60*(12*a^4*c^2*x^4 + 30*a^3*c^2*x^3 - 4*a^2*c^2*x^2 - 75*a*c^2*x - 68*c^2)*sqrt(-a^2*c*x^2 + c)]","A",0
1103,1,313,0,0.751404," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^2,x, algorithm=""fricas"")","\left[\frac{27 \, a c^{\frac{5}{2}} x \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + 24 \, a c^{\frac{5}{2}} x \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - {\left(6 \, a^{4} c^{2} x^{4} + 16 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 64 \, a c^{2} x + 24 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{24 \, x}, -\frac{96 \, a \sqrt{-c} c^{2} x \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - 27 \, a \sqrt{-c} c^{2} x \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 2 \, {\left(6 \, a^{4} c^{2} x^{4} + 16 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 64 \, a c^{2} x + 24 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{48 \, x}\right]"," ",0,"[1/24*(27*a*c^(5/2)*x*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 24*a*c^(5/2)*x*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - (6*a^4*c^2*x^4 + 16*a^3*c^2*x^3 - 3*a^2*c^2*x^2 - 64*a*c^2*x + 24*c^2)*sqrt(-a^2*c*x^2 + c))/x, -1/48*(96*a*sqrt(-c)*c^2*x*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 27*a*sqrt(-c)*c^2*x*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(6*a^4*c^2*x^4 + 16*a^3*c^2*x^3 - 3*a^2*c^2*x^2 - 64*a*c^2*x + 24*c^2)*sqrt(-a^2*c*x^2 + c))/x]","A",0
1104,1,329,0,0.677362," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^3,x, algorithm=""fricas"")","\left[\frac{36 \, a^{2} c^{\frac{5}{2}} x^{2} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + 3 \, a^{2} c^{\frac{5}{2}} x^{2} \log\left(-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 2 \, {\left(2 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 12 \, a c^{2} x + 3 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{12 \, x^{2}}, \frac{3 \, a^{2} \sqrt{-c} c^{2} x^{2} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 9 \, a^{2} \sqrt{-c} c^{2} x^{2} \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - {\left(2 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 12 \, a c^{2} x + 3 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{6 \, x^{2}}\right]"," ",0,"[1/12*(36*a^2*c^(5/2)*x^2*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 3*a^2*c^(5/2)*x^2*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*(2*a^4*c^2*x^4 + 6*a^3*c^2*x^3 - 2*a^2*c^2*x^2 + 12*a*c^2*x + 3*c^2)*sqrt(-a^2*c*x^2 + c))/x^2, 1/6*(3*a^2*sqrt(-c)*c^2*x^2*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 9*a^2*sqrt(-c)*c^2*x^2*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - (2*a^4*c^2*x^4 + 6*a^3*c^2*x^3 - 2*a^2*c^2*x^2 + 12*a*c^2*x + 3*c^2)*sqrt(-a^2*c*x^2 + c))/x^2]","A",0
1105,1,329,0,0.734714," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^4,x, algorithm=""fricas"")","\left[\frac{3 \, a^{3} c^{\frac{5}{2}} x^{3} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + 9 \, a^{3} c^{\frac{5}{2}} x^{3} \log\left(-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - {\left(3 \, a^{4} c^{2} x^{4} + 12 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 6 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{6 \, x^{3}}, \frac{36 \, a^{3} \sqrt{-c} c^{2} x^{3} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 3 \, a^{3} \sqrt{-c} c^{2} x^{3} \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, {\left(3 \, a^{4} c^{2} x^{4} + 12 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 6 \, a c^{2} x + 2 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{12 \, x^{3}}\right]"," ",0,"[1/6*(3*a^3*c^(5/2)*x^3*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 9*a^3*c^(5/2)*x^3*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - (3*a^4*c^2*x^4 + 12*a^3*c^2*x^3 - 2*a^2*c^2*x^2 + 6*a*c^2*x + 2*c^2)*sqrt(-a^2*c*x^2 + c))/x^3, 1/12*(36*a^3*sqrt(-c)*c^2*x^3*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 3*a^3*sqrt(-c)*c^2*x^3*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(3*a^4*c^2*x^4 + 12*a^3*c^2*x^3 - 2*a^2*c^2*x^2 + 6*a*c^2*x + 2*c^2)*sqrt(-a^2*c*x^2 + c))/x^3]","A",0
1106,1,329,0,0.834733," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^5,x, algorithm=""fricas"")","\left[-\frac{96 \, a^{4} c^{\frac{5}{2}} x^{4} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - 27 \, a^{4} c^{\frac{5}{2}} x^{4} \log\left(-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + 2 \, {\left(24 \, a^{4} c^{2} x^{4} - 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} + 16 \, a c^{2} x + 6 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{48 \, x^{4}}, \frac{27 \, a^{4} \sqrt{-c} c^{2} x^{4} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 24 \, a^{4} \sqrt{-c} c^{2} x^{4} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - {\left(24 \, a^{4} c^{2} x^{4} - 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} + 16 \, a c^{2} x + 6 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{24 \, x^{4}}\right]"," ",0,"[-1/48*(96*a^4*c^(5/2)*x^4*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - 27*a^4*c^(5/2)*x^4*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 2*(24*a^4*c^2*x^4 - 64*a^3*c^2*x^3 - 3*a^2*c^2*x^2 + 16*a*c^2*x + 6*c^2)*sqrt(-a^2*c*x^2 + c))/x^4, 1/24*(27*a^4*sqrt(-c)*c^2*x^4*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 24*a^4*sqrt(-c)*c^2*x^4*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - (24*a^4*c^2*x^4 - 64*a^3*c^2*x^3 - 3*a^2*c^2*x^2 + 16*a*c^2*x + 6*c^2)*sqrt(-a^2*c*x^2 + c))/x^4]","A",0
1107,1,286,0,0.842274," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","\left[\frac{315 \, \sqrt{-c} c^{3} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 2 \, {\left(112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right)} \sqrt{-a^{2} c x^{2} + c}}{1792 \, a}, -\frac{315 \, c^{\frac{7}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - {\left(112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right)} \sqrt{-a^{2} c x^{2} + c}}{896 \, a}\right]"," ",0,"[1/1792*(315*sqrt(-c)*c^3*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(112*a^7*c^3*x^7 + 256*a^6*c^3*x^6 - 168*a^5*c^3*x^5 - 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 + 768*a^2*c^3*x^2 + 581*a*c^3*x - 256*c^3)*sqrt(-a^2*c*x^2 + c))/a, -1/896*(315*c^(7/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (112*a^7*c^3*x^7 + 256*a^6*c^3*x^6 - 168*a^5*c^3*x^5 - 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 + 768*a^2*c^3*x^2 + 581*a*c^3*x - 256*c^3)*sqrt(-a^2*c*x^2 + c))/a]","A",0
1108,1,200,0,0.620180," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{9 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, {\left(a^{3} x^{3} + 2 \, a^{2} x^{2} + 5 \, a x - 14\right)} \sqrt{-a^{2} c x^{2} + c}}{6 \, {\left(a^{5} c x - a^{4} c\right)}}, \frac{9 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + {\left(a^{3} x^{3} + 2 \, a^{2} x^{2} + 5 \, a x - 14\right)} \sqrt{-a^{2} c x^{2} + c}}{3 \, {\left(a^{5} c x - a^{4} c\right)}}\right]"," ",0,"[-1/6*(9*(a*x - 1)*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(a^3*x^3 + 2*a^2*x^2 + 5*a*x - 14)*sqrt(-a^2*c*x^2 + c))/(a^5*c*x - a^4*c), 1/3*(9*(a*x - 1)*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (a^3*x^3 + 2*a^2*x^2 + 5*a*x - 14)*sqrt(-a^2*c*x^2 + c))/(a^5*c*x - a^4*c)]","A",0
1109,1,184,0,0.716746," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{5 \, {\left(a x - 1\right)} \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 3 \, a x - 8\right)}}{4 \, {\left(a^{4} c x - a^{3} c\right)}}, \frac{5 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 3 \, a x - 8\right)}}{2 \, {\left(a^{4} c x - a^{3} c\right)}}\right]"," ",0,"[-1/4*(5*(a*x - 1)*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 3*a*x - 8))/(a^4*c*x - a^3*c), 1/2*(5*(a*x - 1)*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 3*a*x - 8))/(a^4*c*x - a^3*c)]","A",0
1110,1,166,0,0.786019," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a x - 1\right)} \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - \sqrt{-a^{2} c x^{2} + c} {\left(a x - 3\right)}}{a^{3} c x - a^{2} c}, \frac{2 \, {\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(a x - 3\right)}}{a^{3} c x - a^{2} c}\right]"," ",0,"[-((a*x - 1)*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - sqrt(-a^2*c*x^2 + c)*(a*x - 3))/(a^3*c*x - a^2*c), (2*(a*x - 1)*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(a*x - 3))/(a^3*c*x - a^2*c)]","A",0
1111,1,152,0,0.671201," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a x - 1\right)} \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 4 \, \sqrt{-a^{2} c x^{2} + c}}{2 \, {\left(a^{2} c x - a c\right)}}, \frac{{\left(a x - 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c}}{a^{2} c x - a c}\right]"," ",0,"[-1/2*((a*x - 1)*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 4*sqrt(-a^2*c*x^2 + c))/(a^2*c*x - a*c), ((a*x - 1)*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - 2*sqrt(-a^2*c*x^2 + c))/(a^2*c*x - a*c)]","A",0
1112,1,147,0,0.924609," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 4 \, \sqrt{-a^{2} c x^{2} + c}}{2 \, {\left(a c x - c\right)}}, -\frac{{\left(a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + 2 \, \sqrt{-a^{2} c x^{2} + c}}{a c x - c}\right]"," ",0,"[1/2*((a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 4*sqrt(-a^2*c*x^2 + c))/(a*c*x - c), -((a*x - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + 2*sqrt(-a^2*c*x^2 + c))/(a*c*x - c)]","A",0
1113,1,178,0,0.701773," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^2/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - a x\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(3 \, a x - 1\right)}}{a c x^{2} - c x}, -\frac{2 \, {\left(a^{2} x^{2} - a x\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(3 \, a x - 1\right)}}{a c x^{2} - c x}\right]"," ",0,"[((a^2*x^2 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - sqrt(-a^2*c*x^2 + c)*(3*a*x - 1))/(a*c*x^2 - c*x), -(2*(a^2*x^2 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(3*a*x - 1))/(a*c*x^2 - c*x)]","A",0
1114,1,208,0,0.646526," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^3/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{5 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} {\left(8 \, a^{2} x^{2} - 3 \, a x - 1\right)}}{4 \, {\left(a c x^{3} - c x^{2}\right)}}, -\frac{5 \, {\left(a^{3} x^{3} - a^{2} x^{2}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(8 \, a^{2} x^{2} - 3 \, a x - 1\right)}}{2 \, {\left(a c x^{3} - c x^{2}\right)}}\right]"," ",0,"[1/4*(5*(a^3*x^3 - a^2*x^2)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*sqrt(-a^2*c*x^2 + c)*(8*a^2*x^2 - 3*a*x - 1))/(a*c*x^3 - c*x^2), -1/2*(5*(a^3*x^3 - a^2*x^2)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(8*a^2*x^2 - 3*a*x - 1))/(a*c*x^3 - c*x^2)]","A",0
1115,1,224,0,0.585249," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^4/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{9 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 2 \, {\left(14 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 2 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c}}{6 \, {\left(a c x^{4} - c x^{3}\right)}}, -\frac{9 \, {\left(a^{4} x^{4} - a^{3} x^{3}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(14 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 2 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c}}{3 \, {\left(a c x^{4} - c x^{3}\right)}}\right]"," ",0,"[1/6*(9*(a^4*x^4 - a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*(14*a^3*x^3 - 5*a^2*x^2 - 2*a*x - 1)*sqrt(-a^2*c*x^2 + c))/(a*c*x^4 - c*x^3), -1/3*(9*(a^4*x^4 - a^3*x^3)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (14*a^3*x^3 - 5*a^2*x^2 - 2*a*x - 1)*sqrt(-a^2*c*x^2 + c))/(a*c*x^4 - c*x^3)]","A",0
1116,1,229,0,0.563796," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + \sqrt{-a^{2} c x^{2} + c} {\left(3 \, a^{2} x^{2} - 14 \, a x + 10\right)}}{3 \, {\left(a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right)}}, -\frac{6 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(3 \, a^{2} x^{2} - 14 \, a x + 10\right)}}{3 \, {\left(a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right)}}\right]"," ",0,"[-1/3*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + sqrt(-a^2*c*x^2 + c)*(3*a^2*x^2 - 14*a*x + 10))/(a^6*c^2*x^2 - 2*a^5*c^2*x + a^4*c^2), -1/3*(6*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(3*a^2*x^2 - 14*a*x + 10))/(a^6*c^2*x^2 - 2*a^5*c^2*x + a^4*c^2)]","A",0
1117,1,215,0,0.650708," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} {\left(5 \, a x - 4\right)}}{6 \, {\left(a^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} c^{2}\right)}}, -\frac{3 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(5 \, a x - 4\right)}}{3 \, {\left(a^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} c^{2}\right)}}\right]"," ",0,"[-1/6*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*sqrt(-a^2*c*x^2 + c)*(5*a*x - 4))/(a^5*c^2*x^2 - 2*a^4*c^2*x + a^3*c^2), -1/3*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - sqrt(-a^2*c*x^2 + c)*(5*a*x - 4))/(a^5*c^2*x^2 - 2*a^4*c^2*x + a^3*c^2)]","A",0
1118,1,50,0,0.682425," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} c x^{2} + c} {\left(2 \, a x - 1\right)}}{3 \, {\left(a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right)}}"," ",0,"1/3*sqrt(-a^2*c*x^2 + c)*(2*a*x - 1)/(a^4*c^2*x^2 - 2*a^3*c^2*x + a^2*c^2)","A",0
1119,1,47,0,0.786152," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a x - 2\right)}}{3 \, {\left(a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"-1/3*sqrt(-a^2*c*x^2 + c)*(a*x - 2)/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)","A",0
1120,1,202,0,0.505317," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} {\left(4 \, a x - 5\right)}}{6 \, {\left(a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right)}}, -\frac{3 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(4 \, a x - 5\right)}}{3 \, {\left(a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right)}}\right]"," ",0,"[1/6*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*sqrt(-a^2*c*x^2 + c)*(4*a*x - 5))/(a^2*c^2*x^2 - 2*a*c^2*x + c^2), -1/3*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(4*a*x - 5))/(a^2*c^2*x^2 - 2*a*c^2*x + c^2)]","A",0
1121,1,238,0,0.678398," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(10 \, a^{2} x^{2} - 14 \, a x + 3\right)}}{3 \, {\left(a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right)}}, -\frac{6 \, {\left(a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(10 \, a^{2} x^{2} - 14 \, a x + 3\right)}}{3 \, {\left(a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right)}}\right]"," ",0,"[1/3*(3*(a^3*x^3 - 2*a^2*x^2 + a*x)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - sqrt(-a^2*c*x^2 + c)*(10*a^2*x^2 - 14*a*x + 3))/(a^2*c^2*x^3 - 2*a*c^2*x^2 + c^2*x), -1/3*(6*(a^3*x^3 - 2*a^2*x^2 + a*x)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(10*a^2*x^2 - 14*a*x + 3))/(a^2*c^2*x^3 - 2*a*c^2*x^2 + c^2*x)]","A",0
1122,1,266,0,0.867996," ","integrate((a*x+1)^2/(-a^2*x^2+1)/x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[\frac{21 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right)} \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 2 \, {\left(32 \, a^{3} x^{3} - 43 \, a^{2} x^{2} + 6 \, a x + 3\right)} \sqrt{-a^{2} c x^{2} + c}}{12 \, {\left(a^{2} c^{2} x^{4} - 2 \, a c^{2} x^{3} + c^{2} x^{2}\right)}}, -\frac{21 \, {\left(a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + {\left(32 \, a^{3} x^{3} - 43 \, a^{2} x^{2} + 6 \, a x + 3\right)} \sqrt{-a^{2} c x^{2} + c}}{6 \, {\left(a^{2} c^{2} x^{4} - 2 \, a c^{2} x^{3} + c^{2} x^{2}\right)}}\right]"," ",0,"[1/12*(21*(a^4*x^4 - 2*a^3*x^3 + a^2*x^2)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*(32*a^3*x^3 - 43*a^2*x^2 + 6*a*x + 3)*sqrt(-a^2*c*x^2 + c))/(a^2*c^2*x^4 - 2*a*c^2*x^3 + c^2*x^2), -1/6*(21*(a^4*x^4 - 2*a^3*x^3 + a^2*x^2)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (32*a^3*x^3 - 43*a^2*x^2 + 6*a*x + 3)*sqrt(-a^2*c*x^2 + c))/(a^2*c^2*x^4 - 2*a*c^2*x^3 + c^2*x^2)]","A",0
1123,1,75,0,0.774078," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 2\right)} \sqrt{-a^{2} c x^{2} + c}}{5 \, {\left(a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/5*(2*a^3*x^3 - 4*a^2*x^2 + a*x + 2)*sqrt(-a^2*c*x^2 + c)/(a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^2*c^3*x - a*c^3)","A",0
1124,1,124,0,0.974910," ","integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","-\frac{{\left(8 \, a^{5} x^{5} - 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 24 \, a^{2} x^{2} - 9 \, a x - 6\right)} \sqrt{-a^{2} c x^{2} + c}}{21 \, {\left(a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/21*(8*a^5*x^5 - 16*a^4*x^4 - 4*a^3*x^3 + 24*a^2*x^2 - 9*a*x - 6)*sqrt(-a^2*c*x^2 + c)/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)","A",0
1125,1,540,0,0.588763," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{{\left({\left(a^{6} c^{3} m^{6} + 21 \, a^{6} c^{3} m^{5} + 175 \, a^{6} c^{3} m^{4} + 735 \, a^{6} c^{3} m^{3} + 1624 \, a^{6} c^{3} m^{2} + 1764 \, a^{6} c^{3} m + 720 \, a^{6} c^{3}\right)} x^{7} + 2 \, {\left(a^{5} c^{3} m^{6} + 22 \, a^{5} c^{3} m^{5} + 190 \, a^{5} c^{3} m^{4} + 820 \, a^{5} c^{3} m^{3} + 1849 \, a^{5} c^{3} m^{2} + 2038 \, a^{5} c^{3} m + 840 \, a^{5} c^{3}\right)} x^{6} - {\left(a^{4} c^{3} m^{6} + 23 \, a^{4} c^{3} m^{5} + 207 \, a^{4} c^{3} m^{4} + 925 \, a^{4} c^{3} m^{3} + 2144 \, a^{4} c^{3} m^{2} + 2412 \, a^{4} c^{3} m + 1008 \, a^{4} c^{3}\right)} x^{5} - 4 \, {\left(a^{3} c^{3} m^{6} + 24 \, a^{3} c^{3} m^{5} + 226 \, a^{3} c^{3} m^{4} + 1056 \, a^{3} c^{3} m^{3} + 2545 \, a^{3} c^{3} m^{2} + 2952 \, a^{3} c^{3} m + 1260 \, a^{3} c^{3}\right)} x^{4} - {\left(a^{2} c^{3} m^{6} + 25 \, a^{2} c^{3} m^{5} + 247 \, a^{2} c^{3} m^{4} + 1219 \, a^{2} c^{3} m^{3} + 3112 \, a^{2} c^{3} m^{2} + 3796 \, a^{2} c^{3} m + 1680 \, a^{2} c^{3}\right)} x^{3} + 2 \, {\left(a c^{3} m^{6} + 26 \, a c^{3} m^{5} + 270 \, a c^{3} m^{4} + 1420 \, a c^{3} m^{3} + 3929 \, a c^{3} m^{2} + 5274 \, a c^{3} m + 2520 \, a c^{3}\right)} x^{2} + {\left(c^{3} m^{6} + 27 \, c^{3} m^{5} + 295 \, c^{3} m^{4} + 1665 \, c^{3} m^{3} + 5104 \, c^{3} m^{2} + 8028 \, c^{3} m + 5040 \, c^{3}\right)} x\right)} x^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040}"," ",0,"((a^6*c^3*m^6 + 21*a^6*c^3*m^5 + 175*a^6*c^3*m^4 + 735*a^6*c^3*m^3 + 1624*a^6*c^3*m^2 + 1764*a^6*c^3*m + 720*a^6*c^3)*x^7 + 2*(a^5*c^3*m^6 + 22*a^5*c^3*m^5 + 190*a^5*c^3*m^4 + 820*a^5*c^3*m^3 + 1849*a^5*c^3*m^2 + 2038*a^5*c^3*m + 840*a^5*c^3)*x^6 - (a^4*c^3*m^6 + 23*a^4*c^3*m^5 + 207*a^4*c^3*m^4 + 925*a^4*c^3*m^3 + 2144*a^4*c^3*m^2 + 2412*a^4*c^3*m + 1008*a^4*c^3)*x^5 - 4*(a^3*c^3*m^6 + 24*a^3*c^3*m^5 + 226*a^3*c^3*m^4 + 1056*a^3*c^3*m^3 + 2545*a^3*c^3*m^2 + 2952*a^3*c^3*m + 1260*a^3*c^3)*x^4 - (a^2*c^3*m^6 + 25*a^2*c^3*m^5 + 247*a^2*c^3*m^4 + 1219*a^2*c^3*m^3 + 3112*a^2*c^3*m^2 + 3796*a^2*c^3*m + 1680*a^2*c^3)*x^3 + 2*(a*c^3*m^6 + 26*a*c^3*m^5 + 270*a*c^3*m^4 + 1420*a*c^3*m^3 + 3929*a*c^3*m^2 + 5274*a*c^3*m + 2520*a*c^3)*x^2 + (c^3*m^6 + 27*c^3*m^5 + 295*c^3*m^4 + 1665*c^3*m^3 + 5104*c^3*m^2 + 8028*c^3*m + 5040*c^3)*x)*x^m/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)","B",0
1126,1,179,0,0.675131," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{{\left({\left(a^{4} c^{2} m^{3} + 7 \, a^{4} c^{2} m^{2} + 14 \, a^{4} c^{2} m + 8 \, a^{4} c^{2}\right)} x^{5} + 2 \, {\left(a^{3} c^{2} m^{3} + 8 \, a^{3} c^{2} m^{2} + 17 \, a^{3} c^{2} m + 10 \, a^{3} c^{2}\right)} x^{4} - 2 \, {\left(a c^{2} m^{3} + 10 \, a c^{2} m^{2} + 29 \, a c^{2} m + 20 \, a c^{2}\right)} x^{2} - {\left(c^{2} m^{3} + 11 \, c^{2} m^{2} + 38 \, c^{2} m + 40 \, c^{2}\right)} x\right)} x^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40}"," ",0,"-((a^4*c^2*m^3 + 7*a^4*c^2*m^2 + 14*a^4*c^2*m + 8*a^4*c^2)*x^5 + 2*(a^3*c^2*m^3 + 8*a^3*c^2*m^2 + 17*a^3*c^2*m + 10*a^3*c^2)*x^4 - 2*(a*c^2*m^3 + 10*a*c^2*m^2 + 29*a*c^2*m + 20*a*c^2)*x^2 - (c^2*m^3 + 11*c^2*m^2 + 38*c^2*m + 40*c^2)*x)*x^m/(m^4 + 12*m^3 + 49*m^2 + 78*m + 40)","B",0
1127,1,82,0,0.546635," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{{\left({\left(a^{2} c m^{2} + 3 \, a^{2} c m + 2 \, a^{2} c\right)} x^{3} + 2 \, {\left(a c m^{2} + 4 \, a c m + 3 \, a c\right)} x^{2} + {\left(c m^{2} + 5 \, c m + 6 \, c\right)} x\right)} x^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6}"," ",0,"((a^2*c*m^2 + 3*a^2*c*m + 2*a^2*c)*x^3 + 2*(a*c*m^2 + 4*a*c*m + 3*a*c)*x^2 + (c*m^2 + 5*c*m + 6*c)*x)*x^m/(m^3 + 6*m^2 + 11*m + 6)","A",0
1128,0,0,0,0.724925," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m/(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(\frac{x^{m}}{a^{2} c x^{2} - 2 \, a c x + c}, x\right)"," ",0,"integral(x^m/(a^2*c*x^2 - 2*a*c*x + c), x)","F",0
1129,0,0,0,0.729071," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{x^{m}}{a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x - c^{2}}, x\right)"," ",0,"integral(-x^m/(a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a*c^2*x - c^2), x)","F",0
1130,0,0,0,0.656813," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","{\rm integral}\left(\frac{x^{m}}{a^{6} c^{3} x^{6} - 2 \, a^{5} c^{3} x^{5} - a^{4} c^{3} x^{4} + 4 \, a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2} - 2 \, a c^{3} x + c^{3}}, x\right)"," ",0,"integral(x^m/(a^6*c^3*x^6 - 2*a^5*c^3*x^5 - a^4*c^3*x^4 + 4*a^3*c^3*x^3 - a^2*c^3*x^2 - 2*a*c^3*x + c^3), x)","F",0
1131,0,0,0,0.705246," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{4} c^{2} x^{4} + 2 \, a^{3} c^{2} x^{3} - 2 \, a c^{2} x - c^{2}\right)} \sqrt{-a^{2} c x^{2} + c} x^{m}, x\right)"," ",0,"integral(-(a^4*c^2*x^4 + 2*a^3*c^2*x^3 - 2*a*c^2*x - c^2)*sqrt(-a^2*c*x^2 + c)*x^m, x)","F",0
1132,0,0,0,0.552553," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a^{2} c x^{2} + 2 \, a c x + c\right)} \sqrt{-a^{2} c x^{2} + c} x^{m}, x\right)"," ",0,"integral((a^2*c*x^2 + 2*a*c*x + c)*sqrt(-a^2*c*x^2 + c)*x^m, x)","F",0
1133,0,0,0,0.652032," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a x + 1\right)} x^{m}}{a x - 1}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*(a*x + 1)*x^m/(a*x - 1), x)","F",0
1134,0,0,0,0.675968," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} x^{m}}{a^{2} c x^{2} - 2 \, a c x + c}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*x^m/(a^2*c*x^2 - 2*a*c*x + c), x)","F",0
1135,0,0,0,0.592702," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^m/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} x^{m}}{a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x - c^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*x^m/(a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a*c^2*x - c^2), x)","F",0
1136,0,0,0,0.635895," ","integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x + 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a x - 1}, x\right)"," ",0,"integral(-(a*x + 1)*(-a^2*c*x^2 + c)^p/(a*x - 1), x)","F",0
1137,1,89,0,0.592090," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{690 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(40 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} + 230 \, a^{3} c x^{3} + 272 \, a^{2} c x^{2} + 345 \, a c x + 544 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{240 \, a^{4}}"," ",0,"-1/240*(690*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (40*a^5*c*x^5 + 144*a^4*c*x^4 + 230*a^3*c*x^3 + 272*a^2*c*x^2 + 345*a*c*x + 544*c)*sqrt(-a^2*x^2 + 1))/a^4","A",0
1138,1,80,0,0.612556," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2*(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{390 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(24 \, a^{4} c x^{4} + 90 \, a^{3} c x^{3} + 152 \, a^{2} c x^{2} + 195 \, a c x + 304 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, a^{3}}"," ",0,"-1/120*(390*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c*x^4 + 90*a^3*c*x^3 + 152*a^2*c*x^2 + 195*a*c*x + 304*c)*sqrt(-a^2*x^2 + 1))/a^3","A",0
1139,1,71,0,0.460508," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{30 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{3} c x^{3} + 8 \, a^{2} c x^{2} + 15 \, a c x + 24 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{8 \, a^{2}}"," ",0,"-1/8*(30*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*c*x^3 + 8*a^2*c*x^2 + 15*a*c*x + 24*c)*sqrt(-a^2*x^2 + 1))/a^2","A",0
1140,1,62,0,0.634773," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{30 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{2} c x^{2} + 9 \, a c x + 22 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a}"," ",0,"-1/6*(30*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^2*c*x^2 + 9*a*c*x + 22*c)*sqrt(-a^2*x^2 + 1))/a","A",0
1141,1,69,0,0.595660," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x,x, algorithm=""fricas"")","-7 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + c \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} {\left(a c x + 6 \, c\right)}"," ",0,"-7*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + c*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 1/2*sqrt(-a^2*x^2 + 1)*(a*c*x + 6*c)","A",0
1142,1,80,0,0.672932," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x^2,x, algorithm=""fricas"")","-\frac{6 \, a c x \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 3 \, a c x \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + a c x + \sqrt{-a^{2} x^{2} + 1} {\left(a c x + c\right)}}{x}"," ",0,"-(6*a*c*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 3*a*c*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) + a*c*x + sqrt(-a^2*x^2 + 1)*(a*c*x + c))/x","A",0
1143,1,85,0,0.511295," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x^3,x, algorithm=""fricas"")","-\frac{4 \, a^{2} c x^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - 7 \, a^{2} c x^{2} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) + \sqrt{-a^{2} x^{2} + 1} {\left(6 \, a c x + c\right)}}{2 \, x^{2}}"," ",0,"-1/2*(4*a^2*c*x^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 7*a^2*c*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*(6*a*c*x + c))/x^2","A",0
1144,1,66,0,0.645244," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x^4,x, algorithm=""fricas"")","\frac{15 \, a^{3} c x^{3} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(22 \, a^{2} c x^{2} + 9 \, a c x + 2 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, x^{3}}"," ",0,"1/6*(15*a^3*c*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (22*a^2*c*x^2 + 9*a*c*x + 2*c)*sqrt(-a^2*x^2 + 1))/x^3","A",0
1145,1,75,0,0.554093," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x^5,x, algorithm=""fricas"")","\frac{15 \, a^{4} c x^{4} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(24 \, a^{3} c x^{3} + 15 \, a^{2} c x^{2} + 8 \, a c x + 2 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{8 \, x^{4}}"," ",0,"1/8*(15*a^4*c*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (24*a^3*c*x^3 + 15*a^2*c*x^2 + 8*a*c*x + 2*c)*sqrt(-a^2*x^2 + 1))/x^4","A",0
1146,1,84,0,0.642468," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x^6,x, algorithm=""fricas"")","\frac{195 \, a^{5} c x^{5} \log\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right) - {\left(304 \, a^{4} c x^{4} + 195 \, a^{3} c x^{3} + 152 \, a^{2} c x^{2} + 90 \, a c x + 24 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, x^{5}}"," ",0,"1/120*(195*a^5*c*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (304*a^4*c*x^4 + 195*a^3*c*x^3 + 152*a^2*c*x^2 + 90*a*c*x + 24*c)*sqrt(-a^2*x^2 + 1))/x^5","A",0
1147,1,93,0,0.832860," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{210 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(24 \, a^{4} c^{2} x^{4} + 90 \, a^{3} c^{2} x^{3} + 112 \, a^{2} c^{2} x^{2} + 15 \, a c^{2} x - 136 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, a}"," ",0,"-1/120*(210*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (24*a^4*c^2*x^4 + 90*a^3*c^2*x^3 + 112*a^2*c^2*x^2 + 15*a*c^2*x - 136*c^2)*sqrt(-a^2*x^2 + 1))/a","A",0
1148,1,114,0,0.635073," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{630 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(80 \, a^{6} c^{3} x^{6} + 280 \, a^{5} c^{3} x^{5} + 208 \, a^{4} c^{3} x^{4} - 350 \, a^{3} c^{3} x^{3} - 656 \, a^{2} c^{3} x^{2} - 245 \, a c^{3} x + 368 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{560 \, a}"," ",0,"-1/560*(630*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (80*a^6*c^3*x^6 + 280*a^5*c^3*x^5 + 208*a^4*c^3*x^4 - 350*a^3*c^3*x^3 - 656*a^2*c^3*x^2 - 245*a*c^3*x + 368*c^3)*sqrt(-a^2*x^2 + 1))/a","A",0
1149,1,137,0,0.650695," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","-\frac{6930 \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(896 \, a^{8} c^{4} x^{8} + 3024 \, a^{7} c^{4} x^{7} + 1024 \, a^{6} c^{4} x^{6} - 7224 \, a^{5} c^{4} x^{5} - 8448 \, a^{4} c^{4} x^{4} + 3066 \, a^{3} c^{4} x^{3} + 10240 \, a^{2} c^{4} x^{2} + 4599 \, a c^{4} x - 3712 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{8064 \, a}"," ",0,"-1/8064*(6930*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (896*a^8*c^4*x^8 + 3024*a^7*c^4*x^7 + 1024*a^6*c^4*x^6 - 7224*a^5*c^4*x^5 - 8448*a^4*c^4*x^4 + 3066*a^3*c^4*x^3 + 10240*a^2*c^4*x^2 + 4599*a*c^4*x - 3712*c^4)*sqrt(-a^2*x^2 + 1))/a","A",0
1150,1,103,0,0.622534," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2/(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(3 \, a^{2} x^{2} - 19 \, a x + 14\right)} \sqrt{-a^{2} x^{2} + 1} + 14}{3 \, {\left(a^{5} c x^{2} - 2 \, a^{4} c x + a^{3} c\right)}}"," ",0,"-1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^5*c*x^2 - 2*a^4*c*x + a^3*c)","A",0
1151,1,96,0,0.658972," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x/(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{5 \, a^{2} x^{2} - 10 \, a x + 6 \, {\left(a^{2} x^{2} - 2 \, a x + 1\right)} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - \sqrt{-a^{2} x^{2} + 1} {\left(7 \, a x - 5\right)} + 5}{3 \, {\left(a^{4} c x^{2} - 2 \, a^{3} c x + a^{2} c\right)}}"," ",0,"-1/3*(5*a^2*x^2 - 10*a*x + 6*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1)*(7*a*x - 5) + 5)/(a^4*c*x^2 - 2*a^3*c*x + a^2*c)","A",0
1152,1,54,0,0.530421," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{a^{2} x^{2} - 2 \, a x + \sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} + 1}{3 \, {\left(a^{3} c x^{2} - 2 \, a^{2} c x + a c\right)}}"," ",0,"1/3*(a^2*x^2 - 2*a*x + sqrt(-a^2*x^2 + 1)*(a*x + 1) + 1)/(a^3*c*x^2 - 2*a^2*c*x + a*c)","A",0
1153,1,91,0,0.553463," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{7 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 21 \, a x - {\left(2 \, a^{2} x^{2} - 6 \, a x + 7\right)} \sqrt{-a^{2} x^{2} + 1} - 7}{15 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}"," ",0,"1/15*(7*a^3*x^3 - 21*a^2*x^2 + 21*a*x - (2*a^2*x^2 - 6*a*x + 7)*sqrt(-a^2*x^2 + 1) - 7)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)","A",0
1154,1,144,0,0.548646," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{13 \, a^{5} x^{5} - 39 \, a^{4} x^{4} + 26 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 39 \, a x - {\left(8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right)} \sqrt{-a^{2} x^{2} + 1} + 13}{35 \, {\left(a^{6} c^{3} x^{5} - 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"1/35*(13*a^5*x^5 - 39*a^4*x^4 + 26*a^3*x^3 + 26*a^2*x^2 - 39*a*x - (8*a^4*x^4 - 24*a^3*x^3 + 20*a^2*x^2 + 4*a*x - 13)*sqrt(-a^2*x^2 + 1) + 13)/(a^6*c^3*x^5 - 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 - 3*a^2*c^3*x + a*c^3)","A",0
1155,1,198,0,0.602530," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","\frac{19 \, a^{7} x^{7} - 57 \, a^{6} x^{6} + 19 \, a^{5} x^{5} + 95 \, a^{4} x^{4} - 95 \, a^{3} x^{3} - 19 \, a^{2} x^{2} + 57 \, a x - {\left(16 \, a^{6} x^{6} - 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} + 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} + 6 \, a x + 19\right)} \sqrt{-a^{2} x^{2} + 1} - 19}{63 \, {\left(a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"1/63*(19*a^7*x^7 - 57*a^6*x^6 + 19*a^5*x^5 + 95*a^4*x^4 - 95*a^3*x^3 - 19*a^2*x^2 + 57*a*x - (16*a^6*x^6 - 48*a^5*x^5 + 24*a^4*x^4 + 56*a^3*x^3 - 66*a^2*x^2 + 6*a*x + 19)*sqrt(-a^2*x^2 + 1) - 19)/(a^8*c^4*x^7 - 3*a^7*c^4*x^6 + a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 - a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)","A",0
1156,1,399,0,0.669288," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{120 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) + {\left(12 \, a^{5} x^{5} + 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} + 120 \, a^{2} x^{2} + 240 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{60 \, {\left(a^{6} x^{2} - a^{4}\right)}}, -\frac{240 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right) - {\left(12 \, a^{5} x^{5} + 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} + 120 \, a^{2} x^{2} + 240 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{60 \, {\left(a^{6} x^{2} - a^{4}\right)}}\right]"," ",0,"[1/60*(120*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) + (12*a^5*x^5 + 45*a^4*x^4 + 80*a^3*x^3 + 120*a^2*x^2 + 240*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*x^2 - a^4), -1/60*(240*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x)) - (12*a^5*x^5 + 45*a^4*x^4 + 80*a^3*x^3 + 120*a^2*x^2 + 240*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*x^2 - a^4)]","A",0
1157,1,381,0,0.741076," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) + {\left(a^{4} x^{4} + 4 \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 16 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{4 \, {\left(a^{5} x^{2} - a^{3}\right)}}, -\frac{16 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right) - {\left(a^{4} x^{4} + 4 \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 16 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{4 \, {\left(a^{5} x^{2} - a^{3}\right)}}\right]"," ",0,"[1/4*(8*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) + (a^4*x^4 + 4*a^3*x^3 + 8*a^2*x^2 + 16*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^5*x^2 - a^3), -1/4*(16*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x)) - (a^4*x^4 + 4*a^3*x^3 + 8*a^2*x^2 + 16*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^5*x^2 - a^3)]","A",0
1158,1,367,0,0.796445," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) + {\left(2 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 24 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{4} x^{2} - a^{2}\right)}}, -\frac{24 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right) - {\left(2 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 24 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{4} x^{2} - a^{2}\right)}}\right]"," ",0,"[1/6*(12*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) + (2*a^3*x^3 + 9*a^2*x^2 + 24*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^4*x^2 - a^2), -1/6*(24*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x)) - (2*a^3*x^3 + 9*a^2*x^2 + 24*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^4*x^2 - a^2)]","A",0
1159,1,345,0,0.678331," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 6 \, a x\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a^{3} x^{2} - a\right)}}, -\frac{8 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 6 \, a x\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a^{3} x^{2} - a\right)}}\right]"," ",0,"[1/2*(4*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)) + sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 6*a*x)*sqrt(-a^2*x^2 + 1))/(a^3*x^2 - a), -1/2*(8*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x)) - sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 6*a*x)*sqrt(-a^2*x^2 + 1))/(a^3*x^2 - a)]","A",0
1160,0,0,0,0.854376," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/x,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)}}{a^{2} x^{3} - 2 \, a x^{2} + x}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x + 1)/(a^2*x^3 - 2*a*x^2 + x), x)","F",0
1161,0,0,0,0.710874," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/x^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)}}{a^{2} x^{4} - 2 \, a x^{3} + x^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x + 1)/(a^2*x^4 - 2*a*x^3 + x^2), x)","F",0
1162,1,450,0,0.728556," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/x^3,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{c} \log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{3} x^{3} - {\left(4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right) - \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(6 \, a + 1\right)} x^{2} - 6 \, a x - 1\right)}}{2 \, {\left(a^{2} x^{4} - x^{2}\right)}}, -\frac{8 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{-c} \arctan\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a^{2} - 2 \, a + 1\right)} x^{2} - 2 \, a x + 1\right)} \sqrt{-c}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right) + \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(6 \, a + 1\right)} x^{2} - 6 \, a x - 1\right)}}{2 \, {\left(a^{2} x^{4} - x^{2}\right)}}\right]"," ",0,"[1/2*(4*(a^4*x^4 - a^2*x^2)*sqrt(c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 - 6*a^2 + 4*a - 1)*x^4 - 6*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) - sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((6*a + 1)*x^2 - 6*a*x - 1))/(a^2*x^4 - x^2), -1/2*(8*(a^4*x^4 - a^2*x^2)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 - 2*a + 1)*x^2 - 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) + sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((6*a + 1)*x^2 - 6*a*x - 1))/(a^2*x^4 - x^2)]","A",0
1163,1,478,0,1.029067," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/x^4,x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{c} \log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{3} x^{3} - {\left(4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(24 \, a^{2} x^{2} - {\left(24 \, a^{2} + 9 \, a + 2\right)} x^{3} + 9 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{2} x^{5} - x^{3}\right)}}, -\frac{24 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{-c} \arctan\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a^{2} - 2 \, a + 1\right)} x^{2} - 2 \, a x + 1\right)} \sqrt{-c}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(24 \, a^{2} x^{2} - {\left(24 \, a^{2} + 9 \, a + 2\right)} x^{3} + 9 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{2} x^{5} - x^{3}\right)}}\right]"," ",0,"[1/6*(12*(a^5*x^5 - a^3*x^3)*sqrt(c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 - 6*a^2 + 4*a - 1)*x^4 - 6*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) + sqrt(-a^2*c*x^2 + c)*(24*a^2*x^2 - (24*a^2 + 9*a + 2)*x^3 + 9*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*x^5 - x^3), -1/6*(24*(a^5*x^5 - a^3*x^3)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 - 2*a + 1)*x^2 - 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) - sqrt(-a^2*c*x^2 + c)*(24*a^2*x^2 - (24*a^2 + 9*a + 2)*x^3 + 9*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*x^5 - x^3)]","A",0
1164,1,504,0,0.884123," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/x^5,x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{6} x^{6} - a^{4} x^{4}\right)} \sqrt{c} \log\left(-\frac{4 \, a^{5} c x^{5} - {\left(2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right)} c x^{6} - {\left(4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{3} x^{3} - {\left(4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right)} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right) + {\left(16 \, a^{3} x^{3} - {\left(16 \, a^{3} + 8 \, a^{2} + 4 \, a + 1\right)} x^{4} + 8 \, a^{2} x^{2} + 4 \, a x + 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{4 \, {\left(a^{2} x^{6} - x^{4}\right)}}, -\frac{16 \, {\left(a^{6} x^{6} - a^{4} x^{4}\right)} \sqrt{-c} \arctan\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a^{2} - 2 \, a + 1\right)} x^{2} - 2 \, a x + 1\right)} \sqrt{-c}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} - a^{2}\right)} c x^{4} - {\left(a^{2} - 2 \, a + 1\right)} c x^{2} - 2 \, a c x + c}\right) - {\left(16 \, a^{3} x^{3} - {\left(16 \, a^{3} + 8 \, a^{2} + 4 \, a + 1\right)} x^{4} + 8 \, a^{2} x^{2} + 4 \, a x + 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{4 \, {\left(a^{2} x^{6} - x^{4}\right)}}\right]"," ",0,"[1/4*(8*(a^6*x^6 - a^4*x^4)*sqrt(c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 - 6*a^2 + 4*a - 1)*x^4 - 6*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) + (16*a^3*x^3 - (16*a^3 + 8*a^2 + 4*a + 1)*x^4 + 8*a^2*x^2 + 4*a*x + 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^2*x^6 - x^4), -1/4*(16*(a^6*x^6 - a^4*x^4)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 - 2*a + 1)*x^2 - 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) - (16*a^3*x^3 - (16*a^3 + 8*a^2 + 4*a + 1)*x^4 + 8*a^2*x^2 + 4*a*x + 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^2*x^6 - x^4)]","A",0
1165,1,67,0,0.637838," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(a^{3} c x^{4} + 4 \, a^{2} c x^{3} + 6 \, a c x^{2} + 4 \, c x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{4 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/4*(a^3*c*x^4 + 4*a^2*c*x^3 + 6*a*c*x^2 + 4*c*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1166,1,98,0,0.604456," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\frac{{\left(5 \, a^{5} c^{2} x^{6} + 18 \, a^{4} c^{2} x^{5} + 15 \, a^{3} c^{2} x^{4} - 20 \, a^{2} c^{2} x^{3} - 45 \, a c^{2} x^{2} - 30 \, c^{2} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{30 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/30*(5*a^5*c^2*x^6 + 18*a^4*c^2*x^5 + 15*a^3*c^2*x^4 - 20*a^2*c^2*x^3 - 45*a*c^2*x^2 - 30*c^2*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1167,1,120,0,0.989810," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","-\frac{{\left(21 \, a^{7} c^{3} x^{8} + 72 \, a^{6} c^{3} x^{7} + 28 \, a^{5} c^{3} x^{6} - 168 \, a^{4} c^{3} x^{5} - 210 \, a^{3} c^{3} x^{4} + 56 \, a^{2} c^{3} x^{3} + 252 \, a c^{3} x^{2} + 168 \, c^{3} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{168 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/168*(21*a^7*c^3*x^8 + 72*a^6*c^3*x^7 + 28*a^5*c^3*x^6 - 168*a^4*c^3*x^5 - 210*a^3*c^3*x^4 + 56*a^2*c^3*x^3 + 252*a*c^3*x^2 + 168*c^3*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1168,1,120,0,0.693475," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(9/2),x, algorithm=""fricas"")","\frac{{\left(21 \, a^{9} c^{4} x^{10} + 70 \, a^{8} c^{4} x^{9} - 240 \, a^{6} c^{4} x^{7} - 210 \, a^{5} c^{4} x^{6} + 252 \, a^{4} c^{4} x^{5} + 420 \, a^{3} c^{4} x^{4} - 315 \, a c^{4} x^{2} - 210 \, c^{4} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{210 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/210*(21*a^9*c^4*x^10 + 70*a^8*c^4*x^9 - 240*a^6*c^4*x^7 - 210*a^5*c^4*x^6 + 252*a^4*c^4*x^5 + 420*a^3*c^4*x^4 - 315*a*c^4*x^2 - 210*c^4*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1169,1,382,0,0.611969," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x + {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x - {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right)}{2 \, {\left(a^{4} c x^{3} - a^{3} c x^{2} - a^{2} c x + a c\right)}}, \frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x + {\left(a^{3} x^{3} - a^{2} x^{2} - a x + 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right)}{a^{4} c x^{3} - a^{3} c x^{2} - a^{2} c x + a c}\right]"," ",0,"[1/2*(4*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x + (a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x - (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)))/(a^4*c*x^3 - a^3*c*x^2 - a^2*c*x + a*c), (2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x + (a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x)))/(a^4*c*x^3 - a^3*c*x^2 - a^2*c*x + a*c)]","B",0
1170,1,72,0,0.693320," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x^{2} - 2 \, x\right)}}{2 \, {\left(a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x - c^{2}\right)}}"," ",0,"1/2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x^2 - 2*x)/(a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a*c^2*x - c^2)","A",0
1171,1,459,0,0.780586," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) + 4 \, {\left(10 \, a^{3} x^{3} - 27 \, a^{2} x^{2} + 21 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{96 \, {\left(a^{6} c^{3} x^{5} - 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x + a c^{3}\right)}}, \frac{3 \, {\left(a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) + 2 \, {\left(10 \, a^{3} x^{3} - 27 \, a^{2} x^{2} + 21 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{48 \, {\left(a^{6} c^{3} x^{5} - 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x + a c^{3}\right)}}\right]"," ",0,"[1/96*(3*(a^5*x^5 - 3*a^4*x^4 + 2*a^3*x^3 + 2*a^2*x^2 - 3*a*x + 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) + 4*(10*a^3*x^3 - 27*a^2*x^2 + 21*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*c^3*x^5 - 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 - 3*a^2*c^3*x + a*c^3), 1/48*(3*(a^5*x^5 - 3*a^4*x^4 + 2*a^3*x^3 + 2*a^2*x^2 - 3*a*x + 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) + 2*(10*a^3*x^3 - 27*a^2*x^2 + 21*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*c^3*x^5 - 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 - 3*a^2*c^3*x + a*c^3)]","A",0
1172,1,565,0,0.742984," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{7} x^{7} - 3 \, a^{6} x^{6} + a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} - a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) + 4 \, {\left(32 \, a^{5} x^{5} - 81 \, a^{4} x^{4} + 19 \, a^{3} x^{3} + 99 \, a^{2} x^{2} - 81 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{384 \, {\left(a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right)}}, \frac{15 \, {\left(a^{7} x^{7} - 3 \, a^{6} x^{6} + a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} - a^{2} x^{2} + 3 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) + 2 \, {\left(32 \, a^{5} x^{5} - 81 \, a^{4} x^{4} + 19 \, a^{3} x^{3} + 99 \, a^{2} x^{2} - 81 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{192 \, {\left(a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right)}}\right]"," ",0,"[1/384*(15*(a^7*x^7 - 3*a^6*x^6 + a^5*x^5 + 5*a^4*x^4 - 5*a^3*x^3 - a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) + 4*(32*a^5*x^5 - 81*a^4*x^4 + 19*a^3*x^3 + 99*a^2*x^2 - 81*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 - 3*a^7*c^4*x^6 + a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 - a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4), 1/192*(15*(a^7*x^7 - 3*a^6*x^6 + a^5*x^5 + 5*a^4*x^4 - 5*a^3*x^3 - a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) + 2*(32*a^5*x^5 - 81*a^4*x^4 + 19*a^3*x^3 + 99*a^2*x^2 - 81*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 - 3*a^7*c^4*x^6 + a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 - a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)]","A",0
1173,0,0,0,0.482524," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} x^{m}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x + 1)*x^m/(a^2*x^2 - 2*a*x + 1), x)","F",0
1174,0,0,0,0.673192," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p} x^{m}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*(-a^2*c*x^2 + c)^p*x^m/(a^2*x^2 - 2*a*x + 1), x)","F",0
1175,0,0,0,0.650072," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a x^{4} + x^{3}\right)} \sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral((a*x^4 + x^3)*sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p/(a^2*x^2 - 2*a*x + 1), x)","F",0
1176,0,0,0,0.986160," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{3} + x^{2}\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x^3 + x^2)*(-a^2*c*x^2 + c)^p/(a^2*x^2 - 2*a*x + 1), x)","F",0
1177,0,0,0,0.833860," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x^{2} + x\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x^2 + x)*(-a^2*c*x^2 + c)^p/(a^2*x^2 - 2*a*x + 1), x)","F",0
1178,0,0,0,0.671005," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*(-a^2*c*x^2 + c)^p/(a^2*x^2 - 2*a*x + 1), x)","F",0
1179,0,0,0,1.257714," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p/x,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a^{2} x^{3} - 2 \, a x^{2} + x}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*(-a^2*c*x^2 + c)^p/(a^2*x^3 - 2*a*x^2 + x), x)","F",0
1180,0,0,0,0.771723," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p/x^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a^{2} x^{4} - 2 \, a x^{3} + x^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*(-a^2*c*x^2 + c)^p/(a^2*x^4 - 2*a*x^3 + x^2), x)","F",0
1181,0,0,0,0.928957," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p/x^3,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x + 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a^{2} x^{5} - 2 \, a x^{4} + x^{3}}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*(-a^2*c*x^2 + c)^p/(a^2*x^5 - 2*a*x^4 + x^3), x)","F",0
1182,1,101,0,0.451432," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^5,x, algorithm=""fricas"")","-\frac{1}{11} \, a^{10} c^{5} x^{11} - \frac{2}{5} \, a^{9} c^{5} x^{10} - \frac{1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac{14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x"," ",0,"-1/11*a^10*c^5*x^11 - 2/5*a^9*c^5*x^10 - 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 + 2*a^6*c^5*x^7 - 14/5*a^4*c^5*x^5 - 2*a^3*c^5*x^4 + a^2*c^5*x^3 + 2*a*c^5*x^2 + c^5*x","A",0
1183,1,92,0,0.649232," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","\frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{2} \, a^{7} c^{4} x^{8} + \frac{4}{7} \, a^{6} c^{4} x^{7} - \frac{2}{3} \, a^{5} c^{4} x^{6} - 2 \, a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac{4}{3} \, a^{2} c^{4} x^{3} + 2 \, a c^{4} x^{2} + c^{4} x"," ",0,"1/9*a^8*c^4*x^9 + 1/2*a^7*c^4*x^8 + 4/7*a^6*c^4*x^7 - 2/3*a^5*c^4*x^6 - 2*a^4*c^4*x^5 - a^3*c^4*x^4 + 4/3*a^2*c^4*x^3 + 2*a*c^4*x^2 + c^4*x","A",0
1184,1,59,0,0.570153," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{1}{7} \, a^{6} c^{3} x^{7} - \frac{2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac{5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x"," ",0,"-1/7*a^6*c^3*x^7 - 2/3*a^5*c^3*x^6 - a^4*c^3*x^5 + 5/3*a^2*c^3*x^3 + 2*a*c^3*x^2 + c^3*x","A",0
1185,1,47,0,0.591942," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{1}{5} \, a^{4} c^{2} x^{5} + a^{3} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{3} + 2 \, a c^{2} x^{2} + c^{2} x"," ",0,"1/5*a^4*c^2*x^5 + a^3*c^2*x^4 + 2*a^2*c^2*x^3 + 2*a*c^2*x^2 + c^2*x","B",0
1186,1,37,0,0.634881," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{a^{3} c x^{3} + 6 \, a^{2} c x^{2} + 21 \, a c x + 24 \, c \log\left(a x - 1\right)}{3 \, a}"," ",0,"-1/3*(a^3*c*x^3 + 6*a^2*c*x^2 + 21*a*c*x + 24*c*log(a*x - 1))/a","A",0
1187,1,19,0,0.542029," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{x}{a^{2} c x^{2} - 2 \, a c x + c}"," ",0,"x/(a^2*c*x^2 - 2*a*c*x + c)","A",0
1188,1,41,0,0.422175," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{1}{3 \, {\left(a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right)}}"," ",0,"-1/3/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)","B",0
1189,1,147,0,0.593240," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{6 \, a^{3} x^{3} - 24 \, a^{2} x^{2} + 38 \, a x - 3 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \log\left(a x + 1\right) + 3 \, {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right)} \log\left(a x - 1\right) - 32}{96 \, {\left(a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"-1/96*(6*a^3*x^3 - 24*a^2*x^2 + 38*a*x - 3*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(a*x + 1) + 3*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(a*x - 1) - 32)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)","B",0
1190,1,191,0,0.751977," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","-\frac{30 \, a^{5} x^{5} - 120 \, a^{4} x^{4} + 160 \, a^{3} x^{3} - 40 \, a^{2} x^{2} - 94 \, a x - 15 \, {\left(a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right)} \log\left(a x + 1\right) + 15 \, {\left(a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right)} \log\left(a x - 1\right) + 96}{320 \, {\left(a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"-1/320*(30*a^5*x^5 - 120*a^4*x^4 + 160*a^3*x^3 - 40*a^2*x^2 - 94*a*x - 15*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(a*x + 1) + 15*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(a*x - 1) + 96)/(a^7*c^4*x^6 - 4*a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^3*c^4*x^2 + 4*a^2*c^4*x - a*c^4)","A",0
1191,0,0,0,0.741918," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} x^{2} + 2 \, a x + 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right)"," ",0,"integral((a^2*x^2 + 2*a*x + 1)*(-a^2*c*x^2 + c)^p/(a^2*x^2 - 2*a*x + 1), x)","F",0
1192,1,137,0,0.655055," ","integrate((-a^2*c*x^2+c)^4/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{630 \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(128 \, a^{8} c^{4} x^{8} - 144 \, a^{7} c^{4} x^{7} - 512 \, a^{6} c^{4} x^{6} + 600 \, a^{5} c^{4} x^{5} + 768 \, a^{4} c^{4} x^{4} - 978 \, a^{3} c^{4} x^{3} - 512 \, a^{2} c^{4} x^{2} + 837 \, a c^{4} x + 128 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{1152 \, a}"," ",0,"-1/1152*(630*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (128*a^8*c^4*x^8 - 144*a^7*c^4*x^7 - 512*a^6*c^4*x^6 + 600*a^5*c^4*x^5 + 768*a^4*c^4*x^4 - 978*a^3*c^4*x^3 - 512*a^2*c^4*x^2 + 837*a*c^4*x + 128*c^4)*sqrt(-a^2*x^2 + 1))/a","A",0
1193,1,114,0,0.723654," ","integrate((-a^2*c*x^2+c)^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{210 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(48 \, a^{6} c^{3} x^{6} - 56 \, a^{5} c^{3} x^{5} - 144 \, a^{4} c^{3} x^{4} + 182 \, a^{3} c^{3} x^{3} + 144 \, a^{2} c^{3} x^{2} - 231 \, a c^{3} x - 48 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{336 \, a}"," ",0,"-1/336*(210*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (48*a^6*c^3*x^6 - 56*a^5*c^3*x^5 - 144*a^4*c^3*x^4 + 182*a^3*c^3*x^3 + 144*a^2*c^3*x^2 - 231*a*c^3*x - 48*c^3)*sqrt(-a^2*x^2 + 1))/a","A",0
1194,1,93,0,0.661432," ","integrate((-a^2*c*x^2+c)^2/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{30 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(8 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 16 \, a^{2} c^{2} x^{2} + 25 \, a c^{2} x + 8 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{40 \, a}"," ",0,"-1/40*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (8*a^4*c^2*x^4 - 10*a^3*c^2*x^3 - 16*a^2*c^2*x^2 + 25*a*c^2*x + 8*c^2)*sqrt(-a^2*x^2 + 1))/a","A",0
1195,1,62,0,0.793213," ","integrate((-a^2*c*x^2+c)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{6 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(2 \, a^{2} c x^{2} - 3 \, a c x - 2 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a}"," ",0,"-1/6*(6*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^2*c*x^2 - 3*a*c*x - 2*c)*sqrt(-a^2*x^2 + 1))/a","A",0
1196,1,31,0,0.593496," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{a x + \sqrt{-a^{2} x^{2} + 1} + 1}{a^{2} c x + a c}"," ",0,"-(a*x + sqrt(-a^2*x^2 + 1) + 1)/(a^2*c*x + a*c)","A",0
1197,1,87,0,0.595534," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{a^{3} x^{3} + a^{2} x^{2} - a x + {\left(2 \, a^{2} x^{2} + 2 \, a x - 1\right)} \sqrt{-a^{2} x^{2} + 1} - 1}{3 \, {\left(a^{4} c^{2} x^{3} + a^{3} c^{2} x^{2} - a^{2} c^{2} x - a c^{2}\right)}}"," ",0,"-1/3*(a^3*x^3 + a^2*x^2 - a*x + (2*a^2*x^2 + 2*a*x - 1)*sqrt(-a^2*x^2 + 1) - 1)/(a^4*c^2*x^3 + a^3*c^2*x^2 - a^2*c^2*x - a*c^2)","A",0
1198,1,141,0,0.836137," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 3 \, a x + {\left(8 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 12 \, a x + 3\right)} \sqrt{-a^{2} x^{2} + 1} + 3}{15 \, {\left(a^{6} c^{3} x^{5} + a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x + a c^{3}\right)}}"," ",0,"-1/15*(3*a^5*x^5 + 3*a^4*x^4 - 6*a^3*x^3 - 6*a^2*x^2 + 3*a*x + (8*a^4*x^4 + 8*a^3*x^3 - 12*a^2*x^2 - 12*a*x + 3)*sqrt(-a^2*x^2 + 1) + 3)/(a^6*c^3*x^5 + a^5*c^3*x^4 - 2*a^4*c^3*x^3 - 2*a^3*c^3*x^2 + a^2*c^3*x + a*c^3)","B",0
1199,1,197,0,0.592941," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","-\frac{5 \, a^{7} x^{7} + 5 \, a^{6} x^{6} - 15 \, a^{5} x^{5} - 15 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 15 \, a^{2} x^{2} - 5 \, a x + {\left(16 \, a^{6} x^{6} + 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} - 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} + 30 \, a x - 5\right)} \sqrt{-a^{2} x^{2} + 1} - 5}{35 \, {\left(a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x - a c^{4}\right)}}"," ",0,"-1/35*(5*a^7*x^7 + 5*a^6*x^6 - 15*a^5*x^5 - 15*a^4*x^4 + 15*a^3*x^3 + 15*a^2*x^2 - 5*a*x + (16*a^6*x^6 + 16*a^5*x^5 - 40*a^4*x^4 - 40*a^3*x^3 + 30*a^2*x^2 + 30*a*x - 5)*sqrt(-a^2*x^2 + 1) - 5)/(a^8*c^4*x^7 + a^7*c^4*x^6 - 3*a^6*c^4*x^5 - 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 + 3*a^3*c^4*x^2 - a^2*c^4*x - a*c^4)","B",0
1200,1,249,0,0.753817," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x, algorithm=""fricas"")","-\frac{35 \, a^{9} x^{9} + 35 \, a^{8} x^{8} - 140 \, a^{7} x^{7} - 140 \, a^{6} x^{6} + 210 \, a^{5} x^{5} + 210 \, a^{4} x^{4} - 140 \, a^{3} x^{3} - 140 \, a^{2} x^{2} + 35 \, a x + {\left(128 \, a^{8} x^{8} + 128 \, a^{7} x^{7} - 448 \, a^{6} x^{6} - 448 \, a^{5} x^{5} + 560 \, a^{4} x^{4} + 560 \, a^{3} x^{3} - 280 \, a^{2} x^{2} - 280 \, a x + 35\right)} \sqrt{-a^{2} x^{2} + 1} + 35}{315 \, {\left(a^{10} c^{5} x^{9} + a^{9} c^{5} x^{8} - 4 \, a^{8} c^{5} x^{7} - 4 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} + 6 \, a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} - 4 \, a^{3} c^{5} x^{2} + a^{2} c^{5} x + a c^{5}\right)}}"," ",0,"-1/315*(35*a^9*x^9 + 35*a^8*x^8 - 140*a^7*x^7 - 140*a^6*x^6 + 210*a^5*x^5 + 210*a^4*x^4 - 140*a^3*x^3 - 140*a^2*x^2 + 35*a*x + (128*a^8*x^8 + 128*a^7*x^7 - 448*a^6*x^6 - 448*a^5*x^5 + 560*a^4*x^4 + 560*a^3*x^3 - 280*a^2*x^2 - 280*a*x + 35)*sqrt(-a^2*x^2 + 1) + 35)/(a^10*c^5*x^9 + a^9*c^5*x^8 - 4*a^8*c^5*x^7 - 4*a^7*c^5*x^6 + 6*a^6*c^5*x^5 + 6*a^5*c^5*x^4 - 4*a^4*c^5*x^3 - 4*a^3*c^5*x^2 + a^2*c^5*x + a*c^5)","B",0
1201,1,80,0,0.712812," ","integrate(x^m*(-a^2*c*x^2+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(a m + a\right)} x^{2} - {\left(m + 2\right)} x\right)} x^{m}}{{\left(a^{2} m^{2} + 3 \, a^{2} m + 2 \, a^{2}\right)} x^{2} - m^{2} - 3 \, m - 2}"," ",0,"sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((a*m + a)*x^2 - (m + 2)*x)*x^m/((a^2*m^2 + 3*a^2*m + 2*a^2)*x^2 - m^2 - 3*m - 2)","A",0
1202,1,50,0,0.495466," ","integrate(x^2*(-a^2*c*x^2+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} c x^{2} + c} {\left(3 \, a x^{4} - 4 \, x^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{12 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/12*sqrt(-a^2*c*x^2 + c)*(3*a*x^4 - 4*x^3)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1203,1,50,0,0.735101," ","integrate(x*(-a^2*c*x^2+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(2 \, a x^{3} - 3 \, x^{2}\right)}}{6 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/6*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(2*a*x^3 - 3*x^2)/(a^2*x^2 - 1)","A",0
1204,1,47,0,0.594675," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x^{2} - 2 \, x\right)}}{2 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x^2 - 2*x)/(a^2*x^2 - 1)","A",0
1205,1,261,0,0.765584," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x,x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x^{4} - 1\right)} \sqrt{c} - c}{a^{2} x^{4} - x^{2}}\right) + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x - a\right)}}{2 \, {\left(a^{2} x^{2} - 1\right)}}, \frac{{\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x^{2} + 1\right)} \sqrt{-c}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x - a\right)}}{a^{2} x^{2} - 1}\right]"," ",0,"[1/2*((a^2*x^2 - 1)*sqrt(c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x^4 - 1)*sqrt(c) - c)/(a^2*x^4 - x^2)) + 2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x - a))/(a^2*x^2 - 1), ((a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x^2 + 1)*sqrt(-c)/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x - a))/(a^2*x^2 - 1)]","B",0
1206,1,263,0,0.790237," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x, algorithm=""fricas"")","\left[\frac{{\left(a^{3} x^{3} - a x\right)} \sqrt{c} \log\left(\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x^{4} - 1\right)} \sqrt{c} - c}{a^{2} x^{4} - x^{2}}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x - 1\right)}}{2 \, {\left(a^{2} x^{3} - x\right)}}, -\frac{{\left(a^{3} x^{3} - a x\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x^{2} + 1\right)} \sqrt{-c}}{a^{2} c x^{4} - {\left(a^{2} + 1\right)} c x^{2} + c}\right) + \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(x - 1\right)}}{a^{2} x^{3} - x}\right]"," ",0,"[1/2*((a^3*x^3 - a*x)*sqrt(c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x^4 - 1)*sqrt(c) - c)/(a^2*x^4 - x^2)) - 2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x - 1))/(a^2*x^3 - x), -((a^3*x^3 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x^2 + 1)*sqrt(-c)/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x - 1))/(a^2*x^3 - x)]","B",0
1207,1,68,0,0.576400," ","integrate((-a^2*c*x^2+c)^(3/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{3} c x^{4} - 4 \, a^{2} c x^{3} - 6 \, a c x^{2} + 12 \, c x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{12 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/12*(3*a^3*c*x^4 - 4*a^2*c*x^3 - 6*a*c*x^2 + 12*c*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1208,1,98,0,0.614138," ","integrate((-a^2*c*x^2+c)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{{\left(5 \, a^{5} c^{2} x^{6} - 6 \, a^{4} c^{2} x^{5} - 15 \, a^{3} c^{2} x^{4} + 20 \, a^{2} c^{2} x^{3} + 15 \, a c^{2} x^{2} - 30 \, c^{2} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{30 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/30*(5*a^5*c^2*x^6 - 6*a^4*c^2*x^5 - 15*a^3*c^2*x^4 + 20*a^2*c^2*x^3 + 15*a*c^2*x^2 - 30*c^2*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1209,1,120,0,0.606752," ","integrate((-a^2*c*x^2+c)^(7/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(35 \, a^{7} c^{3} x^{8} - 40 \, a^{6} c^{3} x^{7} - 140 \, a^{5} c^{3} x^{6} + 168 \, a^{4} c^{3} x^{5} + 210 \, a^{3} c^{3} x^{4} - 280 \, a^{2} c^{3} x^{3} - 140 \, a c^{3} x^{2} + 280 \, c^{3} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{280 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/280*(35*a^7*c^3*x^8 - 40*a^6*c^3*x^7 - 140*a^5*c^3*x^6 + 168*a^4*c^3*x^5 + 210*a^3*c^3*x^4 - 280*a^2*c^3*x^3 - 140*a*c^3*x^2 + 280*c^3*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1210,1,142,0,0.602894," ","integrate((-a^2*c*x^2+c)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","\frac{{\left(63 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} + 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} - 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} + 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} - 630 \, c^{4} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{630 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/630*(63*a^9*c^4*x^10 - 70*a^8*c^4*x^9 - 315*a^7*c^4*x^8 + 360*a^6*c^4*x^7 + 630*a^5*c^4*x^6 - 756*a^4*c^4*x^5 - 630*a^3*c^4*x^4 + 840*a^2*c^4*x^3 + 315*a*c^4*x^2 - 630*c^4*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1211,1,227,0,0.597497," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left(a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right)}{2 \, a \sqrt{c}}, \frac{\sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right)}{a c}\right]"," ",0,"[1/2*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x - (a^4*x^4 + 4*a^3*x^3 + 6*a^2*x^2 + 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1))/(a*sqrt(c)), sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 + 2*a^3*c*x^3 - a^2*c*x^2 - 2*a*c*x))/(a*c)]","B",0
1212,1,343,0,0.714522," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\left[-\frac{4 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x - {\left(a^{3} x^{3} + a^{2} x^{2} - a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right)}{8 \, {\left(a^{4} c^{2} x^{3} + a^{3} c^{2} x^{2} - a^{2} c^{2} x - a c^{2}\right)}}, -\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x - {\left(a^{3} x^{3} + a^{2} x^{2} - a x - 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right)}{4 \, {\left(a^{4} c^{2} x^{3} + a^{3} c^{2} x^{2} - a^{2} c^{2} x - a c^{2}\right)}}\right]"," ",0,"[-1/8*(4*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x - (a^3*x^3 + a^2*x^2 - a*x - 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)))/(a^4*c^2*x^3 + a^3*c^2*x^2 - a^2*c^2*x - a*c^2), -1/4*(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x - (a^3*x^3 + a^2*x^2 - a*x - 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)))/(a^4*c^2*x^3 + a^3*c^2*x^2 - a^2*c^2*x - a*c^2)]","A",0
1213,1,451,0,0.714741," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{5} x^{5} + a^{4} x^{4} - 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) - 4 \, {\left(2 \, a^{3} x^{3} - a^{2} x^{2} - 5 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{32 \, {\left(a^{6} c^{3} x^{5} + a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x + a c^{3}\right)}}, \frac{3 \, {\left(a^{5} x^{5} + a^{4} x^{4} - 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + a x + 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) - 2 \, {\left(2 \, a^{3} x^{3} - a^{2} x^{2} - 5 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{16 \, {\left(a^{6} c^{3} x^{5} + a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x + a c^{3}\right)}}\right]"," ",0,"[1/32*(3*(a^5*x^5 + a^4*x^4 - 2*a^3*x^3 - 2*a^2*x^2 + a*x + 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) - 4*(2*a^3*x^3 - a^2*x^2 - 5*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*c^3*x^5 + a^5*c^3*x^4 - 2*a^4*c^3*x^3 - 2*a^3*c^3*x^2 + a^2*c^3*x + a*c^3), 1/16*(3*(a^5*x^5 + a^4*x^4 - 2*a^3*x^3 - 2*a^2*x^2 + a*x + 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) - 2*(2*a^3*x^3 - a^2*x^2 - 5*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*c^3*x^5 + a^5*c^3*x^4 - 2*a^4*c^3*x^3 - 2*a^3*c^3*x^2 + a^2*c^3*x + a*c^3)]","A",0
1214,1,565,0,0.657676," ","integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{7} x^{7} + a^{6} x^{6} - 3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) - 4 \, {\left(8 \, a^{5} x^{5} - 7 \, a^{4} x^{4} - 31 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 33 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{192 \, {\left(a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x - a c^{4}\right)}}, \frac{15 \, {\left(a^{7} x^{7} + a^{6} x^{6} - 3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - a x - 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) - 2 \, {\left(8 \, a^{5} x^{5} - 7 \, a^{4} x^{4} - 31 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 33 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{96 \, {\left(a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x - a c^{4}\right)}}\right]"," ",0,"[1/192*(15*(a^7*x^7 + a^6*x^6 - 3*a^5*x^5 - 3*a^4*x^4 + 3*a^3*x^3 + 3*a^2*x^2 - a*x - 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) - 4*(8*a^5*x^5 - 7*a^4*x^4 - 31*a^3*x^3 + 9*a^2*x^2 + 33*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 + a^7*c^4*x^6 - 3*a^6*c^4*x^5 - 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 + 3*a^3*c^4*x^2 - a^2*c^4*x - a*c^4), 1/96*(15*(a^7*x^7 + a^6*x^6 - 3*a^5*x^5 - 3*a^4*x^4 + 3*a^3*x^3 + 3*a^2*x^2 - a*x - 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) - 2*(8*a^5*x^5 - 7*a^4*x^4 - 31*a^3*x^3 + 9*a^2*x^2 + 33*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 + a^7*c^4*x^6 - 3*a^6*c^4*x^5 - 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 + 3*a^3*c^4*x^2 - a^2*c^4*x - a*c^4)]","A",0
1215,0,0,0,0.831869," ","integrate(x^m*(-a^2*c*x^2+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p} x^{m}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p*x^m/(a*x + 1), x)","F",0
1216,0,0,0,0.606781," ","integrate(x^3*(-a^2*x^2+1)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p} x^{3}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p*x^3/(a*x + 1), x)","F",0
1217,0,0,0,0.555376," ","integrate(x^2*(-a^2*x^2+1)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p} x^{2}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p*x^2/(a*x + 1), x)","F",0
1218,0,0,0,0.646345," ","integrate(x*(-a^2*x^2+1)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p} x}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p*x/(a*x + 1), x)","F",0
1219,0,0,0,0.720555," ","integrate((-a^2*x^2+1)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p/(a*x + 1), x)","F",0
1220,0,0,0,0.654178," ","integrate((-a^2*x^2+1)^p/(a*x+1)*(-a^2*x^2+1)^(1/2)/x,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p}}{a x^{2} + x}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p/(a*x^2 + x), x)","F",0
1221,0,0,0,0.583371," ","integrate((-a^2*x^2+1)^p/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} x^{2} + 1\right)}^{p}}{a x^{3} + x^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p/(a*x^3 + x^2), x)","F",0
1222,0,0,0,0.568067," ","integrate(x^3*(-a^2*c*x^2+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p} x^{3}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p*x^3/(a*x + 1), x)","F",0
1223,0,0,0,0.617374," ","integrate(x^2*(-a^2*c*x^2+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p} x^{2}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p*x^2/(a*x + 1), x)","F",0
1224,0,0,0,0.608126," ","integrate(x*(-a^2*c*x^2+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p} x}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p*x/(a*x + 1), x)","F",0
1225,0,0,0,0.622166," ","integrate((-a^2*c*x^2+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a x + 1}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p/(a*x + 1), x)","F",0
1226,0,0,0,0.688250," ","integrate((-a^2*c*x^2+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2)/x,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a x^{2} + x}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p/(a*x^2 + x), x)","F",0
1227,0,0,0,0.676360," ","integrate((-a^2*c*x^2+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} x^{2} + 1} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a x^{3} + x^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p/(a*x^3 + x^2), x)","F",0
1228,1,81,0,0.526934," ","integrate((-a^2*c*x^2+c)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{4} \, a^{7} c^{4} x^{8} + \frac{2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac{3}{2} \, a^{3} c^{4} x^{4} - \frac{2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} + c^{4} x"," ",0,"-1/9*a^8*c^4*x^9 + 1/4*a^7*c^4*x^8 + 2/7*a^6*c^4*x^7 - a^5*c^4*x^6 + 3/2*a^3*c^4*x^4 - 2/3*a^2*c^4*x^3 - a*c^4*x^2 + c^4*x","A",0
1229,1,69,0,0.591626," ","integrate((-a^2*c*x^2+c)^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\frac{1}{7} \, a^{6} c^{3} x^{7} - \frac{1}{3} \, a^{5} c^{3} x^{6} - \frac{1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} - \frac{1}{3} \, a^{2} c^{3} x^{3} - a c^{3} x^{2} + c^{3} x"," ",0,"1/7*a^6*c^3*x^7 - 1/3*a^5*c^3*x^6 - 1/5*a^4*c^3*x^5 + a^3*c^3*x^4 - 1/3*a^2*c^3*x^3 - a*c^3*x^2 + c^3*x","A",0
1230,1,37,0,0.589332," ","integrate((-a^2*c*x^2+c)^2/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","-\frac{1}{5} \, a^{4} c^{2} x^{5} + \frac{1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} + c^{2} x"," ",0,"-1/5*a^4*c^2*x^5 + 1/2*a^3*c^2*x^4 - a*c^2*x^2 + c^2*x","A",0
1231,1,20,0,0.674420," ","integrate((-a^2*c*x^2+c)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\frac{1}{3} \, a^{2} c x^{3} - a c x^{2} + c x"," ",0,"1/3*a^2*c*x^3 - a*c*x^2 + c*x","A",0
1232,1,14,0,0.541778," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{1}{a^{2} c x + a c}"," ",0,"-1/(a^2*c*x + a*c)","A",0
1233,1,76,0,1.088903," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{2 \, a x - {\left(a^{2} x^{2} + 2 \, a x + 1\right)} \log\left(a x + 1\right) + {\left(a^{2} x^{2} + 2 \, a x + 1\right)} \log\left(a x - 1\right) + 4}{8 \, {\left(a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"-1/8*(2*a*x - (a^2*x^2 + 2*a*x + 1)*log(a*x + 1) + (a^2*x^2 + 2*a*x + 1)*log(a*x - 1) + 4)/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2)","A",0
1234,1,121,0,0.617995," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 2 \, a x - 3 \, {\left(a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right)} \log\left(a x + 1\right) + 3 \, {\left(a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right)} \log\left(a x - 1\right) - 8}{24 \, {\left(a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/24*(6*a^3*x^3 + 12*a^2*x^2 + 2*a*x - 3*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*log(a*x + 1) + 3*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*log(a*x - 1) - 8)/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x - a*c^3)","A",0
1235,1,217,0,0.891242," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","-\frac{30 \, a^{5} x^{5} + 60 \, a^{4} x^{4} - 20 \, a^{3} x^{3} - 100 \, a^{2} x^{2} - 34 \, a x - 15 \, {\left(a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right)} \log\left(a x + 1\right) + 15 \, {\left(a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right)} \log\left(a x - 1\right) + 32}{128 \, {\left(a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/128*(30*a^5*x^5 + 60*a^4*x^4 - 20*a^3*x^3 - 100*a^2*x^2 - 34*a*x - 15*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*log(a*x + 1) + 15*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*log(a*x - 1) + 32)/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)","B",0
1236,1,184,0,0.599164," ","integrate(x^3*(-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(4 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 15 \, a x + 24\right)} \sqrt{-a^{2} c x^{2} + c} - 15 \, \sqrt{-c} \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right)}{40 \, a^{4}}, -\frac{{\left(4 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 15 \, a x + 24\right)} \sqrt{-a^{2} c x^{2} + c} - 15 \, \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right)}{20 \, a^{4}}\right]"," ",0,"[-1/40*(2*(4*a^4*x^4 - 10*a^3*x^3 + 12*a^2*x^2 - 15*a*x + 24)*sqrt(-a^2*c*x^2 + c) - 15*sqrt(-c)*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a^4, -1/20*((4*a^4*x^4 - 10*a^3*x^3 + 12*a^2*x^2 - 15*a*x + 24)*sqrt(-a^2*c*x^2 + c) - 15*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))/a^4]","A",0
1237,1,168,0,0.732129," ","integrate(x^2*(-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right)} \sqrt{-a^{2} c x^{2} + c} - 21 \, \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right)}{48 \, a^{3}}, -\frac{{\left(6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right)} \sqrt{-a^{2} c x^{2} + c} + 21 \, \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right)}{24 \, a^{3}}\right]"," ",0,"[-1/48*(2*(6*a^3*x^3 - 16*a^2*x^2 + 21*a*x - 32)*sqrt(-a^2*c*x^2 + c) - 21*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a^3, -1/24*((6*a^3*x^3 - 16*a^2*x^2 + 21*a*x - 32)*sqrt(-a^2*c*x^2 + c) + 21*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))/a^3]","A",0
1238,1,150,0,0.607621," ","integrate(x*(-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, \sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 3 \, a x + 5\right)} - 3 \, \sqrt{-c} \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right)}{6 \, a^{2}}, -\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 3 \, a x + 5\right)} - 3 \, \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right)}{3 \, a^{2}}\right]"," ",0,"[-1/6*(2*sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 3*a*x + 5) - 3*sqrt(-c)*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a^2, -1/3*(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 3*a*x + 5) - 3*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))/a^2]","A",0
1239,1,134,0,0.634152," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[-\frac{2 \, \sqrt{-a^{2} c x^{2} + c} {\left(a x - 4\right)} - 3 \, \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right)}{4 \, a}, -\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a x - 4\right)} + 3 \, \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right)}{2 \, a}\right]"," ",0,"[-1/4*(2*sqrt(-a^2*c*x^2 + c)*(a*x - 4) - 3*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a, -1/2*(sqrt(-a^2*c*x^2 + c)*(a*x - 4) + 3*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))/a]","A",0
1240,1,196,0,0.552649," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x,x, algorithm=""fricas"")","\left[2 \, \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + \frac{1}{2} \, \sqrt{c} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - \sqrt{-a^{2} c x^{2} + c}, -\sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + \sqrt{-c} \log\left(2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - \sqrt{-a^{2} c x^{2} + c}\right]"," ",0,"[2*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 1/2*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - sqrt(-a^2*c*x^2 + c), -sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-c)*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - sqrt(-a^2*c*x^2 + c)]","A",0
1241,1,211,0,0.620808," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^2,x, algorithm=""fricas"")","\left[-\frac{a \sqrt{c} x \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - a \sqrt{c} x \log\left(-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + \sqrt{-a^{2} c x^{2} + c}}{x}, \frac{4 \, a \sqrt{-c} x \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) + a \sqrt{-c} x \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, \sqrt{-a^{2} c x^{2} + c}}{2 \, x}\right]"," ",0,"[-(a*sqrt(c)*x*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - a*sqrt(c)*x*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + sqrt(-a^2*c*x^2 + c))/x, 1/2*(4*a*sqrt(-c)*x*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + a*sqrt(-c)*x*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*sqrt(-a^2*c*x^2 + c))/x]","A",0
1242,1,149,0,0.644121," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^3,x, algorithm=""fricas"")","\left[\frac{3 \, a^{2} \sqrt{c} x^{2} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + 2 \, \sqrt{-a^{2} c x^{2} + c} {\left(4 \, a x - 1\right)}}{4 \, x^{2}}, -\frac{3 \, a^{2} \sqrt{-c} x^{2} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(4 \, a x - 1\right)}}{2 \, x^{2}}\right]"," ",0,"[1/4*(3*a^2*sqrt(c)*x^2*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 2*sqrt(-a^2*c*x^2 + c)*(4*a*x - 1))/x^2, -1/2*(3*a^2*sqrt(-c)*x^2*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) - sqrt(-a^2*c*x^2 + c)*(4*a*x - 1))/x^2]","A",0
1243,1,165,0,0.593686," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm=""fricas"")","\left[\frac{3 \, a^{3} \sqrt{c} x^{3} \log\left(-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c} {\left(5 \, a^{2} x^{2} - 3 \, a x + 1\right)}}{6 \, x^{3}}, \frac{3 \, a^{3} \sqrt{-c} x^{3} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(5 \, a^{2} x^{2} - 3 \, a x + 1\right)}}{3 \, x^{3}}\right]"," ",0,"[1/6*(3*a^3*sqrt(c)*x^3*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*sqrt(-a^2*c*x^2 + c)*(5*a^2*x^2 - 3*a*x + 1))/x^3, 1/3*(3*a^3*sqrt(-c)*x^3*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) - sqrt(-a^2*c*x^2 + c)*(5*a^2*x^2 - 3*a*x + 1))/x^3]","A",0
1244,1,181,0,0.699982," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^5,x, algorithm=""fricas"")","\left[\frac{21 \, a^{4} \sqrt{c} x^{4} \log\left(-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right) + 2 \, {\left(32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right)} \sqrt{-a^{2} c x^{2} + c}}{48 \, x^{4}}, -\frac{21 \, a^{4} \sqrt{-c} x^{4} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right) - {\left(32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right)} \sqrt{-a^{2} c x^{2} + c}}{24 \, x^{4}}\right]"," ",0,"[1/48*(21*a^4*sqrt(c)*x^4*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 2*(32*a^3*x^3 - 21*a^2*x^2 + 16*a*x - 6)*sqrt(-a^2*c*x^2 + c))/x^4, -1/24*(21*a^4*sqrt(-c)*x^4*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) - (32*a^3*x^3 - 21*a^2*x^2 + 16*a*x - 6)*sqrt(-a^2*c*x^2 + c))/x^4]","A",0
1245,1,180,0,0.799407," ","integrate((-a^2*c*x^2+c)^(3/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{-c} c \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 2 \, {\left(6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{48 \, a}, -\frac{15 \, c^{\frac{3}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - {\left(6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right)} \sqrt{-a^{2} c x^{2} + c}}{24 \, a}\right]"," ",0,"[1/48*(15*sqrt(-c)*c*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(6*a^3*c*x^3 - 16*a^2*c*x^2 + 9*a*c*x + 16*c)*sqrt(-a^2*c*x^2 + c))/a, -1/24*(15*c^(3/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (6*a^3*c*x^3 - 16*a^2*c*x^2 + 9*a*c*x + 16*c)*sqrt(-a^2*c*x^2 + c))/a]","A",0
1246,1,241,0,0.601709," ","integrate((-a^2*c*x^2+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{105 \, \sqrt{-c} c^{2} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) - 2 \, {\left(40 \, a^{5} c^{2} x^{5} - 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} + 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x - 96 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{480 \, a}, -\frac{105 \, c^{\frac{5}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) + {\left(40 \, a^{5} c^{2} x^{5} - 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} + 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x - 96 \, c^{2}\right)} \sqrt{-a^{2} c x^{2} + c}}{240 \, a}\right]"," ",0,"[1/480*(105*sqrt(-c)*c^2*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(40*a^5*c^2*x^5 - 96*a^4*c^2*x^4 - 10*a^3*c^2*x^3 + 192*a^2*c^2*x^2 - 135*a*c^2*x - 96*c^2)*sqrt(-a^2*c*x^2 + c))/a, -1/240*(105*c^(5/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (40*a^5*c^2*x^5 - 96*a^4*c^2*x^4 - 10*a^3*c^2*x^3 + 192*a^2*c^2*x^2 - 135*a*c^2*x - 96*c^2)*sqrt(-a^2*c*x^2 + c))/a]","A",0
1247,1,286,0,1.353073," ","integrate((-a^2*c*x^2+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","\left[\frac{315 \, \sqrt{-c} c^{3} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 2 \, {\left(112 \, a^{7} c^{3} x^{7} - 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} + 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} - 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x + 256 \, c^{3}\right)} \sqrt{-a^{2} c x^{2} + c}}{1792 \, a}, -\frac{315 \, c^{\frac{7}{2}} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - {\left(112 \, a^{7} c^{3} x^{7} - 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} + 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} - 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x + 256 \, c^{3}\right)} \sqrt{-a^{2} c x^{2} + c}}{896 \, a}\right]"," ",0,"[1/1792*(315*sqrt(-c)*c^3*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(112*a^7*c^3*x^7 - 256*a^6*c^3*x^6 - 168*a^5*c^3*x^5 + 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 - 768*a^2*c^3*x^2 + 581*a*c^3*x + 256*c^3)*sqrt(-a^2*c*x^2 + c))/a, -1/896*(315*c^(7/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (112*a^7*c^3*x^7 - 256*a^6*c^3*x^6 - 168*a^5*c^3*x^5 + 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 - 768*a^2*c^3*x^2 + 581*a*c^3*x + 256*c^3)*sqrt(-a^2*c*x^2 + c))/a]","A",0
1248,1,150,0,0.738856," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a x + 1\right)} \sqrt{-c} \log\left(2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right) + 4 \, \sqrt{-a^{2} c x^{2} + c}}{2 \, {\left(a^{2} c x + a c\right)}}, \frac{{\left(a x + 1\right)} \sqrt{c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right) - 2 \, \sqrt{-a^{2} c x^{2} + c}}{a^{2} c x + a c}\right]"," ",0,"[-1/2*((a*x + 1)*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 4*sqrt(-a^2*c*x^2 + c))/(a^2*c*x + a*c), ((a*x + 1)*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - 2*sqrt(-a^2*c*x^2 + c))/(a^2*c*x + a*c)]","A",0
1249,1,47,0,0.583985," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a x + 2\right)}}{3 \, {\left(a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"-1/3*sqrt(-a^2*c*x^2 + c)*(a*x + 2)/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2)","A",0
1250,1,75,0,0.829903," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x - 2\right)} \sqrt{-a^{2} c x^{2} + c}}{5 \, {\left(a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/5*(2*a^3*x^3 + 4*a^2*x^2 + a*x - 2)*sqrt(-a^2*c*x^2 + c)/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x - a*c^3)","A",0
1251,1,124,0,0.910291," ","integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","-\frac{{\left(8 \, a^{5} x^{5} + 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} - 24 \, a^{2} x^{2} - 9 \, a x + 6\right)} \sqrt{-a^{2} c x^{2} + c}}{21 \, {\left(a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/21*(8*a^5*x^5 + 16*a^4*x^4 - 4*a^3*x^3 - 24*a^2*x^2 - 9*a*x + 6)*sqrt(-a^2*c*x^2 + c)/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)","A",0
1252,0,0,0,1.296369," ","integrate(x^m*(-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a x - 1\right)} x^{m}}{a x + 1}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*(a*x - 1)*x^m/(a*x + 1), x)","F",0
1253,0,0,0,0.656221," ","integrate((-a^2*c*x^2+c)^p/(a*x+1)^2*(-a^2*x^2+1),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a x - 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a x + 1}, x\right)"," ",0,"integral(-(a*x - 1)*(-a^2*c*x^2 + c)^p/(a*x + 1), x)","F",0
1254,1,136,0,0.582857," ","integrate((-a^2*c*x^2+c)^4/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{6930 \, c^{4} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(896 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} + 1024 \, a^{6} c^{4} x^{6} + 7224 \, a^{5} c^{4} x^{5} - 8448 \, a^{4} c^{4} x^{4} - 3066 \, a^{3} c^{4} x^{3} + 10240 \, a^{2} c^{4} x^{2} - 4599 \, a c^{4} x - 3712 \, c^{4}\right)} \sqrt{-a^{2} x^{2} + 1}}{8064 \, a}"," ",0,"-1/8064*(6930*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (896*a^8*c^4*x^8 - 3024*a^7*c^4*x^7 + 1024*a^6*c^4*x^6 + 7224*a^5*c^4*x^5 - 8448*a^4*c^4*x^4 - 3066*a^3*c^4*x^3 + 10240*a^2*c^4*x^2 - 4599*a*c^4*x - 3712*c^4)*sqrt(-a^2*x^2 + 1))/a","A",0
1255,1,115,0,0.582153," ","integrate((-a^2*c*x^2+c)^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{630 \, c^{3} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(80 \, a^{6} c^{3} x^{6} - 280 \, a^{5} c^{3} x^{5} + 208 \, a^{4} c^{3} x^{4} + 350 \, a^{3} c^{3} x^{3} - 656 \, a^{2} c^{3} x^{2} + 245 \, a c^{3} x + 368 \, c^{3}\right)} \sqrt{-a^{2} x^{2} + 1}}{560 \, a}"," ",0,"-1/560*(630*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (80*a^6*c^3*x^6 - 280*a^5*c^3*x^5 + 208*a^4*c^3*x^4 + 350*a^3*c^3*x^3 - 656*a^2*c^3*x^2 + 245*a*c^3*x + 368*c^3)*sqrt(-a^2*x^2 + 1))/a","A",0
1256,1,92,0,0.543981," ","integrate((-a^2*c*x^2+c)^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{210 \, c^{2} \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) + {\left(24 \, a^{4} c^{2} x^{4} - 90 \, a^{3} c^{2} x^{3} + 112 \, a^{2} c^{2} x^{2} - 15 \, a c^{2} x - 136 \, c^{2}\right)} \sqrt{-a^{2} x^{2} + 1}}{120 \, a}"," ",0,"-1/120*(210*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c^2*x^4 - 90*a^3*c^2*x^3 + 112*a^2*c^2*x^2 - 15*a*c^2*x - 136*c^2)*sqrt(-a^2*x^2 + 1))/a","A",0
1257,1,63,0,0.614151," ","integrate((-a^2*c*x^2+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{30 \, c \arctan\left(\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right) - {\left(2 \, a^{2} c x^{2} - 9 \, a c x + 22 \, c\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, a}"," ",0,"-1/6*(30*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^2*c*x^2 - 9*a*c*x + 22*c)*sqrt(-a^2*x^2 + 1))/a","A",0
1258,1,55,0,0.650802," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c),x, algorithm=""fricas"")","-\frac{a^{2} x^{2} + 2 \, a x - \sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)} + 1}{3 \, {\left(a^{3} c x^{2} + 2 \, a^{2} c x + a c\right)}}"," ",0,"-1/3*(a^2*x^2 + 2*a*x - sqrt(-a^2*x^2 + 1)*(a*x - 1) + 1)/(a^3*c*x^2 + 2*a^2*c*x + a*c)","B",0
1259,1,89,0,1.042998," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{7 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 21 \, a x + {\left(2 \, a^{2} x^{2} + 6 \, a x + 7\right)} \sqrt{-a^{2} x^{2} + 1} + 7}{15 \, {\left(a^{4} c^{2} x^{3} + 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x + a c^{2}\right)}}"," ",0,"-1/15*(7*a^3*x^3 + 21*a^2*x^2 + 21*a*x + (2*a^2*x^2 + 6*a*x + 7)*sqrt(-a^2*x^2 + 1) + 7)/(a^4*c^2*x^3 + 3*a^3*c^2*x^2 + 3*a^2*c^2*x + a*c^2)","A",0
1260,1,144,0,0.574970," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{13 \, a^{5} x^{5} + 39 \, a^{4} x^{4} + 26 \, a^{3} x^{3} - 26 \, a^{2} x^{2} - 39 \, a x + {\left(8 \, a^{4} x^{4} + 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} - 4 \, a x - 13\right)} \sqrt{-a^{2} x^{2} + 1} - 13}{35 \, {\left(a^{6} c^{3} x^{5} + 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x - a c^{3}\right)}}"," ",0,"-1/35*(13*a^5*x^5 + 39*a^4*x^4 + 26*a^3*x^3 - 26*a^2*x^2 - 39*a*x + (8*a^4*x^4 + 24*a^3*x^3 + 20*a^2*x^2 - 4*a*x - 13)*sqrt(-a^2*x^2 + 1) - 13)/(a^6*c^3*x^5 + 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^3*c^3*x^2 - 3*a^2*c^3*x - a*c^3)","A",0
1261,1,195,0,0.652508," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","-\frac{19 \, a^{7} x^{7} + 57 \, a^{6} x^{6} + 19 \, a^{5} x^{5} - 95 \, a^{4} x^{4} - 95 \, a^{3} x^{3} + 19 \, a^{2} x^{2} + 57 \, a x + {\left(16 \, a^{6} x^{6} + 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} - 6 \, a x + 19\right)} \sqrt{-a^{2} x^{2} + 1} + 19}{63 \, {\left(a^{8} c^{4} x^{7} + 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} + a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x + a c^{4}\right)}}"," ",0,"-1/63*(19*a^7*x^7 + 57*a^6*x^6 + 19*a^5*x^5 - 95*a^4*x^4 - 95*a^3*x^3 + 19*a^2*x^2 + 57*a*x + (16*a^6*x^6 + 48*a^5*x^5 + 24*a^4*x^4 - 56*a^3*x^3 - 66*a^2*x^2 - 6*a*x + 19)*sqrt(-a^2*x^2 + 1) + 19)/(a^8*c^4*x^7 + 3*a^7*c^4*x^6 + a^6*c^4*x^5 - 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 + a^3*c^4*x^2 + 3*a^2*c^4*x + a*c^4)","A",0
1262,1,399,0,1.117531," ","integrate(x^3*(-a^2*c*x^2+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{120 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left(a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right) - {\left(12 \, a^{5} x^{5} - 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 120 \, a^{2} x^{2} + 240 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{60 \, {\left(a^{6} x^{2} - a^{4}\right)}}, -\frac{240 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right) + {\left(12 \, a^{5} x^{5} - 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 120 \, a^{2} x^{2} + 240 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{60 \, {\left(a^{6} x^{2} - a^{4}\right)}}\right]"," ",0,"[1/60*(120*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x + (a^4*x^4 + 4*a^3*x^3 + 6*a^2*x^2 + 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)) - (12*a^5*x^5 - 45*a^4*x^4 + 80*a^3*x^3 - 120*a^2*x^2 + 240*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*x^2 - a^4), -1/60*(240*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 + 2*a^3*c*x^3 - a^2*c*x^2 - 2*a*c*x)) + (12*a^5*x^5 - 45*a^4*x^4 + 80*a^3*x^3 - 120*a^2*x^2 + 240*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*x^2 - a^4)]","A",0
1263,1,383,0,0.674599," ","integrate(x^2*(-a^2*c*x^2+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left(a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right) - {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 16 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{4 \, {\left(a^{5} x^{2} - a^{3}\right)}}, \frac{16 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right) - {\left(a^{4} x^{4} - 4 \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 16 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{4 \, {\left(a^{5} x^{2} - a^{3}\right)}}\right]"," ",0,"[1/4*(8*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x - (a^4*x^4 + 4*a^3*x^3 + 6*a^2*x^2 + 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)) - (a^4*x^4 - 4*a^3*x^3 + 8*a^2*x^2 - 16*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^5*x^2 - a^3), 1/4*(16*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 + 2*a^3*c*x^3 - a^2*c*x^2 - 2*a*c*x)) - (a^4*x^4 - 4*a^3*x^3 + 8*a^2*x^2 - 16*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^5*x^2 - a^3)]","A",0
1264,1,367,0,0.784919," ","integrate(x*(-a^2*c*x^2+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left(a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right) - {\left(2 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 24 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{4} x^{2} - a^{2}\right)}}, -\frac{24 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right) + {\left(2 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 24 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{4} x^{2} - a^{2}\right)}}\right]"," ",0,"[1/6*(12*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x + (a^4*x^4 + 4*a^3*x^3 + 6*a^2*x^2 + 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)) - (2*a^3*x^3 - 9*a^2*x^2 + 24*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^4*x^2 - a^2), -1/6*(24*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 + 2*a^3*c*x^3 - a^2*c*x^2 - 2*a*c*x)) + (2*a^3*x^3 - 9*a^2*x^2 + 24*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^4*x^2 - a^2)]","A",0
1265,1,347,0,0.730933," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left(a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 6 \, a x\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a^{3} x^{2} - a\right)}}, \frac{8 \, {\left(a^{2} x^{2} - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} - 6 \, a x\right)} \sqrt{-a^{2} x^{2} + 1}}{2 \, {\left(a^{3} x^{2} - a\right)}}\right]"," ",0,"[1/2*(4*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x - (a^4*x^4 + 4*a^3*x^3 + 6*a^2*x^2 + 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)) - sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 6*a*x)*sqrt(-a^2*x^2 + 1))/(a^3*x^2 - a), 1/2*(8*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 + 2*a^3*c*x^3 - a^2*c*x^2 - 2*a*c*x)) - sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 6*a*x)*sqrt(-a^2*x^2 + 1))/(a^3*x^2 - a)]","A",0
1266,0,0,0,0.615432," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)}}{a^{2} x^{3} + 2 \, a x^{2} + x}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x - 1)/(a^2*x^3 + 2*a*x^2 + x), x)","F",0
1267,0,0,0,1.295845," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)}}{a^{2} x^{4} + 2 \, a x^{3} + x^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x - 1)/(a^2*x^4 + 2*a*x^3 + x^2), x)","F",0
1268,1,447,0,0.738381," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^3,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{c} \log\left(\frac{4 \, a^{5} c x^{5} + {\left(2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right)} c x^{6} + {\left(4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right)} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{3} x^{3} - {\left(4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right)} x^{4} + 6 \, a^{2} x^{2} + 4 \, a x + 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right) + \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(6 \, a - 1\right)} x^{2} - 6 \, a x + 1\right)}}{2 \, {\left(a^{2} x^{4} - x^{2}\right)}}, \frac{8 \, {\left(a^{4} x^{4} - a^{2} x^{2}\right)} \sqrt{-c} \arctan\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a^{2} + 2 \, a + 1\right)} x^{2} + 2 \, a x + 1\right)} \sqrt{-c}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + a^{2}\right)} c x^{4} + {\left(a^{2} + 2 \, a + 1\right)} c x^{2} - 2 \, a c x - c}\right) + \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(6 \, a - 1\right)} x^{2} - 6 \, a x + 1\right)}}{2 \, {\left(a^{2} x^{4} - x^{2}\right)}}\right]"," ",0,"[1/2*(4*(a^4*x^4 - a^2*x^2)*sqrt(c)*log((4*a^5*c*x^5 + (2*a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*c*x^6 + (4*a^4 - 4*a^3 - 6*a^2 - 4*a - 1)*c*x^4 - 5*a^2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 + 6*a^2 + 4*a + 1)*x^4 + 6*a^2*x^2 + 4*a*x + 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^4*x^6 + 2*a^3*x^5 - 2*a*x^3 - x^2)) + sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((6*a - 1)*x^2 - 6*a*x + 1))/(a^2*x^4 - x^2), 1/2*(8*(a^4*x^4 - a^2*x^2)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 + 2*a + 1)*x^2 + 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 + a^2)*c*x^4 + (a^2 + 2*a + 1)*c*x^2 - 2*a*c*x - c)) + sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((6*a - 1)*x^2 - 6*a*x + 1))/(a^2*x^4 - x^2)]","A",0
1269,1,477,0,0.780391," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^4,x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{c} \log\left(\frac{4 \, a^{5} c x^{5} + {\left(2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right)} c x^{6} + {\left(4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right)} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x - {\left(4 \, a^{3} x^{3} - {\left(4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right)} x^{4} + 6 \, a^{2} x^{2} + 4 \, a x + 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right) + \sqrt{-a^{2} c x^{2} + c} {\left(24 \, a^{2} x^{2} - {\left(24 \, a^{2} - 9 \, a + 2\right)} x^{3} - 9 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{2} x^{5} - x^{3}\right)}}, -\frac{24 \, {\left(a^{5} x^{5} - a^{3} x^{3}\right)} \sqrt{-c} \arctan\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a^{2} + 2 \, a + 1\right)} x^{2} + 2 \, a x + 1\right)} \sqrt{-c}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + a^{2}\right)} c x^{4} + {\left(a^{2} + 2 \, a + 1\right)} c x^{2} - 2 \, a c x - c}\right) - \sqrt{-a^{2} c x^{2} + c} {\left(24 \, a^{2} x^{2} - {\left(24 \, a^{2} - 9 \, a + 2\right)} x^{3} - 9 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1}}{6 \, {\left(a^{2} x^{5} - x^{3}\right)}}\right]"," ",0,"[1/6*(12*(a^5*x^5 - a^3*x^3)*sqrt(c)*log((4*a^5*c*x^5 + (2*a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*c*x^6 + (4*a^4 - 4*a^3 - 6*a^2 - 4*a - 1)*c*x^4 - 5*a^2*c*x^2 - 4*a*c*x - (4*a^3*x^3 - (4*a^3 + 6*a^2 + 4*a + 1)*x^4 + 6*a^2*x^2 + 4*a*x + 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^4*x^6 + 2*a^3*x^5 - 2*a*x^3 - x^2)) + sqrt(-a^2*c*x^2 + c)*(24*a^2*x^2 - (24*a^2 - 9*a + 2)*x^3 - 9*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*x^5 - x^3), -1/6*(24*(a^5*x^5 - a^3*x^3)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 + 2*a + 1)*x^2 + 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 + a^2)*c*x^4 + (a^2 + 2*a + 1)*c*x^2 - 2*a*c*x - c)) - sqrt(-a^2*c*x^2 + c)*(24*a^2*x^2 - (24*a^2 - 9*a + 2)*x^3 - 9*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*x^5 - x^3)]","A",0
1270,1,503,0,1.021802," ","integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{6} x^{6} - a^{4} x^{4}\right)} \sqrt{c} \log\left(\frac{4 \, a^{5} c x^{5} + {\left(2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right)} c x^{6} + {\left(4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right)} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x + {\left(4 \, a^{3} x^{3} - {\left(4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right)} x^{4} + 6 \, a^{2} x^{2} + 4 \, a x + 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right) - {\left(16 \, a^{3} x^{3} - {\left(16 \, a^{3} - 8 \, a^{2} + 4 \, a - 1\right)} x^{4} - 8 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{4 \, {\left(a^{2} x^{6} - x^{4}\right)}}, \frac{16 \, {\left(a^{6} x^{6} - a^{4} x^{4}\right)} \sqrt{-c} \arctan\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left({\left(2 \, a^{2} + 2 \, a + 1\right)} x^{2} + 2 \, a x + 1\right)} \sqrt{-c}}{2 \, a^{3} c x^{3} - {\left(2 \, a^{3} + a^{2}\right)} c x^{4} + {\left(a^{2} + 2 \, a + 1\right)} c x^{2} - 2 \, a c x - c}\right) - {\left(16 \, a^{3} x^{3} - {\left(16 \, a^{3} - 8 \, a^{2} + 4 \, a - 1\right)} x^{4} - 8 \, a^{2} x^{2} + 4 \, a x - 1\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{4 \, {\left(a^{2} x^{6} - x^{4}\right)}}\right]"," ",0,"[1/4*(8*(a^6*x^6 - a^4*x^4)*sqrt(c)*log((4*a^5*c*x^5 + (2*a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*c*x^6 + (4*a^4 - 4*a^3 - 6*a^2 - 4*a - 1)*c*x^4 - 5*a^2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 + 6*a^2 + 4*a + 1)*x^4 + 6*a^2*x^2 + 4*a*x + 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^4*x^6 + 2*a^3*x^5 - 2*a*x^3 - x^2)) - (16*a^3*x^3 - (16*a^3 - 8*a^2 + 4*a - 1)*x^4 - 8*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^2*x^6 - x^4), 1/4*(16*(a^6*x^6 - a^4*x^4)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 + 2*a + 1)*x^2 + 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 + a^2)*c*x^4 + (a^2 + 2*a + 1)*c*x^2 - 2*a*c*x - c)) - (16*a^3*x^3 - (16*a^3 - 8*a^2 + 4*a - 1)*x^4 - 8*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^2*x^6 - x^4)]","A",0
1271,1,120,0,0.710809," ","integrate((-a^2*c*x^2+c)^(9/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(21 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} + 240 \, a^{6} c^{4} x^{7} - 210 \, a^{5} c^{4} x^{6} - 252 \, a^{4} c^{4} x^{5} + 420 \, a^{3} c^{4} x^{4} - 315 \, a c^{4} x^{2} + 210 \, c^{4} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{210 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/210*(21*a^9*c^4*x^10 - 70*a^8*c^4*x^9 + 240*a^6*c^4*x^7 - 210*a^5*c^4*x^6 - 252*a^4*c^4*x^5 + 420*a^3*c^4*x^4 - 315*a*c^4*x^2 + 210*c^4*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1272,1,120,0,1.096597," ","integrate((-a^2*c*x^2+c)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\frac{{\left(21 \, a^{7} c^{3} x^{8} - 72 \, a^{6} c^{3} x^{7} + 28 \, a^{5} c^{3} x^{6} + 168 \, a^{4} c^{3} x^{5} - 210 \, a^{3} c^{3} x^{4} - 56 \, a^{2} c^{3} x^{3} + 252 \, a c^{3} x^{2} - 168 \, c^{3} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{168 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/168*(21*a^7*c^3*x^8 - 72*a^6*c^3*x^7 + 28*a^5*c^3*x^6 + 168*a^4*c^3*x^5 - 210*a^3*c^3*x^4 - 56*a^2*c^3*x^3 + 252*a*c^3*x^2 - 168*c^3*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1273,1,98,0,0.700440," ","integrate((-a^2*c*x^2+c)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(5 \, a^{5} c^{2} x^{6} - 18 \, a^{4} c^{2} x^{5} + 15 \, a^{3} c^{2} x^{4} + 20 \, a^{2} c^{2} x^{3} - 45 \, a c^{2} x^{2} + 30 \, c^{2} x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{30 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"-1/30*(5*a^5*c^2*x^6 - 18*a^4*c^2*x^5 + 15*a^3*c^2*x^4 + 20*a^2*c^2*x^3 - 45*a*c^2*x^2 + 30*c^2*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1274,1,67,0,0.653521," ","integrate((-a^2*c*x^2+c)^(3/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","\frac{{\left(a^{3} c x^{4} - 4 \, a^{2} c x^{3} + 6 \, a c x^{2} - 4 \, c x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{4 \, {\left(a^{2} x^{2} - 1\right)}}"," ",0,"1/4*(a^3*c*x^4 - 4*a^2*c*x^3 + 6*a*c*x^2 - 4*c*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)","A",0
1275,1,381,0,1.697017," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\left[-\frac{4 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x - {\left(a^{3} x^{3} + a^{2} x^{2} - a x - 1\right)} \sqrt{c} \log\left(\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left(a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right)}{2 \, {\left(a^{4} c x^{3} + a^{3} c x^{2} - a^{2} c x - a c\right)}}, -\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a x + {\left(a^{3} x^{3} + a^{2} x^{2} - a x - 1\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a^{2} x^{2} + 2 \, a x + 2\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right)}{a^{4} c x^{3} + a^{3} c x^{2} - a^{2} c x - a c}\right]"," ",0,"[-1/2*(4*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x - (a^3*x^3 + a^2*x^2 - a*x - 1)*sqrt(c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x + (a^4*x^4 + 4*a^3*x^3 + 6*a^2*x^2 + 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)))/(a^4*c*x^3 + a^3*c*x^2 - a^2*c*x - a*c), -(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*x + (a^3*x^3 + a^2*x^2 - a*x - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 + 2*a^3*c*x^3 - a^2*c*x^2 - 2*a*c*x)))/(a^4*c*x^3 + a^3*c*x^2 - a^2*c*x - a*c)]","B",0
1276,1,72,0,0.561465," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x^{2} + 2 \, x\right)}}{2 \, {\left(a^{4} c^{2} x^{4} + 2 \, a^{3} c^{2} x^{3} - 2 \, a c^{2} x - c^{2}\right)}}"," ",0,"-1/2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x^2 + 2*x)/(a^4*c^2*x^4 + 2*a^3*c^2*x^3 - 2*a*c^2*x - c^2)","A",0
1277,1,461,0,0.653729," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) - 4 \, {\left(10 \, a^{3} x^{3} + 27 \, a^{2} x^{2} + 21 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{96 \, {\left(a^{6} c^{3} x^{5} + 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x - a c^{3}\right)}}, \frac{3 \, {\left(a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) - 2 \, {\left(10 \, a^{3} x^{3} + 27 \, a^{2} x^{2} + 21 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{48 \, {\left(a^{6} c^{3} x^{5} + 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x - a c^{3}\right)}}\right]"," ",0,"[1/96*(3*(a^5*x^5 + 3*a^4*x^4 + 2*a^3*x^3 - 2*a^2*x^2 - 3*a*x - 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) - 4*(10*a^3*x^3 + 27*a^2*x^2 + 21*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*c^3*x^5 + 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^3*c^3*x^2 - 3*a^2*c^3*x - a*c^3), 1/48*(3*(a^5*x^5 + 3*a^4*x^4 + 2*a^3*x^3 - 2*a^2*x^2 - 3*a*x - 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) - 2*(10*a^3*x^3 + 27*a^2*x^2 + 21*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*c^3*x^5 + 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^3*c^3*x^2 - 3*a^2*c^3*x - a*c^3)]","A",0
1278,1,559,0,0.834112," ","integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{7} x^{7} + 3 \, a^{6} x^{6} + a^{5} x^{5} - 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} + a^{2} x^{2} + 3 \, a x + 1\right)} \sqrt{c} \log\left(-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left(a^{3} x^{3} + a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right) - 4 \, {\left(32 \, a^{5} x^{5} + 81 \, a^{4} x^{4} + 19 \, a^{3} x^{3} - 99 \, a^{2} x^{2} - 81 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{384 \, {\left(a^{8} c^{4} x^{7} + 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} + a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x + a c^{4}\right)}}, \frac{15 \, {\left(a^{7} x^{7} + 3 \, a^{6} x^{6} + a^{5} x^{5} - 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} + a^{2} x^{2} + 3 \, a x + 1\right)} \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right) - 2 \, {\left(32 \, a^{5} x^{5} + 81 \, a^{4} x^{4} + 19 \, a^{3} x^{3} - 99 \, a^{2} x^{2} - 81 \, a x\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{192 \, {\left(a^{8} c^{4} x^{7} + 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} + a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x + a c^{4}\right)}}\right]"," ",0,"[1/384*(15*(a^7*x^7 + 3*a^6*x^6 + a^5*x^5 - 5*a^4*x^4 - 5*a^3*x^3 + a^2*x^2 + 3*a*x + 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) - 4*(32*a^5*x^5 + 81*a^4*x^4 + 19*a^3*x^3 - 99*a^2*x^2 - 81*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 + 3*a^7*c^4*x^6 + a^6*c^4*x^5 - 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 + a^3*c^4*x^2 + 3*a^2*c^4*x + a*c^4), 1/192*(15*(a^7*x^7 + 3*a^6*x^6 + a^5*x^5 - 5*a^4*x^4 - 5*a^3*x^3 + a^2*x^2 + 3*a*x + 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) - 2*(32*a^5*x^5 + 81*a^4*x^4 + 19*a^3*x^3 - 99*a^2*x^2 - 81*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 + 3*a^7*c^4*x^6 + a^6*c^4*x^5 - 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 + a^3*c^4*x^2 + 3*a^2*c^4*x + a*c^4)]","A",0
1279,0,0,0,0.806993," ","integrate(x^m*(-a^2*c*x^2+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)} x^{m}}{a^{2} x^{2} + 2 \, a x + 1}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x - 1)*x^m/(a^2*x^2 + 2*a*x + 1), x)","F",0
1280,0,0,0,0.616717," ","integrate((-a^2*c*x^2+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} x^{2} + 1} {\left(a x - 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{a^{2} x^{2} + 2 \, a x + 1}, x\right)"," ",0,"integral(-sqrt(-a^2*x^2 + 1)*(a*x - 1)*(-a^2*c*x^2 + c)^p/(a^2*x^2 + 2*a*x + 1), x)","F",0
1281,1,557,0,0.679072," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-a^2*x^2+1)^(5/2),x, algorithm=""fricas"")","-\frac{13860 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - 1\right) + 13860 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + 1\right) + 3465 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 3465 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, {\left(1280 \, a^{5} x^{5} + 128 \, a^{4} x^{4} - 4144 \, a^{3} x^{3} - 520 \, a^{2} x^{2} + 5174 \, a x + 1547\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{30720 \, a}"," ",0,"-1/30720*(13860*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - 1) + 13860*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + 1) + 3465*sqrt(2)*a*(a^(-4))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 3465*sqrt(2)*a*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*(1280*a^5*x^5 + 128*a^4*x^4 - 4144*a^3*x^3 - 520*a^2*x^2 + 5174*a*x + 1547)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a","B",0
1282,1,541,0,1.658815," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{420 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - 1\right) + 420 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + 1\right) + 105 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 105 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + 4 \, {\left(48 \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 118 \, a x - 43\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{768 \, a}"," ",0,"-1/768*(420*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - 1) + 420*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + 1) + 105*sqrt(2)*a*(a^(-4))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 105*sqrt(2)*a*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*(48*a^3*x^3 + 8*a^2*x^2 - 118*a*x - 43)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a","B",0
1283,1,525,0,0.990236," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-\frac{12 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - 1\right) + 12 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + 1\right) + 3 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 3 \, \sqrt{2} a \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) - 4 \, \sqrt{-a^{2} x^{2} + 1} {\left(2 \, a x + 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, a}"," ",0,"-1/16*(12*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - 1) + 12*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + 1) + 3*sqrt(2)*a*(a^(-4))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 3*sqrt(2)*a*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 4*sqrt(-a^2*x^2 + 1)*(2*a*x + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a","B",0
1284,1,473,0,0.659982," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*x^2+1)^(1/2),x, algorithm=""fricas"")","-2 \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - 1\right) - 2 \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \arctan\left(\sqrt{2} a \sqrt{-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} - \sqrt{2} a \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{1}{4}} + 1\right) - \frac{1}{2} \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right) + \frac{1}{2} \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \log\left(-\frac{\sqrt{2} {\left(a^{4} x - a^{3}\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - {\left(a^{3} x - a^{2}\right)} \sqrt{\frac{1}{a^{4}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right)"," ",0,"-2*sqrt(2)*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - 1) - 2*sqrt(2)*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + 1) - 1/2*sqrt(2)*(a^(-4))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 1/2*sqrt(2)*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))","B",0
1285,1,56,0,0.635240," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*x^2+1)^(3/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-a^{2} x^{2} + 1} {\left(2 \, a x - 1\right)} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{3 \, {\left(a^{3} x^{2} - a\right)}}"," ",0,"-2/3*sqrt(-a^2*x^2 + 1)*(2*a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))/(a^3*x^2 - a)","A",0
1286,1,78,0,0.692396," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*x^2+1)^(5/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(16 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 22 \, a x + 9\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{35 \, {\left(a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right)}}"," ",0,"-2/35*(16*a^3*x^3 - 8*a^2*x^2 - 22*a*x + 9)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))/(a^5*x^4 - 2*a^3*x^2 + a)","A",0
1287,1,104,0,0.586893," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*x^2+1)^(7/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(256 \, a^{5} x^{5} - 128 \, a^{4} x^{4} - 608 \, a^{3} x^{3} + 272 \, a^{2} x^{2} + 422 \, a x - 151\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{693 \, {\left(a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right)}}"," ",0,"-2/693*(256*a^5*x^5 - 128*a^4*x^4 - 608*a^3*x^3 + 272*a^2*x^2 + 422*a*x - 151)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))/(a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)","A",0
1288,1,126,0,1.669662," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*x^2+1)^(9/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(2048 \, a^{7} x^{7} - 1024 \, a^{6} x^{6} - 6912 \, a^{5} x^{5} + 3200 \, a^{4} x^{4} + 8240 \, a^{3} x^{3} - 3384 \, a^{2} x^{2} - 3838 \, a x + 1241\right)} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{6435 \, {\left(a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + a\right)}}"," ",0,"-2/6435*(2048*a^7*x^7 - 1024*a^6*x^6 - 6912*a^5*x^5 + 3200*a^4*x^4 + 8240*a^3*x^3 - 3384*a^2*x^2 - 3838*a*x + 1241)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))/(a^9*x^8 - 4*a^7*x^6 + 6*a^5*x^4 - 4*a^3*x^2 + a)","A",0
1289,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1290,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1291,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1292,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1293,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1294,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1295,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1296,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(9/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1297,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x^3/(-a^2*c*x^2+c)^(5/4),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1298,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x^2/(-a^2*c*x^2+c)^(5/4),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1299,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x/(-a^2*c*x^2+c)^(5/4),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1300,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(5/4),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1301,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(5/4),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1302,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(5/4),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1303,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x^3/(-a^2*c*x^2+c)^(9/8),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1304,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x^2/(-a^2*c*x^2+c)^(9/8),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1305,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x/(-a^2*c*x^2+c)^(9/8),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1306,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(9/8),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1307,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x/(-a^2*c*x^2+c)^(9/8),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1308,0,0,0,1.012074," ","integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{2} c x^{2} - c\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(-(a^2*c*x^2 - c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1309,0,0,0,1.210564," ","integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1310,0,0,0,0.860280," ","integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(-(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1311,0,0,0,0.601306," ","integrate(exp(n*arctanh(a*x))*x^4/(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{x^{4} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(-x^4*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)","F",0
1312,0,0,0,0.905494," ","integrate(exp(n*arctanh(a*x))*x^3/(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{x^{3} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(-x^3*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)","F",0
1313,0,0,0,0.919175," ","integrate(exp(n*arctanh(a*x))*x^2/(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{x^{2} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(-x^2*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)","F",0
1314,0,0,0,0.567319," ","integrate(exp(n*arctanh(a*x))*x/(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{x \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(-x*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)","F",0
1315,1,27,0,0.816232," ","integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c),x, algorithm=""fricas"")","\frac{\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c n}"," ",0,"((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*n)","A",0
1316,0,0,0,0.698871," ","integrate(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{3} - c x}, x\right)"," ",0,"integral(-((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^3 - c*x), x)","F",0
1317,0,0,0,0.782563," ","integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{4} - c x^{2}}, x\right)"," ",0,"integral(-((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^4 - c*x^2), x)","F",0
1318,0,0,0,0.746793," ","integrate(exp(n*arctanh(a*x))*x^4/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{x^{4} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right)"," ",0,"integral(x^4*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)","F",0
1319,0,0,0,0.597347," ","integrate(exp(n*arctanh(a*x))*x^3/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{x^{3} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right)"," ",0,"integral(x^3*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)","F",0
1320,1,91,0,0.472837," ","integrate(exp(n*arctanh(a*x))*x^2/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{{\left(2 \, a n x - {\left(a^{2} n^{2} - 2 \, a^{2}\right)} x^{2} - 2\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{3} c^{2} n^{3} - 4 \, a^{3} c^{2} n - {\left(a^{5} c^{2} n^{3} - 4 \, a^{5} c^{2} n\right)} x^{2}}"," ",0,"-(2*a*n*x - (a^2*n^2 - 2*a^2)*x^2 - 2)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^3*c^2*n^3 - 4*a^3*c^2*n - (a^5*c^2*n^3 - 4*a^5*c^2*n)*x^2)","A",0
1321,1,78,0,0.457268," ","integrate(exp(n*arctanh(a*x))*x/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","-\frac{{\left(a^{2} x^{2} - a n x + 1\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c^{2} n^{2} - 4 \, a^{2} c^{2} - {\left(a^{4} c^{2} n^{2} - 4 \, a^{4} c^{2}\right)} x^{2}}"," ",0,"-(a^2*x^2 - a*n*x + 1)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c^2*n^2 - 4*a^2*c^2 - (a^4*c^2*n^2 - 4*a^4*c^2)*x^2)","A",0
1322,1,79,0,0.460406," ","integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","\frac{{\left(2 \, a^{2} x^{2} - 2 \, a n x + n^{2} - 2\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c^{2} n^{3} - 4 \, a c^{2} n - {\left(a^{3} c^{2} n^{3} - 4 \, a^{3} c^{2} n\right)} x^{2}}"," ",0,"(2*a^2*x^2 - 2*a*n*x + n^2 - 2)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^2*n^3 - 4*a*c^2*n - (a^3*c^2*n^3 - 4*a^3*c^2*n)*x^2)","A",0
1323,0,0,0,0.623828," ","integrate(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{5} - 2 \, a^{2} c^{2} x^{3} + c^{2} x}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^5 - 2*a^2*c^2*x^3 + c^2*x), x)","F",0
1324,0,0,0,0.613137," ","integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{6} - 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}}, x\right)"," ",0,"integral(((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^6 - 2*a^2*c^2*x^4 + c^2*x^2), x)","F",0
1325,1,174,0,0.549494," ","integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","\frac{{\left(24 \, a^{4} x^{4} - 24 \, a^{3} n x^{3} + n^{4} + 12 \, {\left(a^{2} n^{2} - 4 \, a^{2}\right)} x^{2} - 16 \, n^{2} - 4 \, {\left(a n^{3} - 10 \, a n\right)} x + 24\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c^{3} n^{5} - 20 \, a c^{3} n^{3} + 64 \, a c^{3} n + {\left(a^{5} c^{3} n^{5} - 20 \, a^{5} c^{3} n^{3} + 64 \, a^{5} c^{3} n\right)} x^{4} - 2 \, {\left(a^{3} c^{3} n^{5} - 20 \, a^{3} c^{3} n^{3} + 64 \, a^{3} c^{3} n\right)} x^{2}}"," ",0,"(24*a^4*x^4 - 24*a^3*n*x^3 + n^4 + 12*(a^2*n^2 - 4*a^2)*x^2 - 16*n^2 - 4*(a*n^3 - 10*a*n)*x + 24)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^3*n^5 - 20*a*c^3*n^3 + 64*a*c^3*n + (a^5*c^3*n^5 - 20*a^5*c^3*n^3 + 64*a^5*c^3*n)*x^4 - 2*(a^3*c^3*n^5 - 20*a^3*c^3*n^3 + 64*a^3*c^3*n)*x^2)","A",0
1326,1,309,0,0.720445," ","integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^4,x, algorithm=""fricas"")","\frac{{\left(720 \, a^{6} x^{6} - 720 \, a^{5} n x^{5} + n^{6} + 360 \, {\left(a^{4} n^{2} - 6 \, a^{4}\right)} x^{4} - 50 \, n^{4} - 120 \, {\left(a^{3} n^{3} - 16 \, a^{3} n\right)} x^{3} + 30 \, {\left(a^{2} n^{4} - 28 \, a^{2} n^{2} + 72 \, a^{2}\right)} x^{2} + 544 \, n^{2} - 6 \, {\left(a n^{5} - 40 \, a n^{3} + 264 \, a n\right)} x - 720\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c^{4} n^{7} - 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} - {\left(a^{7} c^{4} n^{7} - 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} - 2304 \, a^{7} c^{4} n\right)} x^{6} - 2304 \, a c^{4} n + 3 \, {\left(a^{5} c^{4} n^{7} - 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} - 2304 \, a^{5} c^{4} n\right)} x^{4} - 3 \, {\left(a^{3} c^{4} n^{7} - 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} - 2304 \, a^{3} c^{4} n\right)} x^{2}}"," ",0,"(720*a^6*x^6 - 720*a^5*n*x^5 + n^6 + 360*(a^4*n^2 - 6*a^4)*x^4 - 50*n^4 - 120*(a^3*n^3 - 16*a^3*n)*x^3 + 30*(a^2*n^4 - 28*a^2*n^2 + 72*a^2)*x^2 + 544*n^2 - 6*(a*n^5 - 40*a*n^3 + 264*a*n)*x - 720)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^4*n^7 - 56*a*c^4*n^5 + 784*a*c^4*n^3 - (a^7*c^4*n^7 - 56*a^7*c^4*n^5 + 784*a^7*c^4*n^3 - 2304*a^7*c^4*n)*x^6 - 2304*a*c^4*n + 3*(a^5*c^4*n^7 - 56*a^5*c^4*n^5 + 784*a^5*c^4*n^3 - 2304*a^5*c^4*n)*x^4 - 3*(a^3*c^4*n^7 - 56*a^3*c^4*n^5 + 784*a^3*c^4*n^3 - 2304*a^3*c^4*n)*x^2)","A",0
1327,0,0,0,0.579907," ","integrate(exp(n*arctanh(a*x))*x^3*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-a^{2} c x^{2} + c} x^{3} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*x^3*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1328,0,0,0,0.670557," ","integrate(exp(n*arctanh(a*x))*x^2*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-a^{2} c x^{2} + c} x^{2} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*x^2*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1329,0,0,0,0.478375," ","integrate(exp(n*arctanh(a*x))*x*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-a^{2} c x^{2} + c} x \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*x*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1330,0,0,0,0.853042," ","integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1331,0,0,0,0.581233," ","integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2)/x,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{x}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/x, x)","F",0
1332,0,0,0,0.516897," ","integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2)/x^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{x^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/x^2, x)","F",0
1333,0,0,0,0.657873," ","integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{2} c x^{2} - c\right)} \sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(-(a^2*c*x^2 - c)*sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1334,0,0,0,0.635056," ","integrate(exp(n*arctanh(a*x))*x^3/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} x^{3} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*x^3*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)","F",0
1335,0,0,0,0.654308," ","integrate(exp(n*arctanh(a*x))*x^2/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} x^{2} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*x^2*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)","F",0
1336,0,0,0,0.952294," ","integrate(exp(n*arctanh(a*x))*x/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} x \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*x*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)","F",0
1337,0,0,0,0.538127," ","integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)","F",0
1338,0,0,0,0.655922," ","integrate(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{3} - c x}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^3 - c*x), x)","F",0
1339,0,0,0,0.477957," ","integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{4} - c x^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^4 - c*x^2), x)","F",0
1340,0,0,0,0.503170," ","integrate(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{5} - c x^{3}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^5 - c*x^3), x)","F",0
1341,0,0,0,0.575576," ","integrate(exp(n*arctanh(a*x))*x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} x^{3} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*x^3*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)","F",0
1342,0,0,0,0.675003," ","integrate(exp(n*arctanh(a*x))*x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} x^{2} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*x^2*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)","F",0
1343,1,82,0,0.585637," ","integrate(exp(n*arctanh(a*x))*x/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a n x - 1\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c^{2} n^{2} - a^{2} c^{2} - {\left(a^{4} c^{2} n^{2} - a^{4} c^{2}\right)} x^{2}}"," ",0,"sqrt(-a^2*c*x^2 + c)*(a*n*x - 1)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c^2*n^2 - a^2*c^2 - (a^4*c^2*n^2 - a^4*c^2)*x^2)","A",0
1344,1,80,0,0.412616," ","integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} c x^{2} + c} {\left(a x - n\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c^{2} n^{2} - a c^{2} - {\left(a^{3} c^{2} n^{2} - a^{3} c^{2}\right)} x^{2}}"," ",0,"-sqrt(-a^2*c*x^2 + c)*(a*x - n)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^2*n^2 - a*c^2 - (a^3*c^2*n^2 - a^3*c^2)*x^2)","A",0
1345,0,0,0,0.518185," ","integrate(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{5} - 2 \, a^{2} c^{2} x^{3} + c^{2} x}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^5 - 2*a^2*c^2*x^3 + c^2*x), x)","F",0
1346,0,0,0,0.617103," ","integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{6} - 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^6 - 2*a^2*c^2*x^4 + c^2*x^2), x)","F",0
1347,0,0,0,0.778278," ","integrate(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{7} - 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^7 - 2*a^2*c^2*x^5 + c^2*x^3), x)","F",0
1348,1,175,0,0.557081," ","integrate(exp(n*arctanh(a*x))*x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\frac{\sqrt{-a^{2} c x^{2} + c} {\left({\left(a^{3} n^{3} - 7 \, a^{3} n\right)} x^{3} + 6 \, a n x - 3 \, {\left(a^{2} n^{2} - 3 \, a^{2}\right)} x^{2} - 6\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3} + {\left(a^{8} c^{3} n^{4} - 10 \, a^{8} c^{3} n^{2} + 9 \, a^{8} c^{3}\right)} x^{4} - 2 \, {\left(a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right)} x^{2}}"," ",0,"sqrt(-a^2*c*x^2 + c)*((a^3*n^3 - 7*a^3*n)*x^3 + 6*a*n*x - 3*(a^2*n^2 - 3*a^2)*x^2 - 6)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^3*n^4 - 10*a^4*c^3*n^2 + 9*a^4*c^3 + (a^8*c^3*n^4 - 10*a^8*c^3*n^2 + 9*a^8*c^3)*x^4 - 2*(a^6*c^3*n^4 - 10*a^6*c^3*n^2 + 9*a^6*c^3)*x^2)","A",0
1349,1,180,0,0.502023," ","integrate(exp(n*arctanh(a*x))*x^2/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a^{2} c x^{2} + c} {\left(2 \, a n^{2} x + {\left(a^{3} n^{2} - 3 \, a^{3}\right)} x^{3} - {\left(a^{2} n^{3} - 3 \, a^{2} n\right)} x^{2} - 2 \, n\right)} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{3} c^{3} n^{4} - 10 \, a^{3} c^{3} n^{2} + 9 \, a^{3} c^{3} + {\left(a^{7} c^{3} n^{4} - 10 \, a^{7} c^{3} n^{2} + 9 \, a^{7} c^{3}\right)} x^{4} - 2 \, {\left(a^{5} c^{3} n^{4} - 10 \, a^{5} c^{3} n^{2} + 9 \, a^{5} c^{3}\right)} x^{2}}"," ",0,"-sqrt(-a^2*c*x^2 + c)*(2*a*n^2*x + (a^3*n^2 - 3*a^3)*x^3 - (a^2*n^3 - 3*a^2*n)*x^2 - 2*n)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^3*c^3*n^4 - 10*a^3*c^3*n^2 + 9*a^3*c^3 + (a^7*c^3*n^4 - 10*a^7*c^3*n^2 + 9*a^7*c^3)*x^4 - 2*(a^5*c^3*n^4 - 10*a^5*c^3*n^2 + 9*a^5*c^3)*x^2)","A",0
1350,1,171,0,0.859404," ","integrate(exp(n*arctanh(a*x))*x/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","\frac{{\left(2 \, a^{3} n x^{3} - 2 \, a^{2} n^{2} x^{2} - n^{2} + {\left(a n^{3} - 3 \, a n\right)} x + 3\right)} \sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c^{3} n^{4} - 10 \, a^{2} c^{3} n^{2} + 9 \, a^{2} c^{3} + {\left(a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right)} x^{4} - 2 \, {\left(a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3}\right)} x^{2}}"," ",0,"(2*a^3*n*x^3 - 2*a^2*n^2*x^2 - n^2 + (a*n^3 - 3*a*n)*x + 3)*sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c^3*n^4 - 10*a^2*c^3*n^2 + 9*a^2*c^3 + (a^6*c^3*n^4 - 10*a^6*c^3*n^2 + 9*a^6*c^3)*x^4 - 2*(a^4*c^3*n^4 - 10*a^4*c^3*n^2 + 9*a^4*c^3)*x^2)","A",0
1351,1,165,0,0.520344," ","integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(6 \, a^{3} x^{3} - 6 \, a^{2} n x^{2} - n^{3} + 3 \, {\left(a n^{2} - 3 \, a\right)} x + 7 \, n\right)} \sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c^{3} n^{4} - 10 \, a c^{3} n^{2} + {\left(a^{5} c^{3} n^{4} - 10 \, a^{5} c^{3} n^{2} + 9 \, a^{5} c^{3}\right)} x^{4} + 9 \, a c^{3} - 2 \, {\left(a^{3} c^{3} n^{4} - 10 \, a^{3} c^{3} n^{2} + 9 \, a^{3} c^{3}\right)} x^{2}}"," ",0,"-(6*a^3*x^3 - 6*a^2*n*x^2 - n^3 + 3*(a*n^2 - 3*a)*x + 7*n)*sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^3*n^4 - 10*a*c^3*n^2 + (a^5*c^3*n^4 - 10*a^5*c^3*n^2 + 9*a^5*c^3)*x^4 + 9*a*c^3 - 2*(a^3*c^3*n^4 - 10*a^3*c^3*n^2 + 9*a^3*c^3)*x^2)","A",0
1352,0,0,0,0.624210," ","integrate(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{6} c^{3} x^{7} - 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} - c^{3} x}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^6*c^3*x^7 - 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 - c^3*x), x)","F",0
1353,0,0,0,0.541564," ","integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{6} c^{3} x^{8} - 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} - c^{3} x^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^6*c^3*x^8 - 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 - c^3*x^2), x)","F",0
1354,0,0,0,0.572658," ","integrate(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{6} c^{3} x^{9} - 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} - c^{3} x^{3}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^6*c^3*x^9 - 3*a^4*c^3*x^7 + 3*a^2*c^3*x^5 - c^3*x^3), x)","F",0
1355,1,291,0,0.483112," ","integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^(7/2),x, algorithm=""fricas"")","-\frac{{\left(120 \, a^{5} x^{5} - 120 \, a^{4} n x^{4} - n^{5} + 60 \, {\left(a^{3} n^{2} - 5 \, a^{3}\right)} x^{3} + 30 \, n^{3} - 20 \, {\left(a^{2} n^{3} - 13 \, a^{2} n\right)} x^{2} + 5 \, {\left(a n^{4} - 22 \, a n^{2} + 45 \, a\right)} x - 149 \, n\right)} \sqrt{-a^{2} c x^{2} + c} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a c^{4} n^{6} - 35 \, a c^{4} n^{4} + 259 \, a c^{4} n^{2} - {\left(a^{7} c^{4} n^{6} - 35 \, a^{7} c^{4} n^{4} + 259 \, a^{7} c^{4} n^{2} - 225 \, a^{7} c^{4}\right)} x^{6} - 225 \, a c^{4} + 3 \, {\left(a^{5} c^{4} n^{6} - 35 \, a^{5} c^{4} n^{4} + 259 \, a^{5} c^{4} n^{2} - 225 \, a^{5} c^{4}\right)} x^{4} - 3 \, {\left(a^{3} c^{4} n^{6} - 35 \, a^{3} c^{4} n^{4} + 259 \, a^{3} c^{4} n^{2} - 225 \, a^{3} c^{4}\right)} x^{2}}"," ",0,"-(120*a^5*x^5 - 120*a^4*n*x^4 - n^5 + 60*(a^3*n^2 - 5*a^3)*x^3 + 30*n^3 - 20*(a^2*n^3 - 13*a^2*n)*x^2 + 5*(a*n^4 - 22*a*n^2 + 45*a)*x - 149*n)*sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^4*n^6 - 35*a*c^4*n^4 + 259*a*c^4*n^2 - (a^7*c^4*n^6 - 35*a^7*c^4*n^4 + 259*a^7*c^4*n^2 - 225*a^7*c^4)*x^6 - 225*a*c^4 + 3*(a^5*c^4*n^6 - 35*a^5*c^4*n^4 + 259*a^5*c^4*n^2 - 225*a^5*c^4)*x^4 - 3*(a^3*c^4*n^6 - 35*a^3*c^4*n^4 + 259*a^3*c^4*n^2 - 225*a^3*c^4)*x^2)","A",0
1356,0,0,0,0.548908," ","integrate(exp(n*arctanh(a*x))*x^m*(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right)} x^{m} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*x^m*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1357,0,0,0,1.977578," ","integrate(exp(n*arctanh(a*x))*x^m*(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{2} c x^{2} - c\right)} x^{m} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral(-(a^2*c*x^2 - c)*x^m*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1358,0,0,0,0.493541," ","integrate(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{x^{m} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, x\right)"," ",0,"integral(-x^m*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)","F",0
1359,0,0,0,0.526130," ","integrate(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c)^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{x^{m} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right)"," ",0,"integral(x^m*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)","F",0
1360,0,0,0,0.997615," ","integrate(exp(n*arctanh(a*x))*x^m*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-a^{2} c x^{2} + c\right)}^{p} x^{m} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral((-a^2*c*x^2 + c)^p*x^m*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1361,0,0,0,0.426684," ","integrate(exp(n*arctanh(a*x))*x*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-a^{2} c x^{2} + c\right)}^{p} x \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral((-a^2*c*x^2 + c)^p*x*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1362,0,0,0,0.478121," ","integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-a^{2} c x^{2} + c\right)}^{p} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}, x\right)"," ",0,"integral((-a^2*c*x^2 + c)^p*((a*x + 1)/(a*x - 1))^(1/2*n), x)","F",0
1363,1,80,0,0.534600," ","integrate(exp(2*(1+p)*arctanh(a*x))/((-a^2*x^2+1)^p),x, algorithm=""fricas"")","-\frac{{\left(a^{2} p x^{2} - a x - p + 1\right)} \left(\frac{a x + 1}{a x - 1}\right)^{p + 1}}{{\left(2 \, a p^{2} - a p + {\left(2 \, a^{2} p^{2} - a^{2} p\right)} x\right)} {\left(-a^{2} x^{2} + 1\right)}^{p}}"," ",0,"-(a^2*p*x^2 - a*x - p + 1)*((a*x + 1)/(a*x - 1))^(p + 1)/((2*a*p^2 - a*p + (2*a^2*p^2 - a^2*p)*x)*(-a^2*x^2 + 1)^p)","A",0
1364,1,81,0,0.477189," ","integrate(exp(2*(1+p)*arctanh(a*x))/((-a^2*c*x^2+c)^p),x, algorithm=""fricas"")","-\frac{{\left(a^{2} p x^{2} - a x - p + 1\right)} \left(\frac{a x + 1}{a x - 1}\right)^{p + 1}}{{\left(2 \, a p^{2} - a p + {\left(2 \, a^{2} p^{2} - a^{2} p\right)} x\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}"," ",0,"-(a^2*p*x^2 - a*x - p + 1)*((a*x + 1)/(a*x - 1))^(p + 1)/((2*a*p^2 - a*p + (2*a^2*p^2 - a^2*p)*x)*(-a^2*c*x^2 + c)^p)","A",0
1365,1,42,0,0.455029," ","integrate(exp(2*p*arctanh(a*x))*(-a^2*c*x^2+c)^p,x, algorithm=""fricas"")","\frac{{\left(a x + 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p} \left(\frac{a x + 1}{a x - 1}\right)^{p}}{2 \, a p + a}"," ",0,"(a*x + 1)*(-a^2*c*x^2 + c)^p*((a*x + 1)/(a*x - 1))^p/(2*a*p + a)","A",0
1366,1,44,0,0.544522," ","integrate((-a^2*c*x^2+c)^p/exp(2*p*arctanh(a*x)),x, algorithm=""fricas"")","\frac{{\left(a x - 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{p}}{{\left(2 \, a p + a\right)} \left(\frac{a x + 1}{a x - 1}\right)^{p}}"," ",0,"(a*x - 1)*(-a^2*c*x^2 + c)^p/((2*a*p + a)*((a*x + 1)/(a*x - 1))^p)","A",0
1367,1,77,0,0.479023," ","integrate(exp(n*arctanh(a*x))*x^2*(-a^2*c*x^2+c)^(-1-1/2*n^2),x, algorithm=""fricas"")","-\frac{{\left(a^{3} n x^{3} - a^{2} x^{2} - a n x + 1\right)} {\left(-a^{2} c x^{2} + c\right)}^{-\frac{1}{2} \, n^{2} - 1} \left(\frac{a x + 1}{a x - 1}\right)^{\frac{1}{2} \, n}}{a^{3} n^{3} - a^{3} n}"," ",0,"-(a^3*n*x^3 - a^2*x^2 - a*n*x + 1)*(-a^2*c*x^2 + c)^(-1/2*n^2 - 1)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^3*n^3 - a^3*n)","A",0
1368,1,379,0,1.479661," ","integrate((a*x+1)^6/(-a^2*x^2+1)^3*x^2/(-a^2*c*x^2+c)^19,x, algorithm=""fricas"")","-\frac{6 \, a x - 1}{210 \, {\left(a^{39} c^{19} x^{36} - 6 \, a^{38} c^{19} x^{35} + 70 \, a^{36} c^{19} x^{33} - 105 \, a^{35} c^{19} x^{32} - 336 \, a^{34} c^{19} x^{31} + 896 \, a^{33} c^{19} x^{30} + 720 \, a^{32} c^{19} x^{29} - 3900 \, a^{31} c^{19} x^{28} + 280 \, a^{30} c^{19} x^{27} + 10752 \, a^{29} c^{19} x^{26} - 6552 \, a^{28} c^{19} x^{25} - 20020 \, a^{27} c^{19} x^{24} + 21840 \, a^{26} c^{19} x^{23} + 24960 \, a^{25} c^{19} x^{22} - 43472 \, a^{24} c^{19} x^{21} - 18018 \, a^{23} c^{19} x^{20} + 60060 \, a^{22} c^{19} x^{19} - 60060 \, a^{20} c^{19} x^{17} + 18018 \, a^{19} c^{19} x^{16} + 43472 \, a^{18} c^{19} x^{15} - 24960 \, a^{17} c^{19} x^{14} - 21840 \, a^{16} c^{19} x^{13} + 20020 \, a^{15} c^{19} x^{12} + 6552 \, a^{14} c^{19} x^{11} - 10752 \, a^{13} c^{19} x^{10} - 280 \, a^{12} c^{19} x^{9} + 3900 \, a^{11} c^{19} x^{8} - 720 \, a^{10} c^{19} x^{7} - 896 \, a^{9} c^{19} x^{6} + 336 \, a^{8} c^{19} x^{5} + 105 \, a^{7} c^{19} x^{4} - 70 \, a^{6} c^{19} x^{3} + 6 \, a^{4} c^{19} x - a^{3} c^{19}\right)}}"," ",0,"-1/210*(6*a*x - 1)/(a^39*c^19*x^36 - 6*a^38*c^19*x^35 + 70*a^36*c^19*x^33 - 105*a^35*c^19*x^32 - 336*a^34*c^19*x^31 + 896*a^33*c^19*x^30 + 720*a^32*c^19*x^29 - 3900*a^31*c^19*x^28 + 280*a^30*c^19*x^27 + 10752*a^29*c^19*x^26 - 6552*a^28*c^19*x^25 - 20020*a^27*c^19*x^24 + 21840*a^26*c^19*x^23 + 24960*a^25*c^19*x^22 - 43472*a^24*c^19*x^21 - 18018*a^23*c^19*x^20 + 60060*a^22*c^19*x^19 - 60060*a^20*c^19*x^17 + 18018*a^19*c^19*x^16 + 43472*a^18*c^19*x^15 - 24960*a^17*c^19*x^14 - 21840*a^16*c^19*x^13 + 20020*a^15*c^19*x^12 + 6552*a^14*c^19*x^11 - 10752*a^13*c^19*x^10 - 280*a^12*c^19*x^9 + 3900*a^11*c^19*x^8 - 720*a^10*c^19*x^7 - 896*a^9*c^19*x^6 + 336*a^8*c^19*x^5 + 105*a^7*c^19*x^4 - 70*a^6*c^19*x^3 + 6*a^4*c^19*x - a^3*c^19)","B",0
1369,1,169,0,0.471851," ","integrate((a*x+1)^4/(-a^2*x^2+1)^2*x^2/(-a^2*c*x^2+c)^9,x, algorithm=""fricas"")","\frac{4 \, a x - 1}{60 \, {\left(a^{19} c^{9} x^{16} - 4 \, a^{18} c^{9} x^{15} + 20 \, a^{16} c^{9} x^{13} - 20 \, a^{15} c^{9} x^{12} - 36 \, a^{14} c^{9} x^{11} + 64 \, a^{13} c^{9} x^{10} + 20 \, a^{12} c^{9} x^{9} - 90 \, a^{11} c^{9} x^{8} + 20 \, a^{10} c^{9} x^{7} + 64 \, a^{9} c^{9} x^{6} - 36 \, a^{8} c^{9} x^{5} - 20 \, a^{7} c^{9} x^{4} + 20 \, a^{6} c^{9} x^{3} - 4 \, a^{4} c^{9} x + a^{3} c^{9}\right)}}"," ",0,"1/60*(4*a*x - 1)/(a^19*c^9*x^16 - 4*a^18*c^9*x^15 + 20*a^16*c^9*x^13 - 20*a^15*c^9*x^12 - 36*a^14*c^9*x^11 + 64*a^13*c^9*x^10 + 20*a^12*c^9*x^9 - 90*a^11*c^9*x^8 + 20*a^10*c^9*x^7 + 64*a^9*c^9*x^6 - 36*a^8*c^9*x^5 - 20*a^7*c^9*x^4 + 20*a^6*c^9*x^3 - 4*a^4*c^9*x + a^3*c^9)","B",0
1370,1,49,0,0.648131," ","integrate((a*x+1)^2/(-a^2*x^2+1)*x^2/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{2 \, a x - 1}{6 \, {\left(a^{7} c^{3} x^{4} - 2 \, a^{6} c^{3} x^{3} + 2 \, a^{4} c^{3} x - a^{3} c^{3}\right)}}"," ",0,"-1/6*(2*a*x - 1)/(a^7*c^3*x^4 - 2*a^6*c^3*x^3 + 2*a^4*c^3*x - a^3*c^3)","A",0
1371,1,49,0,0.605703," ","integrate(x^2/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^3,x, algorithm=""fricas"")","-\frac{2 \, a x + 1}{6 \, {\left(a^{7} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{3} - 2 \, a^{4} c^{3} x - a^{3} c^{3}\right)}}"," ",0,"-1/6*(2*a*x + 1)/(a^7*c^3*x^4 + 2*a^6*c^3*x^3 - 2*a^4*c^3*x - a^3*c^3)","A",0
1372,1,169,0,0.633807," ","integrate(x^2/(a*x+1)^4*(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^9,x, algorithm=""fricas"")","\frac{4 \, a x + 1}{60 \, {\left(a^{19} c^{9} x^{16} + 4 \, a^{18} c^{9} x^{15} - 20 \, a^{16} c^{9} x^{13} - 20 \, a^{15} c^{9} x^{12} + 36 \, a^{14} c^{9} x^{11} + 64 \, a^{13} c^{9} x^{10} - 20 \, a^{12} c^{9} x^{9} - 90 \, a^{11} c^{9} x^{8} - 20 \, a^{10} c^{9} x^{7} + 64 \, a^{9} c^{9} x^{6} + 36 \, a^{8} c^{9} x^{5} - 20 \, a^{7} c^{9} x^{4} - 20 \, a^{6} c^{9} x^{3} + 4 \, a^{4} c^{9} x + a^{3} c^{9}\right)}}"," ",0,"1/60*(4*a*x + 1)/(a^19*c^9*x^16 + 4*a^18*c^9*x^15 - 20*a^16*c^9*x^13 - 20*a^15*c^9*x^12 + 36*a^14*c^9*x^11 + 64*a^13*c^9*x^10 - 20*a^12*c^9*x^9 - 90*a^11*c^9*x^8 - 20*a^10*c^9*x^7 + 64*a^9*c^9*x^6 + 36*a^8*c^9*x^5 - 20*a^7*c^9*x^4 - 20*a^6*c^9*x^3 + 4*a^4*c^9*x + a^3*c^9)","B",0
1373,1,496,0,0.978701," ","integrate((a*x+1)^5/(-a^2*x^2+1)^(5/2)*x^2/(-a^2*c*x^2+c)^(27/2),x, algorithm=""fricas"")","-\frac{{\left(a^{22} x^{25} - 5 \, a^{21} x^{24} + 40 \, a^{19} x^{22} - 50 \, a^{18} x^{21} - 126 \, a^{17} x^{20} + 280 \, a^{16} x^{19} + 160 \, a^{15} x^{18} - 765 \, a^{14} x^{17} + 105 \, a^{13} x^{16} + 1248 \, a^{12} x^{15} - 720 \, a^{11} x^{14} - 1260 \, a^{10} x^{13} + 1260 \, a^{9} x^{12} + 720 \, a^{8} x^{11} - 1248 \, a^{7} x^{10} - 105 \, a^{6} x^{9} + 765 \, a^{5} x^{8} - 160 \, a^{4} x^{7} - 280 \, a^{3} x^{6} + 126 \, a^{2} x^{5} + 50 \, a x^{4} - 40 \, x^{3}\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{120 \, {\left(a^{27} c^{14} x^{27} - 5 \, a^{26} c^{14} x^{26} - a^{25} c^{14} x^{25} + 45 \, a^{24} c^{14} x^{24} - 50 \, a^{23} c^{14} x^{23} - 166 \, a^{22} c^{14} x^{22} + 330 \, a^{21} c^{14} x^{21} + 286 \, a^{20} c^{14} x^{20} - 1045 \, a^{19} c^{14} x^{19} - 55 \, a^{18} c^{14} x^{18} + 2013 \, a^{17} c^{14} x^{17} - 825 \, a^{16} c^{14} x^{16} - 2508 \, a^{15} c^{14} x^{15} + 1980 \, a^{14} c^{14} x^{14} + 1980 \, a^{13} c^{14} x^{13} - 2508 \, a^{12} c^{14} x^{12} - 825 \, a^{11} c^{14} x^{11} + 2013 \, a^{10} c^{14} x^{10} - 55 \, a^{9} c^{14} x^{9} - 1045 \, a^{8} c^{14} x^{8} + 286 \, a^{7} c^{14} x^{7} + 330 \, a^{6} c^{14} x^{6} - 166 \, a^{5} c^{14} x^{5} - 50 \, a^{4} c^{14} x^{4} + 45 \, a^{3} c^{14} x^{3} - a^{2} c^{14} x^{2} - 5 \, a c^{14} x + c^{14}\right)}}"," ",0,"-1/120*(a^22*x^25 - 5*a^21*x^24 + 40*a^19*x^22 - 50*a^18*x^21 - 126*a^17*x^20 + 280*a^16*x^19 + 160*a^15*x^18 - 765*a^14*x^17 + 105*a^13*x^16 + 1248*a^12*x^15 - 720*a^11*x^14 - 1260*a^10*x^13 + 1260*a^9*x^12 + 720*a^8*x^11 - 1248*a^7*x^10 - 105*a^6*x^9 + 765*a^5*x^8 - 160*a^4*x^7 - 280*a^3*x^6 + 126*a^2*x^5 + 50*a*x^4 - 40*x^3)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^27*c^14*x^27 - 5*a^26*c^14*x^26 - a^25*c^14*x^25 + 45*a^24*c^14*x^24 - 50*a^23*c^14*x^23 - 166*a^22*c^14*x^22 + 330*a^21*c^14*x^21 + 286*a^20*c^14*x^20 - 1045*a^19*c^14*x^19 - 55*a^18*c^14*x^18 + 2013*a^17*c^14*x^17 - 825*a^16*c^14*x^16 - 2508*a^15*c^14*x^15 + 1980*a^14*c^14*x^14 + 1980*a^13*c^14*x^13 - 2508*a^12*c^14*x^12 - 825*a^11*c^14*x^11 + 2013*a^10*c^14*x^10 - 55*a^9*c^14*x^9 - 1045*a^8*c^14*x^8 + 286*a^7*c^14*x^7 + 330*a^6*c^14*x^6 - 166*a^5*c^14*x^5 - 50*a^4*c^14*x^4 + 45*a^3*c^14*x^3 - a^2*c^14*x^2 - 5*a*c^14*x + c^14)","B",0
1374,1,193,0,0.540067," ","integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2/(-a^2*c*x^2+c)^(11/2),x, algorithm=""fricas"")","-\frac{{\left(a^{6} x^{9} - 3 \, a^{5} x^{8} + 8 \, a^{3} x^{6} - 6 \, a^{2} x^{5} - 6 \, a x^{4} + 8 \, x^{3}\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{24 \, {\left(a^{11} c^{6} x^{11} - 3 \, a^{10} c^{6} x^{10} - a^{9} c^{6} x^{9} + 11 \, a^{8} c^{6} x^{8} - 6 \, a^{7} c^{6} x^{7} - 14 \, a^{6} c^{6} x^{6} + 14 \, a^{5} c^{6} x^{5} + 6 \, a^{4} c^{6} x^{4} - 11 \, a^{3} c^{6} x^{3} + a^{2} c^{6} x^{2} + 3 \, a c^{6} x - c^{6}\right)}}"," ",0,"-1/24*(a^6*x^9 - 3*a^5*x^8 + 8*a^3*x^6 - 6*a^2*x^5 - 6*a*x^4 + 8*x^3)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^11*c^6*x^11 - 3*a^10*c^6*x^10 - a^9*c^6*x^9 + 11*a^8*c^6*x^8 - 6*a^7*c^6*x^7 - 14*a^6*c^6*x^6 + 14*a^5*c^6*x^5 + 6*a^4*c^6*x^4 - 11*a^3*c^6*x^3 + a^2*c^6*x^2 + 3*a*c^6*x - c^6)","B",0
1375,0,0,0,0.614327," ","integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\right)"," ",0,"integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^2/(a^5*c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)","F",0
1376,0,0,0,0.757851," ","integrate(x^2/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{5} c^{2} x^{5} + a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + a c^{2} x + c^{2}}, x\right)"," ",0,"integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^2/(a^5*c^2*x^5 + a^4*c^2*x^4 - 2*a^3*c^2*x^3 - 2*a^2*c^2*x^2 + a*c^2*x + c^2), x)","F",0
1377,1,192,0,0.951926," ","integrate(x^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(11/2),x, algorithm=""fricas"")","\frac{{\left(a^{6} x^{9} + 3 \, a^{5} x^{8} - 8 \, a^{3} x^{6} - 6 \, a^{2} x^{5} + 6 \, a x^{4} + 8 \, x^{3}\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{24 \, {\left(a^{11} c^{6} x^{11} + 3 \, a^{10} c^{6} x^{10} - a^{9} c^{6} x^{9} - 11 \, a^{8} c^{6} x^{8} - 6 \, a^{7} c^{6} x^{7} + 14 \, a^{6} c^{6} x^{6} + 14 \, a^{5} c^{6} x^{5} - 6 \, a^{4} c^{6} x^{4} - 11 \, a^{3} c^{6} x^{3} - a^{2} c^{6} x^{2} + 3 \, a c^{6} x + c^{6}\right)}}"," ",0,"1/24*(a^6*x^9 + 3*a^5*x^8 - 8*a^3*x^6 - 6*a^2*x^5 + 6*a*x^4 + 8*x^3)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^11*c^6*x^11 + 3*a^10*c^6*x^10 - a^9*c^6*x^9 - 11*a^8*c^6*x^8 - 6*a^7*c^6*x^7 + 14*a^6*c^6*x^6 + 14*a^5*c^6*x^5 - 6*a^4*c^6*x^4 - 11*a^3*c^6*x^3 - a^2*c^6*x^2 + 3*a*c^6*x + c^6)","B",0
1378,1,497,0,0.950685," ","integrate(x^2/(a*x+1)^5*(-a^2*x^2+1)^(5/2)/(-a^2*c*x^2+c)^(27/2),x, algorithm=""fricas"")","\frac{{\left(a^{22} x^{25} + 5 \, a^{21} x^{24} - 40 \, a^{19} x^{22} - 50 \, a^{18} x^{21} + 126 \, a^{17} x^{20} + 280 \, a^{16} x^{19} - 160 \, a^{15} x^{18} - 765 \, a^{14} x^{17} - 105 \, a^{13} x^{16} + 1248 \, a^{12} x^{15} + 720 \, a^{11} x^{14} - 1260 \, a^{10} x^{13} - 1260 \, a^{9} x^{12} + 720 \, a^{8} x^{11} + 1248 \, a^{7} x^{10} - 105 \, a^{6} x^{9} - 765 \, a^{5} x^{8} - 160 \, a^{4} x^{7} + 280 \, a^{3} x^{6} + 126 \, a^{2} x^{5} - 50 \, a x^{4} - 40 \, x^{3}\right)} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{120 \, {\left(a^{27} c^{14} x^{27} + 5 \, a^{26} c^{14} x^{26} - a^{25} c^{14} x^{25} - 45 \, a^{24} c^{14} x^{24} - 50 \, a^{23} c^{14} x^{23} + 166 \, a^{22} c^{14} x^{22} + 330 \, a^{21} c^{14} x^{21} - 286 \, a^{20} c^{14} x^{20} - 1045 \, a^{19} c^{14} x^{19} + 55 \, a^{18} c^{14} x^{18} + 2013 \, a^{17} c^{14} x^{17} + 825 \, a^{16} c^{14} x^{16} - 2508 \, a^{15} c^{14} x^{15} - 1980 \, a^{14} c^{14} x^{14} + 1980 \, a^{13} c^{14} x^{13} + 2508 \, a^{12} c^{14} x^{12} - 825 \, a^{11} c^{14} x^{11} - 2013 \, a^{10} c^{14} x^{10} - 55 \, a^{9} c^{14} x^{9} + 1045 \, a^{8} c^{14} x^{8} + 286 \, a^{7} c^{14} x^{7} - 330 \, a^{6} c^{14} x^{6} - 166 \, a^{5} c^{14} x^{5} + 50 \, a^{4} c^{14} x^{4} + 45 \, a^{3} c^{14} x^{3} + a^{2} c^{14} x^{2} - 5 \, a c^{14} x - c^{14}\right)}}"," ",0,"1/120*(a^22*x^25 + 5*a^21*x^24 - 40*a^19*x^22 - 50*a^18*x^21 + 126*a^17*x^20 + 280*a^16*x^19 - 160*a^15*x^18 - 765*a^14*x^17 - 105*a^13*x^16 + 1248*a^12*x^15 + 720*a^11*x^14 - 1260*a^10*x^13 - 1260*a^9*x^12 + 720*a^8*x^11 + 1248*a^7*x^10 - 105*a^6*x^9 - 765*a^5*x^8 - 160*a^4*x^7 + 280*a^3*x^6 + 126*a^2*x^5 - 50*a*x^4 - 40*x^3)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^27*c^14*x^27 + 5*a^26*c^14*x^26 - a^25*c^14*x^25 - 45*a^24*c^14*x^24 - 50*a^23*c^14*x^23 + 166*a^22*c^14*x^22 + 330*a^21*c^14*x^21 - 286*a^20*c^14*x^20 - 1045*a^19*c^14*x^19 + 55*a^18*c^14*x^18 + 2013*a^17*c^14*x^17 + 825*a^16*c^14*x^16 - 2508*a^15*c^14*x^15 - 1980*a^14*c^14*x^14 + 1980*a^13*c^14*x^13 + 2508*a^12*c^14*x^12 - 825*a^11*c^14*x^11 - 2013*a^10*c^14*x^10 - 55*a^9*c^14*x^9 + 1045*a^8*c^14*x^8 + 286*a^7*c^14*x^7 - 330*a^6*c^14*x^6 - 166*a^5*c^14*x^5 + 50*a^4*c^14*x^4 + 45*a^3*c^14*x^3 + a^2*c^14*x^2 - 5*a*c^14*x - c^14)","B",0
