1,1,94,0,0.1127564,"\int \frac{(A+B x) \left(a+b x+c x^2\right)}{d+f x^2} \, dx","Int[((A + B*x)*(a + b*x + c*x^2))/(d + f*x^2),x]","-\frac{\log \left(d+f x^2\right) (-a B f-A b f+B c d)}{2 f^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) (-a A f+A c d+b B d)}{\sqrt{d} f^{3/2}}+\frac{x (A c+b B)}{f}+\frac{B c x^2}{2 f}","-\frac{\log \left(d+f x^2\right) (-a B f-A b f+B c d)}{2 f^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) (-a A f+A c d+b B d)}{\sqrt{d} f^{3/2}}+\frac{x (A c+b B)}{f}+\frac{B c x^2}{2 f}",1,"((b*B + A*c)*x)/f + (B*c*x^2)/(2*f) - ((b*B*d + A*c*d - a*A*f)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*f^(3/2)) - ((B*c*d - A*b*f - a*B*f)*Log[d + f*x^2])/(2*f^2)","A",5,4,25,0.1600,1,"{1629, 635, 205, 260}"
2,1,228,0,0.3296416,"\int \frac{(A+B x) \left(a+b x+c x^2\right)^2}{d+f x^2} \, dx","Int[((A + B*x)*(a + b*x + c*x^2)^2)/(d + f*x^2),x]","-\frac{\log \left(d+f x^2\right) \left(2 A b f (c d-a f)-B \left(-f \left(b^2 d-a^2 f\right)-2 a c d f+c^2 d^2\right)\right)}{2 f^3}+\frac{x^2 \left(2 A b c f-B \left(-2 a c f+b^2 (-f)+c^2 d\right)\right)}{2 f^2}+\frac{x \left(-A c (c d-2 a f)-b B (2 c d-2 a f)+A b^2 f\right)}{f^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) \left(-A (c d-a f)^2-2 b B d (c d-a f)+A b^2 d f\right)}{\sqrt{d} f^{5/2}}+\frac{c x^3 (A c+2 b B)}{3 f}+\frac{B c^2 x^4}{4 f}","-\frac{\log \left(d+f x^2\right) \left(2 A b f (c d-a f)-B \left(-f \left(b^2 d-a^2 f\right)-2 a c d f+c^2 d^2\right)\right)}{2 f^3}+\frac{x^2 \left(2 A b c f-B \left(-2 a c f+b^2 (-f)+c^2 d\right)\right)}{2 f^2}+\frac{x \left(-A c (c d-2 a f)-b B (2 c d-2 a f)+A b^2 f\right)}{f^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) \left(-A (c d-a f)^2-2 b B d (c d-a f)+A b^2 d f\right)}{\sqrt{d} f^{5/2}}+\frac{c x^3 (A c+2 b B)}{3 f}+\frac{B c^2 x^4}{4 f}",1,"((A*b^2*f - A*c*(c*d - 2*a*f) - b*B*(2*c*d - 2*a*f))*x)/f^2 + ((2*A*b*c*f - B*(c^2*d - b^2*f - 2*a*c*f))*x^2)/(2*f^2) + (c*(2*b*B + A*c)*x^3)/(3*f) + (B*c^2*x^4)/(4*f) - ((A*b^2*d*f - 2*b*B*d*(c*d - a*f) - A*(c*d - a*f)^2)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*f^(5/2)) - ((2*A*b*f*(c*d - a*f) - B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)))*Log[d + f*x^2])/(2*f^3)","A",5,4,27,0.1481,1,"{1012, 635, 205, 260}"
3,1,441,0,0.6210308,"\int \frac{(A+B x) \left(a+b x+c x^2\right)^3}{d+f x^2} \, dx","Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + f*x^2),x]","-\frac{x^2 \left(A b f \left(-6 a c f+b^2 (-f)+3 c^2 d\right)-B \left(-3 c f \left(b^2 d-a^2 f\right)+3 a b^2 f^2-3 a c^2 d f+c^3 d^2\right)\right)}{2 f^3}+\frac{\log \left(d+f x^2\right) \left(A b f \left(-f \left(b^2 d-3 a^2 f\right)-6 a c d f+3 c^2 d^2\right)-B (c d-a f) \left(-f \left(3 b^2 d-a^2 f\right)-2 a c d f+c^2 d^2\right)\right)}{2 f^4}-\frac{x \left(-A c \left(3 a^2 f^2-3 a c d f+c^2 d^2\right)+3 A b^2 f (c d-a f)-3 b B (c d-a f)^2+b^3 B d f\right)}{f^3}+\frac{c x^4 \left(3 A b c f-B \left(-3 a c f-3 b^2 f+c^2 d\right)\right)}{4 f^2}+\frac{x^3 \left(-A c^2 (c d-3 a f)-3 b B c (c d-2 a f)+3 A b^2 c f+b^3 B f\right)}{3 f^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) \left(3 A b^2 d f (c d-a f)-A (c d-a f)^3-3 b B d (c d-a f)^2+b^3 B d^2 f\right)}{\sqrt{d} f^{7/2}}+\frac{c^2 x^5 (A c+3 b B)}{5 f}+\frac{B c^3 x^6}{6 f}","-\frac{x^2 \left(A b f \left(-6 a c f+b^2 (-f)+3 c^2 d\right)-B \left(-3 c f \left(b^2 d-a^2 f\right)+3 a b^2 f^2-3 a c^2 d f+c^3 d^2\right)\right)}{2 f^3}+\frac{\log \left(d+f x^2\right) \left(A b f \left(-f \left(b^2 d-3 a^2 f\right)-6 a c d f+3 c^2 d^2\right)-B (c d-a f) \left(-f \left(3 b^2 d-a^2 f\right)-2 a c d f+c^2 d^2\right)\right)}{2 f^4}-\frac{x \left(-A c \left(3 a^2 f^2-3 a c d f+c^2 d^2\right)+3 A b^2 f (c d-a f)-3 b B (c d-a f)^2+b^3 B d f\right)}{f^3}+\frac{c x^4 \left(3 A b c f-B \left(-3 a c f-3 b^2 f+c^2 d\right)\right)}{4 f^2}+\frac{x^3 \left(-A c^2 (c d-3 a f)-3 b B c (c d-2 a f)+3 A b^2 c f+b^3 B f\right)}{3 f^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) \left(3 A b^2 d f (c d-a f)-A (c d-a f)^3-3 b B d (c d-a f)^2+b^3 B d^2 f\right)}{\sqrt{d} f^{7/2}}+\frac{c^2 x^5 (A c+3 b B)}{5 f}+\frac{B c^3 x^6}{6 f}",1,"-(((b^3*B*d*f + 3*A*b^2*f*(c*d - a*f) - 3*b*B*(c*d - a*f)^2 - A*c*(c^2*d^2 - 3*a*c*d*f + 3*a^2*f^2))*x)/f^3) - ((A*b*f*(3*c^2*d - b^2*f - 6*a*c*f) - B*(c^3*d^2 - 3*a*c^2*d*f + 3*a*b^2*f^2 - 3*c*f*(b^2*d - a^2*f)))*x^2)/(2*f^3) + ((b^3*B*f + 3*A*b^2*c*f - A*c^2*(c*d - 3*a*f) - 3*b*B*c*(c*d - 2*a*f))*x^3)/(3*f^2) + (c*(3*A*b*c*f - B*(c^2*d - 3*b^2*f - 3*a*c*f))*x^4)/(4*f^2) + (c^2*(3*b*B + A*c)*x^5)/(5*f) + (B*c^3*x^6)/(6*f) + ((b^3*B*d^2*f + 3*A*b^2*d*f*(c*d - a*f) - 3*b*B*d*(c*d - a*f)^2 - A*(c*d - a*f)^3)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*f^(7/2)) + ((A*b*f*(3*c^2*d^2 - 6*a*c*d*f - f*(b^2*d - 3*a^2*f)) - B*(c*d - a*f)*(c^2*d^2 - 2*a*c*d*f - f*(3*b^2*d - a^2*f)))*Log[d + f*x^2])/(2*f^4)","A",5,4,27,0.1481,1,"{1012, 635, 205, 260}"
4,1,274,0,0.2819074,"\int \frac{A+B x}{\left(a+b x+c x^2\right) \left(d+f x^2\right)} \, dx","Int[(A + B*x)/((a + b*x + c*x^2)*(d + f*x^2)),x]","\frac{\log \left(a+b x+c x^2\right) (-a B f+A b f+B c d)}{2 \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)}-\frac{\log \left(d+f x^2\right) (-a B f+A b f+B c d)}{2 \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)}+\frac{\sqrt{f} \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) (a A f-A c d+b B d)}{\sqrt{d} \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right)}{\sqrt{b^2-4 a c} \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)}","\frac{\log \left(a+b x+c x^2\right) (-a B f+A b f+B c d)}{2 \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)}-\frac{\log \left(d+f x^2\right) (-a B f+A b f+B c d)}{2 \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)}+\frac{\sqrt{f} \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) (a A f-A c d+b B d)}{\sqrt{d} \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right)}{\sqrt{b^2-4 a c} \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)}",1,"(Sqrt[f]*(b*B*d - A*c*d + a*A*f)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))) - ((A*b^2*f + 2*A*c*(c*d - a*f) - b*B*(c*d + a*f))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))) + ((B*c*d + A*b*f - a*B*f)*Log[a + b*x + c*x^2])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))) - ((B*c*d + A*b*f - a*B*f)*Log[d + f*x^2])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f)))","A",8,8,27,0.2963,1,"{1023, 634, 618, 206, 628, 635, 205, 260}"
5,1,596,0,1.771683,"\int \frac{A+B x}{\left(a+b x+c x^2\right)^2 \left(d+f x^2\right)} \, dx","Int[(A + B*x)/((a + b*x + c*x^2)^2*(d + f*x^2)),x]","-\frac{f^{3/2} \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) \left(-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right)}{\sqrt{d} \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)^2}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-4 A b^2 c f \left(3 a^2 f^2-3 a c d f+2 c^2 d^2\right)+b^3 B f \left(-a^2 f^2-4 a c d f+5 c^2 d^2\right)+2 b B c \left(3 a^2 c d f^2+3 a^3 f^3-7 a c^2 d^2 f+c^3 d^3\right)-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right)}{\left(b^2-4 a c\right)^{3/2} \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)^2}-\frac{f \log \left(a+b x+c x^2\right) \left(B \left(-f \left(b^2 d-a^2 f\right)-2 a c d f+c^2 d^2\right)+2 A b f (c d-a f)\right)}{2 \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)^2}+\frac{f \log \left(d+f x^2\right) \left(B \left(-f \left(b^2 d-a^2 f\right)-2 a c d f+c^2 d^2\right)+2 A b f (c d-a f)\right)}{2 \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)^2}+\frac{-(A b-a B) \left(-2 a c f+b^2 f+2 c^2 d\right)-c x \left(2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right)+A b c (a f+c d)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right) \left((c d-a f)^2+b^2 d f\right)}","-\frac{f^{3/2} \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) \left(-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right)}{\sqrt{d} \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)^2}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-4 A b^2 c f \left(3 a^2 f^2-3 a c d f+2 c^2 d^2\right)+b^3 B f \left(-a^2 f^2-4 a c d f+5 c^2 d^2\right)+2 b B c \left(3 a^2 c d f^2+3 a^3 f^3-7 a c^2 d^2 f+c^3 d^3\right)-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right)}{\left(b^2-4 a c\right)^{3/2} \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)^2}-\frac{f \log \left(a+b x+c x^2\right) \left(B \left(-f \left(b^2 d-a^2 f\right)-2 a c d f+c^2 d^2\right)+2 A b f (c d-a f)\right)}{2 \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)^2}+\frac{f \log \left(d+f x^2\right) \left(B \left(-f \left(b^2 d-a^2 f\right)-2 a c d f+c^2 d^2\right)+2 A b f (c d-a f)\right)}{2 \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)^2}+\frac{-(A b-a B) \left(-2 a c f+b^2 f+2 c^2 d\right)-c x \left(2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right)+A b c (a f+c d)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right) \left((c d-a f)^2+b^2 d f\right)}",1,"(A*b*c*(c*d + a*f) - (A*b - a*B)*(2*c^2*d + b^2*f - 2*a*c*f) - c*(A*b^2*f + 2*A*c*(c*d - a*f) - b*B*(c*d + a*f))*x)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(a + b*x + c*x^2)) - (f^(3/2)*(A*b^2*d*f + 2*b*B*d*(c*d - a*f) - A*(c*d - a*f)^2)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2) - ((b^5*B*d*f^2 - 2*A*b^4*f^2*(c*d - a*f) - 4*A*c^2*(c*d - 3*a*f)*(c*d - a*f)^2 + b^3*B*f*(5*c^2*d^2 - 4*a*c*d*f - a^2*f^2) - 4*A*b^2*c*f*(2*c^2*d^2 - 3*a*c*d*f + 3*a^2*f^2) + 2*b*B*c*(c^3*d^3 - 7*a*c^2*d^2*f + 3*a^2*c*d*f^2 + 3*a^3*f^3))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(3/2)*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2) - (f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)))*Log[a + b*x + c*x^2])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2) + (f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)))*Log[d + f*x^2])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2)","A",9,9,27,0.3333,1,"{1018, 1074, 634, 618, 206, 628, 635, 205, 260}"
6,1,331,0,0.6005823,"\int \frac{(A+B x) \sqrt{a+b x+c x^2}}{d-f x^2} \, dx","Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]","-\frac{\left(B \sqrt{d}-A \sqrt{f}\right) \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} f^{3/2}}+\frac{\left(A \sqrt{f}+B \sqrt{d}\right) \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} f^{3/2}}-\frac{(2 A c+b B) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} f}-\frac{B \sqrt{a+b x+c x^2}}{f}","-\frac{\left(B \sqrt{d}-A \sqrt{f}\right) \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} f^{3/2}}+\frac{\left(A \sqrt{f}+B \sqrt{d}\right) \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} f^{3/2}}-\frac{(2 A c+b B) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} f}-\frac{B \sqrt{a+b x+c x^2}}{f}",1,"-((B*Sqrt[a + b*x + c*x^2])/f) - ((b*B + 2*A*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f) - ((B*Sqrt[d] - A*Sqrt[f])*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*f^(3/2)) + ((B*Sqrt[d] + A*Sqrt[f])*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*f^(3/2))","A",9,6,30,0.2000,1,"{1021, 1078, 621, 206, 1033, 724}"
7,1,249,0,0.2043151,"\int \frac{A+B x}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Int[(A + B*x)/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{\left(\frac{A \sqrt{f}}{\sqrt{d}}+B\right) \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{f} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\left(B-\frac{A \sqrt{f}}{\sqrt{d}}\right) \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{f} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}","\frac{\left(\frac{A \sqrt{f}}{\sqrt{d}}+B\right) \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{f} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\left(B-\frac{A \sqrt{f}}{\sqrt{d}}\right) \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{f} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}",1,"-((B - (A*Sqrt[f])/Sqrt[d])*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[f]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ((B + (A*Sqrt[f])/Sqrt[d])*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[f]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])","A",5,3,30,0.1000,1,"{1033, 724, 206}"
8,1,380,0,0.7980828,"\int \frac{A+B x}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Int[(A + B*x)/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","-\frac{2 \left(c x \left(-2 A c (a f+c d)+b B (c d-a f)+A b^2 f\right)-A b c (3 a f+c d)+a B \left(2 a c f+b^2 (-f)+2 c^2 d\right)+A b^3 f\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}-\frac{\sqrt{f} \left(B \sqrt{d}-A \sqrt{f}\right) \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{\sqrt{f} \left(A \sqrt{f}+B \sqrt{d}\right) \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}","-\frac{2 \left(c x \left(-2 A c (a f+c d)+b B (c d-a f)+A b^2 f\right)+A \left(b^3 f-b c (3 a f+c d)\right)+a B \left(2 a c f+b^2 (-f)+2 c^2 d\right)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}-\frac{\sqrt{f} \left(B \sqrt{d}-A \sqrt{f}\right) \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{\sqrt{f} \left(A \sqrt{f}+B \sqrt{d}\right) \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}",1,"(-2*(A*b^3*f - A*b*c*(c*d + 3*a*f) + a*B*(2*c^2*d - b^2*f + 2*a*c*f) + c*(A*b^2*f + b*B*(c*d - a*f) - 2*A*c*(c*d + a*f))*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) - ((B*Sqrt[d] - A*Sqrt[f])*Sqrt[f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + ((B*Sqrt[d] + A*Sqrt[f])*Sqrt[f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))","A",6,4,30,0.1333,1,"{1018, 1033, 724, 206}"
9,1,796,0,1.8700755,"\int \frac{A+B x}{\left(a+b x+c x^2\right)^{5/2} \left(d-f x^2\right)} \, dx","Int[(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^2)),x]","-\frac{\left(B \sqrt{d}-A \sqrt{f}\right) \tanh ^{-1}\left(\frac{-2 \sqrt{f} a+\left(2 c \sqrt{d}-b \sqrt{f}\right) x+b \sqrt{d}}{2 \sqrt{-\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right) f^{3/2}}{2 \sqrt{d} \left(-\sqrt{d} \sqrt{f} b+c d+a f\right)^{5/2}}+\frac{\left(\sqrt{f} A+B \sqrt{d}\right) \tanh ^{-1}\left(\frac{2 \sqrt{f} a+\left(\sqrt{f} b+2 c \sqrt{d}\right) x+b \sqrt{d}}{2 \sqrt{\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right) f^{3/2}}{2 \sqrt{d} \left(\sqrt{d} \sqrt{f} b+c d+a f\right)^{5/2}}-\frac{2 \left(3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left(7 c^2 d^2+14 a c f d-3 a^2 f^2\right) b^4+A c f \left(15 c^2 d^2+46 a c f d+43 a^2 f^2\right) b^3+2 B c \left(2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right) b^2-4 A c^2 \left(2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right) b+24 a^2 B c^2 f (c d+a f)^2+c \left(3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left(17 c^2 d^2+10 a c f d-3 a^2 f^2\right) b^3+2 A c f \left(15 c^2 d^2+22 a c f d+19 a^2 f^2\right) b^2+4 B c \left(2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right) x\right)}{3 \left(b^2-4 a c\right)^2 \left(c^2 d^2+2 a c f d-f \left(b^2 d-a^2 f\right)\right)^2 \sqrt{c x^2+b x+a}}-\frac{2 \left(A f b^3-A c (c d+3 a f) b+a B \left(-f b^2+2 c^2 d+2 a c f\right)+c \left(A f b^2+B (c d-a f) b-2 A c (c d+a f)\right) x\right)}{3 \left(b^2-4 a c\right) \left(b^2 d f-(c d+a f)^2\right) \left(c x^2+b x+a\right)^{3/2}}","-\frac{\left(B \sqrt{d}-A \sqrt{f}\right) \tanh ^{-1}\left(\frac{-2 \sqrt{f} a+\left(2 c \sqrt{d}-b \sqrt{f}\right) x+b \sqrt{d}}{2 \sqrt{-\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right) f^{3/2}}{2 \sqrt{d} \left(-\sqrt{d} \sqrt{f} b+c d+a f\right)^{5/2}}+\frac{\left(\sqrt{f} A+B \sqrt{d}\right) \tanh ^{-1}\left(\frac{2 \sqrt{f} a+\left(\sqrt{f} b+2 c \sqrt{d}\right) x+b \sqrt{d}}{2 \sqrt{\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right) f^{3/2}}{2 \sqrt{d} \left(\sqrt{d} \sqrt{f} b+c d+a f\right)^{5/2}}-\frac{2 \left(3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left(7 c^2 d^2+14 a c f d-3 a^2 f^2\right) b^4+A c f \left(15 c^2 d^2+46 a c f d+43 a^2 f^2\right) b^3+2 B c \left(2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right) b^2-4 A c^2 \left(2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right) b+24 a^2 B c^2 f (c d+a f)^2+c \left(3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left(17 c^2 d^2+10 a c f d-3 a^2 f^2\right) b^3+2 A c f \left(15 c^2 d^2+22 a c f d+19 a^2 f^2\right) b^2+4 B c \left(2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right) x\right)}{3 \left(b^2-4 a c\right)^2 \left(c^2 d^2+2 a c f d-f \left(b^2 d-a^2 f\right)\right)^2 \sqrt{c x^2+b x+a}}-\frac{2 \left(a B \left(-f b^2+2 c^2 d+2 a c f\right)+A \left(b^3 f-b c (c d+3 a f)\right)+c \left(A f b^2+B (c d-a f) b-2 A c (c d+a f)\right) x\right)}{3 \left(b^2-4 a c\right) \left(b^2 d f-(c d+a f)^2\right) \left(c x^2+b x+a\right)^{3/2}}",1,"(-2*(A*b^3*f - A*b*c*(c*d + 3*a*f) + a*B*(2*c^2*d - b^2*f + 2*a*c*f) + c*(A*b^2*f + b*B*(c*d - a*f) - 2*A*c*(c*d + a*f))*x))/(3*(b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*(a + b*x + c*x^2)^(3/2)) - (2*(3*b^6*B*d*f^2 + 24*a^2*B*c^2*f*(c*d + a*f)^2 - A*b^5*f^2*(7*c*d + 6*a*f) - b^4*B*f*(7*c^2*d^2 + 14*a*c*d*f - 3*a^2*f^2) + A*b^3*c*f*(15*c^2*d^2 + 46*a*c*d*f + 43*a^2*f^2) + 2*b^2*B*c*(2*c^3*d^3 + 5*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 11*a^3*f^3) - 4*A*b*c^2*(2*c^3*d^3 + 9*a*c^2*d^2*f + 24*a^2*c*d*f^2 + 17*a^3*f^3) + c*(3*b^5*B*d*f^2 - 2*A*b^4*f^2*(4*c*d + 3*a*f) - 8*A*c^2*(c*d + a*f)^2*(2*c*d + 5*a*f) - b^3*B*f*(17*c^2*d^2 + 10*a*c*d*f - 3*a^2*f^2) + 2*A*b^2*c*f*(15*c^2*d^2 + 22*a*c*d*f + 19*a^2*f^2) + 4*b*B*c*(2*c^3*d^3 + 11*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 5*a^3*f^3))*x))/(3*(b^2 - 4*a*c)^2*(c^2*d^2 + 2*a*c*d*f - f*(b^2*d - a^2*f))^2*Sqrt[a + b*x + c*x^2]) - ((B*Sqrt[d] - A*Sqrt[f])*f^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(5/2)) + ((B*Sqrt[d] + A*Sqrt[f])*f^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(5/2))","A",7,5,30,0.1667,1,"{1018, 1064, 1033, 724, 206}"
10,1,47,0,0.0342737,"\int \frac{1+2 x}{\left(-1+x^2\right) \sqrt{-1+x+x^2}} \, dx","Int[(1 + 2*x)/((-1 + x^2)*Sqrt[-1 + x + x^2]),x]","\frac{3}{2} \tanh ^{-1}\left(\frac{1-3 x}{2 \sqrt{x^2+x-1}}\right)-\frac{1}{2} \tan ^{-1}\left(\frac{x+3}{2 \sqrt{x^2+x-1}}\right)","\frac{3}{2} \tanh ^{-1}\left(\frac{1-3 x}{2 \sqrt{x^2+x-1}}\right)-\frac{1}{2} \tan ^{-1}\left(\frac{x+3}{2 \sqrt{x^2+x-1}}\right)",1,"-ArcTan[(3 + x)/(2*Sqrt[-1 + x + x^2])]/2 + (3*ArcTanh[(1 - 3*x)/(2*Sqrt[-1 + x + x^2])])/2","A",5,4,23,0.1739,1,"{1033, 724, 206, 204}"
11,1,117,0,0.1688858,"\int \frac{1+2 x}{\left(1+x^2\right) \sqrt{-1+x+x^2}} \, dx","Int[(1 + 2*x)/((1 + x^2)*Sqrt[-1 + x + x^2]),x]","\sqrt{\frac{1}{2} \left(\sqrt{5}-2\right)} \tanh ^{-1}\left(\frac{\sqrt{5} x-2 \sqrt{5}+5}{\sqrt{10 \left(\sqrt{5}-2\right)} \sqrt{x^2+x-1}}\right)-\sqrt{\frac{1}{2} \left(2+\sqrt{5}\right)} \tan ^{-1}\left(\frac{-\sqrt{5} x+2 \sqrt{5}+5}{\sqrt{10 \left(2+\sqrt{5}\right)} \sqrt{x^2+x-1}}\right)","\sqrt{\frac{1}{2} \left(\sqrt{5}-2\right)} \tanh ^{-1}\left(\frac{\sqrt{5} x-2 \sqrt{5}+5}{\sqrt{10 \left(\sqrt{5}-2\right)} \sqrt{x^2+x-1}}\right)-\sqrt{\frac{1}{2} \left(2+\sqrt{5}\right)} \tan ^{-1}\left(\frac{-\sqrt{5} x+2 \sqrt{5}+5}{\sqrt{10 \left(2+\sqrt{5}\right)} \sqrt{x^2+x-1}}\right)",1,"-(Sqrt[(2 + Sqrt[5])/2]*ArcTan[(5 + 2*Sqrt[5] - Sqrt[5]*x)/(Sqrt[10*(2 + Sqrt[5])]*Sqrt[-1 + x + x^2])]) + Sqrt[(-2 + Sqrt[5])/2]*ArcTanh[(5 - 2*Sqrt[5] + Sqrt[5]*x)/(Sqrt[10*(-2 + Sqrt[5])]*Sqrt[-1 + x + x^2])]","A",5,4,23,0.1739,1,"{1036, 1030, 207, 203}"
12,1,484,0,23.5809919,"\int \frac{a-c+b x}{\left(1+x^2\right) \sqrt{a+b x+c x^2}} \, dx","Int[(a - c + b*x)/((1 + x^2)*Sqrt[a + b*x + c*x^2]),x]","-\frac{\sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \tan ^{-1}\left(\frac{b \sqrt{a^2-2 a c+b^2+c^2}-x \left((a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2\right)}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \sqrt{a+b x+c x^2}}\right)}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \tanh ^{-1}\left(\frac{x \left((a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2\right)+b \sqrt{a^2-2 a c+b^2+c^2}}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \sqrt{a+b x+c x^2}}\right)}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2}}","-\frac{\sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \tan ^{-1}\left(\frac{b \sqrt{a^2-2 a c+b^2+c^2}-x \left((a-c) \left(\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2\right)}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(2 c-\sqrt{a^2-2 a c+b^2+c^2}\right)+c \left(c-\sqrt{a^2-2 a c+b^2+c^2}\right)+a^2+b^2} \sqrt{a+b x+c x^2}}\right)}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \tanh ^{-1}\left(\frac{x \left((a-c) \left(-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right)+b^2\right)+b \sqrt{a^2-2 a c+b^2+c^2}}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left(\sqrt{a^2-2 a c+b^2+c^2}+2 c\right)+c \left(\sqrt{a^2-2 a c+b^2+c^2}+c\right)+a^2+b^2} \sqrt{a+b x+c x^2}}\right)}{\sqrt{2} \sqrt[4]{a^2-2 a c+b^2+c^2}}",1,"-((Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] - (b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*x)/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4))) - (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] + (b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*x)/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4))","A",5,4,30,0.1333,1,"{1036, 1030, 208, 205}"
13,1,182,0,0.3455928,"\int \frac{(A+B x) \left(a+b x+c x^2\right)}{d+e x+f x^2} \, dx","Int[((A + B*x)*(a + b*x + c*x^2))/(d + e*x + f*x^2),x]","-\frac{\log \left(d+e x+f x^2\right) \left(B f (b e-a f)+A f (c e-b f)-B c \left(e^2-d f\right)\right)}{2 f^3}-\frac{\tanh ^{-1}\left(\frac{e+2 f x}{\sqrt{e^2-4 d f}}\right) \left(A f \left(2 a f^2-b e f-2 c d f+c e^2\right)+B f \left(-a e f-2 b d f+b e^2\right)-B c \left(e^3-3 d e f\right)\right)}{f^3 \sqrt{e^2-4 d f}}-\frac{x (-A c f-b B f+B c e)}{f^2}+\frac{B c x^2}{2 f}","-\frac{\log \left(d+e x+f x^2\right) \left(A f (c e-b f)-B \left(a f^2-b e f-c d f+c e^2\right)\right)}{2 f^3}-\frac{\tanh ^{-1}\left(\frac{e+2 f x}{\sqrt{e^2-4 d f}}\right) \left(A f \left(2 a f^2-b e f-2 c d f+c e^2\right)+B \left(f \left(-a e f-2 b d f+b e^2\right)-c \left(e^3-3 d e f\right)\right)\right)}{f^3 \sqrt{e^2-4 d f}}-\frac{x (-A c f-b B f+B c e)}{f^2}+\frac{B c x^2}{2 f}",1,"-(((B*c*e - b*B*f - A*c*f)*x)/f^2) + (B*c*x^2)/(2*f) - ((B*f*(b*e^2 - 2*b*d*f - a*e*f) - B*c*(e^3 - 3*d*e*f) + A*f*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2))*ArcTanh[(e + 2*f*x)/Sqrt[e^2 - 4*d*f]])/(f^3*Sqrt[e^2 - 4*d*f]) - ((B*f*(b*e - a*f) + A*f*(c*e - b*f) - B*c*(e^2 - d*f))*Log[d + e*x + f*x^2])/(2*f^3)","A",6,5,28,0.1786,1,"{1628, 634, 618, 206, 628}"
14,1,542,0,1.1024969,"\int \frac{(A+B x) \left(a+b x+c x^2\right)^2}{d+e x+f x^2} \, dx","Int[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x + f*x^2),x]","\frac{\log \left(d+e x+f x^2\right) \left(B \left(-f^2 \left(-a^2 f^2+2 a b e f+b^2 \left(-\left(e^2-d f\right)\right)\right)+2 c f \left(a f \left(e^2-d f\right)-b \left(e^3-2 d e f\right)\right)+c^2 \left(d^2 f^2-3 d e^2 f+e^4\right)\right)+A f (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)\right)}{2 f^5}-\frac{\tanh ^{-1}\left(\frac{e+2 f x}{\sqrt{e^2-4 d f}}\right) \left(A f \left(-f^2 \left(-2 a^2 f^2+2 a b e f+b^2 \left(-\left(e^2-2 d f\right)\right)\right)+2 c f \left(a f \left(e^2-2 d f\right)-b \left(e^3-3 d e f\right)\right)+c^2 \left(2 d^2 f^2-4 d e^2 f+e^4\right)\right)-B \left(f^2 \left(a^2 e f^2-2 a b f \left(e^2-2 d f\right)+b^2 \left(e^3-3 d e f\right)\right)+2 c f \left(a e f \left(e^2-3 d f\right)-b \left(2 d^2 f^2-4 d e^2 f+e^4\right)\right)+c^2 \left(5 d^2 e f^2-5 d e^3 f+e^5\right)\right)\right)}{f^5 \sqrt{e^2-4 d f}}-\frac{x^2 \left(A c f (c e-2 b f)-B \left(-2 c f (b e-a f)+b^2 f^2+c^2 \left(e^2-d f\right)\right)\right)}{2 f^3}+\frac{x \left(A f \left(-2 c f (b e-a f)+b^2 f^2+c^2 \left(e^2-d f\right)\right)+B (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)\right)}{f^4}-\frac{c x^3 (-A c f-2 b B f+B c e)}{3 f^2}+\frac{B c^2 x^4}{4 f}","\frac{\log \left(d+e x+f x^2\right) \left(B \left(-f^2 \left(-a^2 f^2+2 a b e f+b^2 \left(-\left(e^2-d f\right)\right)\right)+2 c f \left(a f \left(e^2-d f\right)-b \left(e^3-2 d e f\right)\right)+c^2 \left(d^2 f^2-3 d e^2 f+e^4\right)\right)+A f (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)\right)}{2 f^5}-\frac{\tanh ^{-1}\left(\frac{e+2 f x}{\sqrt{e^2-4 d f}}\right) \left(A f \left(-f^2 \left(-2 a^2 f^2+2 a b e f+b^2 \left(-\left(e^2-2 d f\right)\right)\right)+2 c f \left(a f \left(e^2-2 d f\right)-b \left(e^3-3 d e f\right)\right)+c^2 \left(2 d^2 f^2-4 d e^2 f+e^4\right)\right)-B \left(f^2 \left(a^2 e f^2-2 a b f \left(e^2-2 d f\right)+b^2 \left(e^3-3 d e f\right)\right)+2 c f \left(a e f \left(e^2-3 d f\right)-b \left(2 d^2 f^2-4 d e^2 f+e^4\right)\right)+c^2 \left(5 d^2 e f^2-5 d e^3 f+e^5\right)\right)\right)}{f^5 \sqrt{e^2-4 d f}}-\frac{x^2 \left(A c f (c e-2 b f)-B \left(-2 c f (b e-a f)+b^2 f^2+c^2 \left(e^2-d f\right)\right)\right)}{2 f^3}+\frac{x \left(A f \left(-2 c f (b e-a f)+b^2 f^2+c^2 \left(e^2-d f\right)\right)+B (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)\right)}{f^4}-\frac{c x^3 (-A c f-2 b B f+B c e)}{3 f^2}+\frac{B c^2 x^4}{4 f}",1,"((B*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) + A*f*(b^2*f^2 - 2*c*f*(b*e - a*f) + c^2*(e^2 - d*f)))*x)/f^4 - ((A*c*f*(c*e - 2*b*f) - B*(b^2*f^2 - 2*c*f*(b*e - a*f) + c^2*(e^2 - d*f)))*x^2)/(2*f^3) - (c*(B*c*e - 2*b*B*f - A*c*f)*x^3)/(3*f^2) + (B*c^2*x^4)/(4*f) - ((A*f*(c^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2) - f^2*(2*a*b*e*f - 2*a^2*f^2 - b^2*(e^2 - 2*d*f)) + 2*c*f*(a*f*(e^2 - 2*d*f) - b*(e^3 - 3*d*e*f))) - B*(c^2*(e^5 - 5*d*e^3*f + 5*d^2*e*f^2) + f^2*(a^2*e*f^2 - 2*a*b*f*(e^2 - 2*d*f) + b^2*(e^3 - 3*d*e*f)) + 2*c*f*(a*e*f*(e^2 - 3*d*f) - b*(e^4 - 4*d*e^2*f + 2*d^2*f^2))))*ArcTanh[(e + 2*f*x)/Sqrt[e^2 - 4*d*f]])/(f^5*Sqrt[e^2 - 4*d*f]) + ((A*f*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) + B*(c^2*(e^4 - 3*d*e^2*f + d^2*f^2) - f^2*(2*a*b*e*f - a^2*f^2 - b^2*(e^2 - d*f)) + 2*c*f*(a*f*(e^2 - d*f) - b*(e^3 - 2*d*e*f))))*Log[d + e*x + f*x^2])/(2*f^5)","A",6,5,30,0.1667,1,"{1011, 634, 618, 206, 628}"
15,1,398,0,0.4775532,"\int \frac{A+B x}{\left(a+b x+c x^2\right) \left(d+e x+f x^2\right)} \, dx","Int[(A + B*x)/((a + b*x + c*x^2)*(d + e*x + f*x^2)),x]","\frac{\tanh ^{-1}\left(\frac{e+2 f x}{\sqrt{e^2-4 d f}}\right) \left(B (a e f-2 b d f+c d e)-A \left(2 a f^2-b e f-2 c d f+c e^2\right)\right)}{\sqrt{e^2-4 d f} \left(f \left(a^2 f-a b e+b^2 d\right)+a c \left(e^2-2 d f\right)-b c d e+c^2 d^2\right)}+\frac{\log \left(a+b x+c x^2\right) (-a B f+A b f-A c e+B c d)}{2 \left(f \left(a^2 f-a b e+b^2 d\right)+a c \left(e^2-2 d f\right)-b c d e+c^2 d^2\right)}-\frac{\log \left(d+e x+f x^2\right) (-a B f+A b f-A c e+B c d)}{2 \left(f \left(a^2 f-a b e+b^2 d\right)+a c \left(e^2-2 d f\right)-b c d e+c^2 d^2\right)}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-b (a B f+A c e+B c d)+2 c (-a A f+a B e+A c d)+A b^2 f\right)}{\sqrt{b^2-4 a c} \left(f \left(a^2 f-a b e+b^2 d\right)+a c \left(e^2-2 d f\right)-b c d e+c^2 d^2\right)}","\frac{\tanh ^{-1}\left(\frac{e+2 f x}{\sqrt{e^2-4 d f}}\right) \left(B (a e f-2 b d f+c d e)-A \left(2 a f^2-b e f-2 c d f+c e^2\right)\right)}{\sqrt{e^2-4 d f} \left(f \left(a^2 f-a b e+b^2 d\right)-c \left(b d e-a \left(e^2-2 d f\right)\right)+c^2 d^2\right)}+\frac{\log \left(a+b x+c x^2\right) (-a B f+A b f-A c e+B c d)}{2 \left(f \left(a^2 f-a b e+b^2 d\right)-c \left(b d e-a \left(e^2-2 d f\right)\right)+c^2 d^2\right)}-\frac{\log \left(d+e x+f x^2\right) (-a B f+A b f-A c e+B c d)}{2 \left(f \left(a^2 f-a b e+b^2 d\right)-c \left(b d e-a \left(e^2-2 d f\right)\right)+c^2 d^2\right)}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-b (a B f+A c e+B c d)+2 c (-a A f+a B e+A c d)+A b^2 f\right)}{\sqrt{b^2-4 a c} \left(f \left(a^2 f-a b e+b^2 d\right)-c \left(b d e-a \left(e^2-2 d f\right)\right)+c^2 d^2\right)}",1,"-(((A*b^2*f + 2*c*(A*c*d + a*B*e - a*A*f) - b*(B*c*d + A*c*e + a*B*f))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f)))) + ((B*(c*d*e - 2*b*d*f + a*e*f) - A*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2))*ArcTanh[(e + 2*f*x)/Sqrt[e^2 - 4*d*f]])/(Sqrt[e^2 - 4*d*f]*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))) + ((B*c*d - A*c*e + A*b*f - a*B*f)*Log[a + b*x + c*x^2])/(2*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))) - ((B*c*d - A*c*e + A*b*f - a*B*f)*Log[d + e*x + f*x^2])/(2*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f)))","A",9,5,30,0.1667,1,"{1022, 634, 618, 206, 628}"
16,1,1067,0,4.1768714,"\int \frac{A+B x}{\left(a+b x+c x^2\right)^2 \left(d+e x+f x^2\right)} \, dx","Int[(A + B*x)/((a + b*x + c*x^2)^2*(d + e*x + f*x^2)),x]","-\frac{A c (2 a c e-b (c d+a f))+(A b-a B) \left(f b^2+2 c^2 d-c (b e+2 a f)\right)+c \left(A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right) x}{\left(b^2-4 a c\right) \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \left(c x^2+b x+a\right)}-\frac{\left((B d-A e) f^2 b^5-2 f \left(-a A f^2+B c d e-A c \left(e^2-d f\right)\right) b^4-\left(A c e \left(c e^2-4 a f^2-2 c d f\right)+B \left(a^2 f^3+4 a c d f^2-c^2 d \left(e^2+5 d f\right)\right)\right) b^3-4 \left(B d^2 e c^3+A f \left(2 c^2 d^2+3 a^2 f^2+3 a c \left(e^2-d f\right)\right) c\right) b^2+2 c \left(B \left(c^3 d^3+a c^2 \left(e^2-7 d f\right) d+3 a^3 f^3+3 a^2 c f \left(e^2+d f\right)\right)+A c e \left(3 c^2 d^2+3 a^2 f^2+a c \left(3 e^2+2 d f\right)\right)\right) b-4 c^2 \left(A \left(c^3 d^3+a c^2 \left(3 e^2-5 d f\right) d-3 a^3 f^3-a^2 c f \left(e^2-7 d f\right)\right)-a B e \left(c^2 d^2-3 a^2 f^2-a c \left(e^2-2 d f\right)\right)\right)\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2} \left(c^2 d^2-b c e d+f \left(f a^2-b e a+b^2 d\right)+a c \left(e^2-2 d f\right)\right)^2}+\frac{\left(B \left(d e \left(e^2-3 d f\right) c^2-2 d f \left(b e^2-a f e-2 b d f\right) c+f^2 \left(e f a^2-4 b d f a+b^2 d e\right)\right)-A \left(\left(e^4-4 d f e^2+2 d^2 f^2\right) c^2+2 f \left(a f \left(e^2-2 d f\right)-b \left(e^3-3 d e f\right)\right) c-f^2 \left(-\left(e^2-2 d f\right) b^2+2 a e f b-2 a^2 f^2\right)\right)\right) \tanh ^{-1}\left(\frac{e+2 f x}{\sqrt{e^2-4 d f}}\right)}{\sqrt{e^2-4 d f} \left(c^2 d^2-b c e d+f \left(f a^2-b e a+b^2 d\right)+a c \left(e^2-2 d f\right)\right)^2}+\frac{\left(A (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)-B \left(-d \left(e^2-d f\right) c^2+2 d f (b e-a f) c-f^2 \left(b^2 d-a^2 f\right)\right)\right) \log \left(c x^2+b x+a\right)}{2 \left(c^2 d^2-b c e d+f \left(f a^2-b e a+b^2 d\right)+a c \left(e^2-2 d f\right)\right)^2}-\frac{\left(A (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)-B \left(-d \left(e^2-d f\right) c^2+2 d f (b e-a f) c-f^2 \left(b^2 d-a^2 f\right)\right)\right) \log \left(f x^2+e x+d\right)}{2 \left(c^2 d^2-b c e d+f \left(f a^2-b e a+b^2 d\right)+a c \left(e^2-2 d f\right)\right)^2}","-\frac{A c (2 a c e-b (c d+a f))+(A b-a B) \left(f b^2+2 c^2 d-c (b e+2 a f)\right)+c \left(A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right) x}{\left(b^2-4 a c\right) \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \left(c x^2+b x+a\right)}-\frac{\left((B d-A e) f^2 b^5-2 f \left(B c d e-A \left(c e^2+a f^2-c d f\right)\right) b^4-\left(A c e \left(c e^2-4 a f^2-2 c d f\right)+B \left(a^2 f^3+4 a c d f^2-c^2 d \left(e^2+5 d f\right)\right)\right) b^3-4 c \left(B c^2 e d^2+A f \left(2 c^2 d^2+3 a^2 f^2+3 a c \left(e^2-d f\right)\right)\right) b^2+2 c \left(B \left(c^3 d^3+a c^2 \left(e^2-7 d f\right) d+3 a^3 f^3+3 a^2 c f \left(e^2+d f\right)\right)+A c e \left(3 c^2 d^2+3 a^2 f^2+a c \left(3 e^2+2 d f\right)\right)\right) b-4 c^2 \left(A \left(c^3 d^3+a c^2 \left(3 e^2-5 d f\right) d-3 a^3 f^3-a^2 c f \left(e^2-7 d f\right)\right)-a B e \left(c^2 d^2-3 a^2 f^2-a c \left(e^2-2 d f\right)\right)\right)\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2} \left(c^2 d^2+f \left(f a^2-b e a+b^2 d\right)-c \left(b d e-a \left(e^2-2 d f\right)\right)\right)^2}+\frac{\left(B \left(d e \left(e^2-3 d f\right) c^2-2 d f \left(b e^2-a f e-2 b d f\right) c+f^2 \left(e f a^2-4 b d f a+b^2 d e\right)\right)-A \left(\left(e^4-4 d f e^2+2 d^2 f^2\right) c^2+2 f \left(a f \left(e^2-2 d f\right)-b \left(e^3-3 d e f\right)\right) c-f^2 \left(-\left(e^2-2 d f\right) b^2+2 a e f b-2 a^2 f^2\right)\right)\right) \tanh ^{-1}\left(\frac{e+2 f x}{\sqrt{e^2-4 d f}}\right)}{\sqrt{e^2-4 d f} \left(c^2 d^2+f \left(f a^2-b e a+b^2 d\right)-c \left(b d e-a \left(e^2-2 d f\right)\right)\right)^2}+\frac{\left(A (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)-B \left(-d \left(e^2-d f\right) c^2+2 d f (b e-a f) c-f^2 \left(b^2 d-a^2 f\right)\right)\right) \log \left(c x^2+b x+a\right)}{2 \left(c^2 d^2+f \left(f a^2-b e a+b^2 d\right)-c \left(b d e-a \left(e^2-2 d f\right)\right)\right)^2}-\frac{\left(A (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)-B \left(-d \left(e^2-d f\right) c^2+2 d f (b e-a f) c-f^2 \left(b^2 d-a^2 f\right)\right)\right) \log \left(f x^2+e x+d\right)}{2 \left(c^2 d^2+f \left(f a^2-b e a+b^2 d\right)-c \left(b d e-a \left(e^2-2 d f\right)\right)\right)^2}",1,"-((A*c*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*b^2*f + 2*c*(A*c*d + a*B*e - a*A*f) - b*(B*c*d + A*c*e + a*B*f))*x)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(a + b*x + c*x^2))) - ((b^5*(B*d - A*e)*f^2 - 2*b^4*f*(B*c*d*e - a*A*f^2 - A*c*(e^2 - d*f)) - 4*c^2*(A*(c^3*d^3 - 3*a^3*f^3 - a^2*c*f*(e^2 - 7*d*f) + a*c^2*d*(3*e^2 - 5*d*f)) - a*B*e*(c^2*d^2 - 3*a^2*f^2 - a*c*(e^2 - 2*d*f))) - 4*b^2*(B*c^3*d^2*e + A*c*f*(2*c^2*d^2 + 3*a^2*f^2 + 3*a*c*(e^2 - d*f))) + 2*b*c*(B*(c^3*d^3 + 3*a^3*f^3 + a*c^2*d*(e^2 - 7*d*f) + 3*a^2*c*f*(e^2 + d*f)) + A*c*e*(3*c^2*d^2 + 3*a^2*f^2 + a*c*(3*e^2 + 2*d*f))) - b^3*(A*c*e*(c*e^2 - 2*c*d*f - 4*a*f^2) + B*(4*a*c*d*f^2 + a^2*f^3 - c^2*d*(e^2 + 5*d*f))))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(3/2)*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))^2) + ((B*(c^2*d*e*(e^2 - 3*d*f) - 2*c*d*f*(b*e^2 - 2*b*d*f - a*e*f) + f^2*(b^2*d*e - 4*a*b*d*f + a^2*e*f)) - A*(c^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2) - f^2*(2*a*b*e*f - 2*a^2*f^2 - b^2*(e^2 - 2*d*f)) + 2*c*f*(a*f*(e^2 - 2*d*f) - b*(e^3 - 3*d*e*f))))*ArcTanh[(e + 2*f*x)/Sqrt[e^2 - 4*d*f]])/(Sqrt[e^2 - 4*d*f]*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))^2) + ((A*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) - B*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f)))*Log[a + b*x + c*x^2])/(2*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))^2) - ((A*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) - B*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f)))*Log[d + e*x + f*x^2])/(2*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))^2)","A",10,6,30,0.2000,1,"{1016, 1072, 634, 618, 206, 628}"
17,1,140,0,0.1314804,"\int \frac{g+h x}{\left(a+b x+c x^2\right) \left(a d+b d x+c d x^2\right)^2} \, dx","Int[(g + h*x)/((a + b*x + c*x^2)*(a*d + b*d*x + c*d*x^2)^2),x]","\frac{3 (b+2 c x) (2 c g-b h)}{2 d^2 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{-2 a h+x (2 c g-b h)+b g}{2 d^2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}-\frac{6 c (2 c g-b h) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{d^2 \left(b^2-4 a c\right)^{5/2}}","\frac{3 (b+2 c x) (2 c g-b h)}{2 d^2 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{-2 a h+x (2 c g-b h)+b g}{2 d^2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}-\frac{6 c (2 c g-b h) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{d^2 \left(b^2-4 a c\right)^{5/2}}",1,"-(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*(b^2 - 4*a*c)*d^2*(a + b*x + c*x^2)^2) + (3*(2*c*g - b*h)*(b + 2*c*x))/(2*(b^2 - 4*a*c)^2*d^2*(a + b*x + c*x^2)) - (6*c*(2*c*g - b*h)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*d^2)","A",5,5,34,0.1471,1,"{998, 638, 614, 618, 206}"
18,1,140,0,0.103563,"\int \frac{g+h x}{\left(a+b x+c x^2\right)^2 \left(a d+b d x+c d x^2\right)} \, dx","Int[(g + h*x)/((a + b*x + c*x^2)^2*(a*d + b*d*x + c*d*x^2)),x]","\frac{3 (b+2 c x) (2 c g-b h)}{2 d \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{-2 a h+x (2 c g-b h)+b g}{2 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}-\frac{6 c (2 c g-b h) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{d \left(b^2-4 a c\right)^{5/2}}","\frac{3 (b+2 c x) (2 c g-b h)}{2 d \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{-2 a h+x (2 c g-b h)+b g}{2 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}-\frac{6 c (2 c g-b h) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{d \left(b^2-4 a c\right)^{5/2}}",1,"-(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*(b^2 - 4*a*c)*d*(a + b*x + c*x^2)^2) + (3*(2*c*g - b*h)*(b + 2*c*x))/(2*(b^2 - 4*a*c)^2*d*(a + b*x + c*x^2)) - (6*c*(2*c*g - b*h)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*d)","A",5,5,34,0.1471,1,"{998, 638, 614, 618, 206}"
19,1,615,0,8.9975622,"\int \frac{(A+B x) \sqrt{a+b x+c x^2}}{d+e x+f x^2} \, dx","Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/(d + e*x + f*x^2),x]","\frac{\left(2 f (A f (c d-a f)-B d (c e-b f))-\left(e-\sqrt{e^2-4 d f}\right) \left(B f (b e-a f)+A f (c e-b f)-B c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left(2 f (A f (c d-a f)-B d (c e-b f))-\left(\sqrt{e^2-4 d f}+e\right) \left(B f (b e-a f)+A f (c e-b f)-B c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) (-2 A c f-b B f+2 B c e)}{2 \sqrt{c} f^2}+\frac{B \sqrt{a+b x+c x^2}}{f}","\frac{\left(2 f (A f (c d-a f)-B d (c e-b f))-\left(e-\sqrt{e^2-4 d f}\right) \left(B \left(f (b e-a f)-c \left(e^2-d f\right)\right)+A f (c e-b f)\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left(2 f (A f (c d-a f)-B d (c e-b f))-\left(\sqrt{e^2-4 d f}+e\right) \left(B \left(f (b e-a f)-c \left(e^2-d f\right)\right)+A f (c e-b f)\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) (-2 A c f-b B f+2 B c e)}{2 \sqrt{c} f^2}+\frac{B \sqrt{a+b x+c x^2}}{f}",1,"(B*Sqrt[a + b*x + c*x^2])/f - ((2*B*c*e - b*B*f - 2*A*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f^2) + ((2*f*(A*f*(c*d - a*f) - B*d*(c*e - b*f)) - (e - Sqrt[e^2 - 4*d*f])*(B*f*(b*e - a*f) + A*f*(c*e - b*f) - B*c*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - ((2*f*(A*f*(c*d - a*f) - B*d*(c*e - b*f)) - (e + Sqrt[e^2 - 4*d*f])*(B*f*(b*e - a*f) + A*f*(c*e - b*f) - B*c*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",9,6,32,0.1875,1,"{1019, 1076, 621, 206, 1032, 724}"
20,1,1092,0,18.8666442,"\int \frac{(A+B x) \left(a+b x+c x^2\right)^{3/2}}{d+e x+f x^2} \, dx","Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x + f*x^2),x]","\frac{B \left(c x^2+b x+a\right)^{3/2}}{3 f}-\frac{\left(2 A c f (4 c e-5 b f)-B \left(8 \left(e^2-d f\right) c^2-2 f (5 b e-4 a f) c+b^2 f^2\right)+2 c f (2 B c e-b B f-2 A c f) x\right) \sqrt{c x^2+b x+a}}{8 c f^3}+\frac{\left(2 A c f \left(8 \left(e^2-d f\right) c^2-12 f (b e-a f) c+3 b^2 f^2\right)-B \left(16 \left(e^3-2 d e f\right) c^3-24 f \left(b e^2-a f e-b d f\right) c^2+6 b f^2 (b e-2 a f) c+b^3 f^3\right)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{16 c^{3/2} f^4}-\frac{\left(2 c f \left(B d (c e-b f) \left(c e^2-b f e+2 a f^2-2 c d f\right)+A f \left(-d \left(e^2-d f\right) c^2+2 d f (b e-a f) c-f^2 \left(b^2 d-a^2 f\right)\right)\right)-c \left(e-\sqrt{e^2-4 d f}\right) \left(A f (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)+B \left(\left(e^4-3 d f e^2+d^2 f^2\right) c^2+2 f \left(a f \left(e^2-d f\right)-b \left(e^3-2 d e f\right)\right) c-f^2 \left(-\left(e^2-d f\right) b^2+2 a e f b-a^2 f^2\right)\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e-\sqrt{e^2-4 d f}\right)+2 \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right)}{\sqrt{2} c f^4 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt{e^2-4 d f}}}+\frac{\left(2 f \left(B d (c e-b f) \left(c e^2-b f e+2 a f^2-2 c d f\right)+A f \left(-d \left(e^2-d f\right) c^2+2 d f (b e-a f) c-f^2 \left(b^2 d-a^2 f\right)\right)\right)-\left(e+\sqrt{e^2-4 d f}\right) \left(A f (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)+B \left(\left(e^4-3 d f e^2+d^2 f^2\right) c^2+2 f \left(a f \left(e^2-d f\right)-b \left(e^3-2 d e f\right)\right) c-f^2 \left(-\left(e^2-d f\right) b^2+2 a e f b-a^2 f^2\right)\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e+\sqrt{e^2-4 d f}\right)+2 \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right)}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt{e^2-4 d f}}}","\frac{B \left(c x^2+b x+a\right)^{3/2}}{3 f}-\frac{\left(2 A c f (4 c e-5 b f)-B \left(8 \left(e^2-d f\right) c^2-2 f (5 b e-4 a f) c+b^2 f^2\right)+2 c f (2 B c e-b B f-2 A c f) x\right) \sqrt{c x^2+b x+a}}{8 c f^3}+\frac{\left(2 A c f \left(8 \left(e^2-d f\right) c^2-12 f (b e-a f) c+3 b^2 f^2\right)-B \left(16 \left(e^3-2 d e f\right) c^3-24 f \left(b e^2-a f e-b d f\right) c^2+6 b f^2 (b e-2 a f) c+b^3 f^3\right)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{16 c^{3/2} f^4}-\frac{\left(2 c f \left(B d (c e-b f) \left(c e^2-b f e+2 a f^2-2 c d f\right)+A f \left(-d \left(e^2-d f\right) c^2+2 d f (b e-a f) c-f^2 \left(b^2 d-a^2 f\right)\right)\right)-c \left(e-\sqrt{e^2-4 d f}\right) \left(A f (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)+B \left(\left(e^4-3 d f e^2+d^2 f^2\right) c^2+2 f \left(a f \left(e^2-d f\right)-b \left(e^3-2 d e f\right)\right) c-f^2 \left(-\left(e^2-d f\right) b^2+2 a e f b-a^2 f^2\right)\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e-\sqrt{e^2-4 d f}\right)+2 \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right)}{\sqrt{2} c f^4 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt{e^2-4 d f}}}+\frac{\left(2 f \left(B d (c e-b f) \left(c e^2-b f e+2 a f^2-2 c d f\right)+A f \left(-d \left(e^2-d f\right) c^2+2 d f (b e-a f) c-f^2 \left(b^2 d-a^2 f\right)\right)\right)-\left(e+\sqrt{e^2-4 d f}\right) \left(A f (c e-b f) \left(f (b e-2 a f)-c \left(e^2-2 d f\right)\right)+B \left(\left(e^4-3 d f e^2+d^2 f^2\right) c^2+2 f \left(a f \left(e^2-d f\right)-b \left(e^3-2 d e f\right)\right) c-f^2 \left(-\left(e^2-d f\right) b^2+2 a e f b-a^2 f^2\right)\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e+\sqrt{e^2-4 d f}\right)+2 \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right)}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt{e^2-4 d f}}}",1,"-((2*A*c*f*(4*c*e - 5*b*f) - B*(b^2*f^2 - 2*c*f*(5*b*e - 4*a*f) + 8*c^2*(e^2 - d*f)) + 2*c*f*(2*B*c*e - b*B*f - 2*A*c*f)*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^3) + (B*(a + b*x + c*x^2)^(3/2))/(3*f) + ((2*A*c*f*(3*b^2*f^2 - 12*c*f*(b*e - a*f) + 8*c^2*(e^2 - d*f)) - B*(b^3*f^3 + 6*b*c*f^2*(b*e - 2*a*f) - 24*c^2*f*(b*e^2 - b*d*f - a*e*f) + 16*c^3*(e^3 - 2*d*e*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*f^4) - ((2*c*f*(B*d*(c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2) + A*f*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f))) - c*(e - Sqrt[e^2 - 4*d*f])*(A*f*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) + B*(c^2*(e^4 - 3*d*e^2*f + d^2*f^2) - f^2*(2*a*b*e*f - a^2*f^2 - b^2*(e^2 - d*f)) + 2*c*f*(a*f*(e^2 - d*f) - b*(e^3 - 2*d*e*f)))))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*c*f^4*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) + ((2*f*(B*d*(c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2) + A*f*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f))) - (e + Sqrt[e^2 - 4*d*f])*(A*f*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) + B*(c^2*(e^4 - 3*d*e^2*f + d^2*f^2) - f^2*(2*a*b*e*f - a^2*f^2 - b^2*(e^2 - d*f)) + 2*c*f*(a*f*(e^2 - d*f) - b*(e^3 - 2*d*e*f)))))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^4*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",10,7,32,0.2188,1,"{1019, 1066, 1076, 621, 206, 1032, 724}"
21,1,416,0,2.7021095,"\int \frac{A+B x}{\left(a+b x+c x^2\right) \sqrt{d+e x+f x^2}} \, dx","Int[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]),x]","\frac{\left(-B \sqrt{b^2-4 a c}-2 A c+b B\right) \tanh ^{-1}\left(\frac{2 x \left(c e-f \left(b-\sqrt{b^2-4 a c}\right)\right)-e \left(b-\sqrt{b^2-4 a c}\right)+4 c d}{2 \sqrt{2} \sqrt{d+e x+f x^2} \sqrt{\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}+\frac{\left(2 A c-B \left(\sqrt{b^2-4 a c}+b\right)\right) \tanh ^{-1}\left(\frac{2 x \left(c e-f \left(\sqrt{b^2-4 a c}+b\right)\right)-e \left(\sqrt{b^2-4 a c}+b\right)+4 c d}{2 \sqrt{2} \sqrt{d+e x+f x^2} \sqrt{-\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}","\frac{\left(-B \sqrt{b^2-4 a c}-2 A c+b B\right) \tanh ^{-1}\left(\frac{2 x \left(c e-f \left(b-\sqrt{b^2-4 a c}\right)\right)-e \left(b-\sqrt{b^2-4 a c}\right)+4 c d}{2 \sqrt{2} \sqrt{d+e x+f x^2} \sqrt{\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}+\frac{\left(2 A c-B \left(\sqrt{b^2-4 a c}+b\right)\right) \tanh ^{-1}\left(\frac{2 x \left(c e-f \left(\sqrt{b^2-4 a c}+b\right)\right)-e \left(\sqrt{b^2-4 a c}+b\right)+4 c d}{2 \sqrt{2} \sqrt{d+e x+f x^2} \sqrt{-\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}",1,"((b*B - 2*A*c - B*Sqrt[b^2 - 4*a*c])*ArcTanh[(4*c*d - (b - Sqrt[b^2 - 4*a*c])*e + 2*(c*e - (b - Sqrt[b^2 - 4*a*c])*f)*x)/(2*Sqrt[2]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f + Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*Sqrt[d + e*x + f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f + Sqrt[b^2 - 4*a*c]*(c*e - b*f)]) + ((2*A*c - B*(b + Sqrt[b^2 - 4*a*c]))*ArcTanh[(4*c*d - (b + Sqrt[b^2 - 4*a*c])*e + 2*(c*e - (b + Sqrt[b^2 - 4*a*c])*f)*x)/(2*Sqrt[2]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*Sqrt[d + e*x + f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)])","A",5,3,32,0.09375,1,"{1032, 724, 206}"
22,1,780,0,5.1622928,"\int \frac{A+B x}{\left(a+c x^2\right) \sqrt{d+e x+f x^2}} \, dx","Int[(A + B*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]),x]","\frac{\sqrt{A \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e} \sqrt{B \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)-A c e} \tanh ^{-1}\left(\frac{\sqrt{e} \left(a \left(A c e-B \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)\right)-c x \left(A \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e\right)\right)}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{A \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e} \sqrt{B \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)-A c e}}\right)}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}}-\frac{\sqrt{B \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)-A c e} \sqrt{A \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e} \tanh ^{-1}\left(\frac{\sqrt{e} \left(a \left(A c e-B \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)\right)-c x \left(A \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e\right)\right)}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{B \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)-A c e} \sqrt{A \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e}}\right)}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}}","\frac{\sqrt{A \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e} \sqrt{B \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)-A c e} \tanh ^{-1}\left(\frac{\sqrt{e} \left(a \left(A c e-B \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)\right)-c x \left(A \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e\right)\right)}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{A \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e} \sqrt{B \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)-A c e}}\right)}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}}-\frac{\sqrt{B \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)-A c e} \sqrt{A \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e} \tanh ^{-1}\left(\frac{\sqrt{e} \left(a \left(A c e-B \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)\right)-c x \left(A \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e\right)\right)}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{B \left(-\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)-A c e} \sqrt{A \left(\sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}-a f+c d\right)+a B e}}\right)}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2}}",1,"(Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[-(A*c*e) + B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*ArcTanh[(Sqrt[e]*(a*(A*c*e - B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])) - c*(a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[-(A*c*e) + B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[d + e*x + f*x^2])])/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[e]*Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]) - (Sqrt[-(A*c*e) + B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*ArcTanh[(Sqrt[e]*(a*(A*c*e - B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])) - c*(a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[-(A*c*e) + B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[d + e*x + f*x^2])])/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[e]*Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])","A",5,3,29,0.1034,1,"{1036, 1030, 208}"
23,1,302,0,0.8426356,"\int \frac{A+B x}{\left(a+b x+c x^2\right) \sqrt{d+f x^2}} \, dx","Int[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]),x]","\frac{\left(-B \sqrt{b^2-4 a c}-2 A c+b B\right) \tanh ^{-1}\left(\frac{2 c d-f x \left(b-\sqrt{b^2-4 a c}\right)}{\sqrt{2} \sqrt{d+f x^2} \sqrt{b f \left(b-\sqrt{b^2-4 a c}\right)-2 a c f+2 c^2 d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b f \left(b-\sqrt{b^2-4 a c}\right)-2 a c f+2 c^2 d}}+\frac{\left(2 A c-B \left(\sqrt{b^2-4 a c}+b\right)\right) \tanh ^{-1}\left(\frac{2 c d-f x \left(\sqrt{b^2-4 a c}+b\right)}{\sqrt{2} \sqrt{d+f x^2} \sqrt{b f \left(\sqrt{b^2-4 a c}+b\right)-2 a c f+2 c^2 d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b f \left(\sqrt{b^2-4 a c}+b\right)-2 a c f+2 c^2 d}}","\frac{\left(-B \sqrt{b^2-4 a c}-2 A c+b B\right) \tanh ^{-1}\left(\frac{2 c d-f x \left(b-\sqrt{b^2-4 a c}\right)}{\sqrt{2} \sqrt{d+f x^2} \sqrt{b f \left(b-\sqrt{b^2-4 a c}\right)-2 a c f+2 c^2 d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b f \left(b-\sqrt{b^2-4 a c}\right)-2 a c f+2 c^2 d}}+\frac{\left(2 A c-B \left(\sqrt{b^2-4 a c}+b\right)\right) \tanh ^{-1}\left(\frac{2 c d-f x \left(\sqrt{b^2-4 a c}+b\right)}{\sqrt{2} \sqrt{d+f x^2} \sqrt{b f \left(\sqrt{b^2-4 a c}+b\right)-2 a c f+2 c^2 d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b f \left(\sqrt{b^2-4 a c}+b\right)-2 a c f+2 c^2 d}}",1,"((b*B - 2*A*c - B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*d - (b - Sqrt[b^2 - 4*a*c])*f*x)/(Sqrt[2]*Sqrt[2*c^2*d - 2*a*c*f + b*(b - Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - 2*a*c*f + b*(b - Sqrt[b^2 - 4*a*c])*f]) + ((2*A*c - B*(b + Sqrt[b^2 - 4*a*c]))*ArcTanh[(2*c*d - (b + Sqrt[b^2 - 4*a*c])*f*x)/(Sqrt[2]*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f])","A",5,3,29,0.1034,1,"{1034, 725, 206}"
24,1,101,0,0.1270999,"\int \frac{A+B x}{\left(a+c x^2\right) \sqrt{d+f x^2}} \, dx","Int[(A + B*x)/((a + c*x^2)*Sqrt[d + f*x^2]),x]","\frac{A \tan ^{-1}\left(\frac{x \sqrt{c d-a f}}{\sqrt{a} \sqrt{d+f x^2}}\right)}{\sqrt{a} \sqrt{c d-a f}}-\frac{B \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+f x^2}}{\sqrt{c d-a f}}\right)}{\sqrt{c} \sqrt{c d-a f}}","\frac{A \tan ^{-1}\left(\frac{x \sqrt{c d-a f}}{\sqrt{a} \sqrt{d+f x^2}}\right)}{\sqrt{a} \sqrt{c d-a f}}-\frac{B \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+f x^2}}{\sqrt{c d-a f}}\right)}{\sqrt{c} \sqrt{c d-a f}}",1,"(A*ArcTan[(Sqrt[c*d - a*f]*x)/(Sqrt[a]*Sqrt[d + f*x^2])])/(Sqrt[a]*Sqrt[c*d - a*f]) - (B*ArcTanh[(Sqrt[c]*Sqrt[d + f*x^2])/Sqrt[c*d - a*f]])/(Sqrt[c]*Sqrt[c*d - a*f])","A",6,6,26,0.2308,1,"{1010, 377, 205, 444, 63, 208}"
25,1,139,0,0.2200487,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \sqrt{1+3 x-2 x^2}} \, dx","Int[(2 + x)/((2 + 4*x - 3*x^2)*Sqrt[1 + 3*x - 2*x^2]),x]","\frac{1}{2} \sqrt{\sqrt{10}-\frac{13}{5}} \tan ^{-1}\left(\frac{\left(1+4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right)+\frac{1}{2} \sqrt{\frac{13}{5}+\sqrt{10}} \tanh ^{-1}\left(\frac{\left(1-4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right)","\frac{1}{2} \sqrt{\sqrt{10}-\frac{13}{5}} \tan ^{-1}\left(\frac{\left(1+4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right)+\frac{1}{2} \sqrt{\frac{13}{5}+\sqrt{10}} \tanh ^{-1}\left(\frac{\left(1-4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right)",1,"(Sqrt[-13/5 + Sqrt[10]]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/2 + (Sqrt[13/5 + Sqrt[10]]*ArcTanh[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/2","A",5,4,30,0.1333,1,"{1032, 724, 204, 206}"
26,1,166,0,0.2210624,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \left(1+3 x-2 x^2\right)^{3/2}} \, dx","Int[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x - 2*x^2)^(3/2)),x]","-\frac{2 (14 x+15)}{17 \sqrt{-2 x^2+3 x+1}}-\frac{9}{2} \sqrt{\frac{1}{5} \left(\sqrt{10}-3\right)} \tan ^{-1}\left(\frac{\left(1+4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right)+\frac{9}{2} \sqrt{\frac{1}{5} \left(3+\sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(1-4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right)","-\frac{2 (14 x+15)}{17 \sqrt{-2 x^2+3 x+1}}-\frac{9}{2} \sqrt{\frac{1}{5} \left(\sqrt{10}-3\right)} \tan ^{-1}\left(\frac{\left(1+4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right)+\frac{9}{2} \sqrt{\frac{1}{5} \left(3+\sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(1-4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right)",1,"(-2*(15 + 14*x))/(17*Sqrt[1 + 3*x - 2*x^2]) - (9*Sqrt[(-3 + Sqrt[10])/5]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/2 + (9*Sqrt[(3 + Sqrt[10])/5]*ArcTanh[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/2","A",7,6,30,0.2000,1,"{1016, 12, 1032, 724, 204, 206}"
27,1,193,0,0.2659461,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \left(1+3 x-2 x^2\right)^{5/2}} \, dx","Int[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x - 2*x^2)^(5/2)),x]","-\frac{2 (14 x+15)}{51 \left(-2 x^2+3 x+1\right)^{3/2}}-\frac{2 (4814 x+291)}{867 \sqrt{-2 x^2+3 x+1}}+\frac{9}{2} \sqrt{\frac{1}{5} \left(17 \sqrt{10}-53\right)} \tan ^{-1}\left(\frac{\left(1+4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right)+\frac{9}{2} \sqrt{\frac{1}{5} \left(53+17 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(1-4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right)","-\frac{2 (14 x+15)}{51 \left(-2 x^2+3 x+1\right)^{3/2}}-\frac{2 (4814 x+291)}{867 \sqrt{-2 x^2+3 x+1}}+\frac{9}{2} \sqrt{\frac{1}{5} \left(17 \sqrt{10}-53\right)} \tan ^{-1}\left(\frac{\left(1+4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right)+\frac{9}{2} \sqrt{\frac{1}{5} \left(53+17 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(1-4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right)",1,"(-2*(15 + 14*x))/(51*(1 + 3*x - 2*x^2)^(3/2)) - (2*(291 + 4814*x))/(867*Sqrt[1 + 3*x - 2*x^2]) + (9*Sqrt[(-53 + 17*Sqrt[10])/5]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/2 + (9*Sqrt[(53 + 17*Sqrt[10])/5]*ArcTanh[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/2","A",7,6,30,0.2000,1,"{1016, 1060, 1032, 724, 204, 206}"
28,1,151,0,0.2285552,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \sqrt{1+3 x+2 x^2}} \, dx","Int[(2 + x)/((2 + 4*x - 3*x^2)*Sqrt[1 + 3*x + 2*x^2]),x]","\frac{1}{2} \sqrt{1-\frac{7 \sqrt{\frac{2}{5}}}{5}} \tanh ^{-1}\left(\frac{\left(17+4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)-\frac{1}{2} \sqrt{1+\frac{7 \sqrt{\frac{2}{5}}}{5}} \tanh ^{-1}\left(\frac{\left(17-4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)","\frac{1}{2} \sqrt{1-\frac{7 \sqrt{\frac{2}{5}}}{5}} \tanh ^{-1}\left(\frac{\left(17+4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)-\frac{1}{2} \sqrt{1+\frac{7 \sqrt{\frac{2}{5}}}{5}} \tanh ^{-1}\left(\frac{\left(17-4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)",1,"-(Sqrt[1 + (7*Sqrt[2/5])/5]*ArcTanh[(3*(4 - Sqrt[10]) + (17 - 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/2 + (Sqrt[1 - (7*Sqrt[2/5])/5]*ArcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/2","A",5,3,30,0.1000,1,"{1032, 724, 206}"
29,1,174,0,0.2547017,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \left(1+3 x+2 x^2\right)^{3/2}} \, dx","Int[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x + 2*x^2)^(3/2)),x]","\frac{2 (22 x+21)}{5 \sqrt{2 x^2+3 x+1}}-\frac{1}{10} \sqrt{\frac{3}{5} \left(2065+653 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(17-4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)+\frac{1}{10} \sqrt{\frac{3}{5} \left(2065-653 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(17+4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)","\frac{2 (22 x+21)}{5 \sqrt{2 x^2+3 x+1}}-\frac{1}{10} \sqrt{\frac{3}{5} \left(2065+653 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(17-4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)+\frac{1}{10} \sqrt{\frac{3}{5} \left(2065-653 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(17+4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)",1,"(2*(21 + 22*x))/(5*Sqrt[1 + 3*x + 2*x^2]) - (Sqrt[(3*(2065 + 653*Sqrt[10]))/5]*ArcTanh[(3*(4 - Sqrt[10]) + (17 - 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/10 + (Sqrt[(3*(2065 - 653*Sqrt[10]))/5]*ArcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/10","A",6,4,30,0.1333,1,"{1016, 1032, 724, 206}"
30,1,197,0,0.3033024,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \left(1+3 x+2 x^2\right)^{5/2}} \, dx","Int[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x + 2*x^2)^(5/2)),x]","\frac{2 (22 x+21)}{15 \left(2 x^2+3 x+1\right)^{3/2}}+\frac{2 (230 x+273)}{15 \sqrt{2 x^2+3 x+1}}-\frac{1}{50} \sqrt{\frac{1}{3} \left(4885115+1544809 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(17-4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)+\frac{1}{50} \sqrt{\frac{1}{3} \left(4885115-1544809 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(17+4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)","\frac{2 (22 x+21)}{15 \left(2 x^2+3 x+1\right)^{3/2}}+\frac{2 (230 x+273)}{15 \sqrt{2 x^2+3 x+1}}-\frac{1}{50} \sqrt{\frac{1}{3} \left(4885115+1544809 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(17-4 \sqrt{10}\right) x+3 \left(4-\sqrt{10}\right)}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)+\frac{1}{50} \sqrt{\frac{1}{3} \left(4885115-1544809 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{\left(17+4 \sqrt{10}\right) x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)",1,"(2*(21 + 22*x))/(15*(1 + 3*x + 2*x^2)^(3/2)) + (2*(273 + 230*x))/(15*Sqrt[1 + 3*x + 2*x^2]) - (Sqrt[(4885115 + 1544809*Sqrt[10])/3]*ArcTanh[(3*(4 - Sqrt[10]) + (17 - 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/50 + (Sqrt[(4885115 - 1544809*Sqrt[10])/3]*ArcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/50","A",7,5,30,0.1667,1,"{1016, 1060, 1032, 724, 206}"
31,1,15,0,0.016535,"\int \frac{1+x}{\left(4+2 x+x^2\right) \sqrt{5+2 x+x^2}} \, dx","Int[(1 + x)/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]","-\tanh ^{-1}\left(\sqrt{x^2+2 x+5}\right)","-\tanh ^{-1}\left(\sqrt{x^2+2 x+5}\right)",1,"-ArcTanh[Sqrt[5 + 2*x + x^2]]","A",2,2,26,0.07692,1,"{1024, 206}"
32,1,44,0,0.0522627,"\int \frac{4+x}{\left(4+2 x+x^2\right) \sqrt{5+2 x+x^2}} \, dx","Int[(4 + x)/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]","\sqrt{3} \tan ^{-1}\left(\frac{x+1}{\sqrt{3} \sqrt{x^2+2 x+5}}\right)-\tanh ^{-1}\left(\sqrt{x^2+2 x+5}\right)","\sqrt{3} \tan ^{-1}\left(\frac{x+1}{\sqrt{3} \sqrt{x^2+2 x+5}}\right)-\tanh ^{-1}\left(\sqrt{x^2+2 x+5}\right)",1,"Sqrt[3]*ArcTan[(1 + x)/(Sqrt[3]*Sqrt[5 + 2*x + x^2])] - ArcTanh[Sqrt[5 + 2*x + x^2]]","A",5,5,26,0.1923,1,"{1025, 982, 204, 1024, 206}"
33,1,24,0,0.0187828,"\int \frac{1+2 x}{\left(3+x+x^2\right) \sqrt{5+x+x^2}} \, dx","Int[(1 + 2*x)/((3 + x + x^2)*Sqrt[5 + x + x^2]),x]","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{x^2+x+5}}{\sqrt{2}}\right)","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{x^2+x+5}}{\sqrt{2}}\right)",1,"-(Sqrt[2]*ArcTanh[Sqrt[5 + x + x^2]/Sqrt[2]])","A",2,2,24,0.08333,1,"{1024, 206}"
34,1,56,0,0.0501775,"\int \frac{x}{\left(3+x+x^2\right) \sqrt{5+x+x^2}} \, dx","Int[x/((3 + x + x^2)*Sqrt[5 + x + x^2]),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{2}{11}} (2 x+1)}{\sqrt{x^2+x+5}}\right)}{\sqrt{22}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{x^2+x+5}}{\sqrt{2}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{2}{11}} (2 x+1)}{\sqrt{x^2+x+5}}\right)}{\sqrt{22}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{x^2+x+5}}{\sqrt{2}}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2/11]*(1 + 2*x))/Sqrt[5 + x + x^2]]/Sqrt[22]) - ArcTanh[Sqrt[5 + x + x^2]/Sqrt[2]]/Sqrt[2]","A",5,5,20,0.2500,1,"{1025, 982, 204, 1024, 206}"
35,1,249,0,0.9097219,"\int \frac{A+B x}{\sqrt{d+e x+f x^2} \left(a e+b e x+b f x^2\right)^2} \, dx","Int[(A + B*x)/(Sqrt[d + e*x + f*x^2]*(a*e + b*e*x + b*f*x^2)^2),x]","\frac{(B e-2 A f) \left(8 a e f-b \left(4 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{(e+2 f x) \sqrt{b d-a e}}{\sqrt{e} \sqrt{b e-4 a f} \sqrt{d+e x+f x^2}}\right)}{2 e^{3/2} f (b d-a e)^{3/2} (b e-4 a f)^{3/2}}-\frac{\sqrt{d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left(a e+b e x+b f x^2\right)}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x+f x^2}}{\sqrt{b d-a e}}\right)}{2 \sqrt{b} f (b d-a e)^{3/2}}","\frac{(B e-2 A f) \left(8 a e f-b \left(4 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{(e+2 f x) \sqrt{b d-a e}}{\sqrt{e} \sqrt{b e-4 a f} \sqrt{d+e x+f x^2}}\right)}{2 e^{3/2} f (b d-a e)^{3/2} (b e-4 a f)^{3/2}}-\frac{\sqrt{d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left(a e+b e x+b f x^2\right)}+\frac{B \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x+f x^2}}{\sqrt{b d-a e}}\right)}{2 \sqrt{b} f (b d-a e)^{3/2}}",1,"-((((A*b - 2*a*B)*e - b*(B*e - 2*A*f)*x)*Sqrt[d + e*x + f*x^2])/(e*(b*d - a*e)*(b*e - 4*a*f)*(a*e + b*e*x + b*f*x^2))) + ((B*e - 2*A*f)*(8*a*e*f - b*(e^2 + 4*d*f))*ArcTanh[(Sqrt[b*d - a*e]*(e + 2*f*x))/(Sqrt[e]*Sqrt[b*e - 4*a*f]*Sqrt[d + e*x + f*x^2])])/(2*e^(3/2)*(b*d - a*e)^(3/2)*f*(b*e - 4*a*f)^(3/2)) + (B*ArcTanh[(Sqrt[b]*Sqrt[d + e*x + f*x^2])/Sqrt[b*d - a*e]])/(2*Sqrt[b]*(b*d - a*e)^(3/2)*f)","A",6,5,36,0.1389,1,"{1016, 1025, 982, 208, 1024}"
36,1,48,0,0.0498577,"\int \frac{(g+h x) \sqrt{a+b x+c x^2}}{\left(a d+b d x+c d x^2\right)^2} \, dx","Int[((g + h*x)*Sqrt[a + b*x + c*x^2])/(a*d + b*d*x + c*d*x^2)^2,x]","-\frac{2 (-2 a h+x (2 c g-b h)+b g)}{d^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}","-\frac{2 (-2 a h+x (2 c g-b h)+b g)}{d^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}",1,"(-2*(b*g - 2*a*h + (2*c*g - b*h)*x))/((b^2 - 4*a*c)*d^2*Sqrt[a + b*x + c*x^2])","A",2,2,36,0.05556,1,"{998, 636}"
37,1,17,0,0.0224239,"\int \frac{3+2 x}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Int[(3 + 2*x)/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)","\tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)",1,"ArcTanh[x/Sqrt[-3 - 4*x - x^2]]","A",2,2,32,0.06250,1,"{1027, 206}"
38,1,86,0,0.1845976,"\int \frac{3+4 x}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Int[(3 + 4*x)/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\sqrt{2} \tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)","\sqrt{2} \tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)",1,"Sqrt[2]*ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]] - Sqrt[2]*ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]] + ArcTanh[x/Sqrt[-3 - 4*x - x^2]]","A",13,9,32,0.2812,1,"{1028, 986, 12, 1026, 1161, 618, 204, 1027, 206}"
39,1,136,0,0.1142303,"\int \frac{(g+h x) \sqrt{a+b x+c x^2}}{\left(a d+b d x+c d x^2\right)^{3/2}} \, dx","Int[((g + h*x)*Sqrt[a + b*x + c*x^2])/(a*d + b*d*x + c*d*x^2)^(3/2),x]","\frac{h \sqrt{a+b x+c x^2} \log \left(a+b x+c x^2\right)}{2 c d \sqrt{a d+b d x+c d x^2}}-\frac{\sqrt{a+b x+c x^2} (2 c g-b h) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c d \sqrt{b^2-4 a c} \sqrt{a d+b d x+c d x^2}}","\frac{h \sqrt{a+b x+c x^2} \log \left(a+b x+c x^2\right)}{2 c d \sqrt{a d+b d x+c d x^2}}-\frac{\sqrt{a+b x+c x^2} (2 c g-b h) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c d \sqrt{b^2-4 a c} \sqrt{a d+b d x+c d x^2}}",1,"-(((2*c*g - b*h)*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c*Sqrt[b^2 - 4*a*c]*d*Sqrt[a*d + b*d*x + c*d*x^2])) + (h*Sqrt[a + b*x + c*x^2]*Log[a + b*x + c*x^2])/(2*c*d*Sqrt[a*d + b*d*x + c*d*x^2])","A",5,5,38,0.1316,1,"{999, 634, 618, 206, 628}"
40,1,212,0,0.1160953,"\int x^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2} \, dx","Int[x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2],x]","-\frac{a c^2 \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{3/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2\right)^{3/2} (8 b c-15 a d x)}{60 d^2 (a+b x)}-\frac{a c x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{8 d (a+b x)}+\frac{b x^2 \sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2\right)^{3/2}}{5 d (a+b x)}","-\frac{a c^2 \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{3/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2\right)^{3/2} (8 b c-15 a d x)}{60 d^2 (a+b x)}-\frac{a c x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{8 d (a+b x)}+\frac{b x^2 \sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2\right)^{3/2}}{5 d (a+b x)}",1,"-(a*c*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/(8*d*(a + b*x)) + (b*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + d*x^2)^(3/2))/(5*d*(a + b*x)) - ((8*b*c - 15*a*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + d*x^2)^(3/2))/(60*d^2*(a + b*x)) - (a*c^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(3/2)*(a + b*x))","A",6,6,35,0.1714,1,"{1001, 833, 780, 195, 217, 206}"
41,1,161,0,0.0673629,"\int x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2} \, dx","Int[x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2],x]","-\frac{b c^2 \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{3/2} (a+b x)}-\frac{b c x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{8 d (a+b x)}+\frac{(4 a+3 b x) \sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2\right)^{3/2}}{12 d (a+b x)}","-\frac{b c^2 \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{8 d^{3/2} (a+b x)}-\frac{b c x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{8 d (a+b x)}+\frac{(4 a+3 b x) \sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2\right)^{3/2}}{12 d (a+b x)}",1,"-(b*c*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/(8*d*(a + b*x)) + ((4*a + 3*b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + d*x^2)^(3/2))/(12*d*(a + b*x)) - (b*c^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(3/2)*(a + b*x))","A",5,5,33,0.1515,1,"{1001, 780, 195, 217, 206}"
42,1,148,0,0.0567116,"\int \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2} \, dx","Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2],x]","\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2\right)^{3/2}}{3 d (a+b x)}+\frac{a x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{2 (a+b x)}+\frac{a c \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 \sqrt{d} (a+b x)}","\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2\right)^{3/2}}{3 d (a+b x)}+\frac{a x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{2 (a+b x)}+\frac{a c \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 \sqrt{d} (a+b x)}",1,"(a*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/(2*(a + b*x)) + (b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + d*x^2)^(3/2))/(3*d*(a + b*x)) + (a*c*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*Sqrt[d]*(a + b*x))","A",5,5,32,0.1562,1,"{970, 641, 195, 217, 206}"
43,1,160,0,0.1194172,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{x} \, dx","Int[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/x,x]","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 a+b x) \sqrt{c+d x^2}}{2 (a+b x)}+\frac{b c \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 \sqrt{d} (a+b x)}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a+b x}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 a+b x) \sqrt{c+d x^2}}{2 (a+b x)}+\frac{b c \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{2 \sqrt{d} (a+b x)}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a+b x}",1,"((2*a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/(2*(a + b*x)) + (b*c*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*Sqrt[d]*(a + b*x)) - (a*Sqrt[c]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(a + b*x)","A",8,8,35,0.2286,1,"{1001, 815, 844, 217, 206, 266, 63, 208}"
44,1,156,0,0.1131908,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{x^2} \, dx","Int[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/x^2,x]","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a-b x) \sqrt{c+d x^2}}{x (a+b x)}+\frac{a \sqrt{d} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{a+b x}-\frac{b \sqrt{c} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a+b x}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a-b x) \sqrt{c+d x^2}}{x (a+b x)}+\frac{a \sqrt{d} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{a+b x}-\frac{b \sqrt{c} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{a+b x}",1,"-(((a - b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/(x*(a + b*x))) + (a*Sqrt[d]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(a + b*x) - (b*Sqrt[c]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(a + b*x)","A",8,8,35,0.2286,1,"{1001, 813, 844, 217, 206, 266, 63, 208}"
45,1,161,0,0.1162689,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{x^3} \, dx","Int[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/x^3,x]","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+2 b x) \sqrt{c+d x^2}}{2 x^2 (a+b x)}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{a+b x}-\frac{a d \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 \sqrt{c} (a+b x)}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+2 b x) \sqrt{c+d x^2}}{2 x^2 (a+b x)}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right)}{a+b x}-\frac{a d \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)}{2 \sqrt{c} (a+b x)}",1,"-((a + 2*b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/(2*x^2*(a + b*x)) + (b*Sqrt[d]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(a + b*x) - (a*d*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*Sqrt[c]*(a + b*x))","A",8,8,35,0.2286,1,"{1001, 811, 844, 217, 206, 266, 63, 208}"
46,1,317,0,0.3330164,"\int x^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2} \, dx","Int[x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2],x]","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2+e x\right)^{3/2} \left(-6 d x (10 a d-7 b e)+50 a d e+32 b c d-35 b e^2\right)}{240 d^3 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt{c+d x^2+e x} \left(2 a d \left(4 c d-5 e^2\right)-b \left(12 c d e-7 e^3\right)\right)}{128 d^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(4 c d-e^2\right) \left(8 a c d^2-10 a d e^2-12 b c d e+7 b e^3\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{256 d^{9/2} (a+b x)}+\frac{b x^2 \sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2+e x\right)^{3/2}}{5 d (a+b x)}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2+e x\right)^{3/2} \left(-6 d x (10 a d-7 b e)+50 a d e+32 b c d-35 b e^2\right)}{240 d^3 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt{c+d x^2+e x} \left(2 a d \left(4 c d-5 e^2\right)-b \left(12 c d e-7 e^3\right)\right)}{128 d^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(4 c d-e^2\right) \left(8 a c d^2-10 a d e^2-12 b c d e+7 b e^3\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{256 d^{9/2} (a+b x)}+\frac{b x^2 \sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2+e x\right)^{3/2}}{5 d (a+b x)}",1,"-((2*a*d*(4*c*d - 5*e^2) - b*(12*c*d*e - 7*e^3))*(e + 2*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/(128*d^4*(a + b*x)) + (b*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(5*d*(a + b*x)) - ((32*b*c*d + 50*a*d*e - 35*b*e^2 - 6*d*(10*a*d - 7*b*e)*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(240*d^3*(a + b*x)) - ((4*c*d - e^2)*(8*a*c*d^2 - 12*b*c*d*e - 10*a*d*e^2 + 7*b*e^3)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(256*d^(9/2)*(a + b*x))","A",6,6,38,0.1579,1,"{1000, 832, 779, 612, 621, 206}"
47,1,227,0,0.126839,"\int x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2} \, dx","Int[x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2],x]","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt{c+d x^2+e x} \left(8 a d e+4 b c d-5 b e^2\right)}{64 d^3 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(4 c d-e^2\right) \left(8 a d e+4 b c d-5 b e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{128 d^{7/2} (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2+e x\right)^{3/2} (8 a d+6 b d x-5 b e)}{24 d^2 (a+b x)}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt{c+d x^2+e x} \left(8 a d e+4 b c d-5 b e^2\right)}{64 d^3 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(4 c d-e^2\right) \left(8 a d e+4 b c d-5 b e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{128 d^{7/2} (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2+e x\right)^{3/2} (8 a d+6 b d x-5 b e)}{24 d^2 (a+b x)}",1,"-((4*b*c*d + 8*a*d*e - 5*b*e^2)*(e + 2*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/(64*d^3*(a + b*x)) + ((8*a*d - 5*b*e + 6*b*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(24*d^2*(a + b*x)) - ((4*c*d - e^2)*(4*b*c*d + 8*a*d*e - 5*b*e^2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(128*d^(7/2)*(a + b*x))","A",5,5,36,0.1389,1,"{1000, 779, 612, 621, 206}"
48,1,198,0,0.1010757,"\int \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2} \, dx","Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2],x]","\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(4 c d-e^2\right) (2 a d-b e) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{16 d^{5/2} (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) (2 a d-b e) \sqrt{c+d x^2+e x}}{8 d^2 (a+b x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2+e x\right)^{3/2}}{3 d (a+b x)}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(4 c d-e^2\right) (2 a d-b e) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{16 d^{5/2} (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 d x+e) (2 a d-b e) \sqrt{c+d x^2+e x}}{8 d^2 (a+b x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \left(c+d x^2+e x\right)^{3/2}}{3 d (a+b x)}",1,"((2*a*d - b*e)*(e + 2*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/(8*d^2*(a + b*x)) + (b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(c + e*x + d*x^2)^(3/2))/(3*d*(a + b*x)) + ((2*a*d - b*e)*(4*c*d - e^2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(16*d^(5/2)*(a + b*x))","A",5,5,35,0.1429,1,"{969, 640, 612, 621, 206}"
49,1,211,0,0.2214341,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2}}{x} \, dx","Int[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/x,x]","\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(4 a d e+4 b c d-b e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{8 d^{3/2} (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2+e x} (4 a d+2 b d x+b e)}{4 d (a+b x)}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{a+b x}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(4 a d e+4 b c d-b e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{8 d^{3/2} (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2+e x} (4 a d+2 b d x+b e)}{4 d (a+b x)}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{a+b x}",1,"((4*a*d + b*e + 2*b*d*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/(4*d*(a + b*x)) + ((4*b*c*d + 4*a*d*e - b*e^2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(8*d^(3/2)*(a + b*x)) - (a*Sqrt[c]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x + d*x^2])])/(a + b*x)","A",7,6,38,0.1579,1,"{1000, 814, 843, 621, 206, 724}"
50,1,202,0,0.1944113,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2}}{x^2} \, dx","Int[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/x^2,x]","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a-b x) \sqrt{c+d x^2+e x}}{x (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 a d+b e) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{2 \sqrt{d} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a e+2 b c) \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{2 \sqrt{c} (a+b x)}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a-b x) \sqrt{c+d x^2+e x}}{x (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 a d+b e) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{2 \sqrt{d} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a e+2 b c) \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{2 \sqrt{c} (a+b x)}",1,"-(((a - b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/(x*(a + b*x))) + ((2*a*d + b*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(2*Sqrt[d]*(a + b*x)) - ((2*b*c + a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x + d*x^2])])/(2*Sqrt[c]*(a + b*x))","A",7,6,38,0.1579,1,"{1000, 812, 843, 621, 206, 724}"
51,1,215,0,0.1834166,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2}}{x^3} \, dx","Int[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/x^3,x]","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(4 a c d-a e^2+4 b c e\right) \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{8 c^{3/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2 (a+b x)}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{a+b x}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left(4 a c d-a e^2+4 b c e\right) \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right)}{8 c^{3/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2 (a+b x)}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right)}{a+b x}",1,"-((2*a*c + (4*b*c + a*e)*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/(4*c*x^2*(a + b*x)) + (b*Sqrt[d]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + e*x + d*x^2])])/(a + b*x) - ((4*a*c*d + 4*b*c*e - a*e^2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + e*x + d*x^2])])/(8*c^(3/2)*(a + b*x))","A",7,6,38,0.1579,1,"{1000, 810, 843, 621, 206, 724}"
52,1,452,0,1.9673923,"\int \frac{x^2 \sqrt{a+c x^2}}{d+e x+f x^2} \, dx","Int[(x^2*Sqrt[a + c*x^2])/(d + e*x + f*x^2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(a f^2+2 c \left(e^2-d f\right)\right)}{2 \sqrt{c} f^3}-\frac{\left(e \left(e-\sqrt{e^2-4 d f}\right) \left(a f^2+c \left(e^2-2 d f\right)\right)-2 d f \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(e \left(\sqrt{e^2-4 d f}+e\right) \left(a f^2+c \left(e^2-2 d f\right)\right)-2 d f \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{a+c x^2} (2 e-f x)}{2 f^2}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(a f^2+2 c \left(e^2-d f\right)\right)}{2 \sqrt{c} f^3}-\frac{\left(e \left(e-\sqrt{e^2-4 d f}\right) \left(a f^2+c \left(e^2-2 d f\right)\right)-2 d f \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(e \left(\sqrt{e^2-4 d f}+e\right) \left(a f^2+c \left(e^2-2 d f\right)\right)-2 d f \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{a+c x^2} (2 e-f x)}{2 f^2}",1,"-((2*e - f*x)*Sqrt[a + c*x^2])/(2*f^2) + ((a*f^2 + 2*c*(e^2 - d*f))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*Sqrt[c]*f^3) - ((e*(e - Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - 2*d*f)) - 2*d*f*(a*f^2 + c*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((e*(e + Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - 2*d*f)) - 2*d*f*(a*f^2 + c*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",9,6,27,0.2222,1,"{1069, 1080, 217, 206, 1034, 725}"
53,1,395,0,0.9314063,"\int \frac{x \sqrt{a+c x^2}}{d+e x+f x^2} \, dx","Int[(x*Sqrt[a + c*x^2])/(d + e*x + f*x^2),x]","-\frac{\left(2 c d e f-\left(e-\sqrt{e^2-4 d f}\right) \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(2 c d e f-\left(\sqrt{e^2-4 d f}+e\right) \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{c} e \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{f^2}+\frac{\sqrt{a+c x^2}}{f}","-\frac{\left(2 c d e f-\left(e-\sqrt{e^2-4 d f}\right) \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(2 c d e f-\left(\sqrt{e^2-4 d f}+e\right) \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{c} e \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{f^2}+\frac{\sqrt{a+c x^2}}{f}",1,"Sqrt[a + c*x^2]/f - (Sqrt[c]*e*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/f^2 - ((2*c*d*e*f - (e - Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((2*c*d*e*f - (e + Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",9,6,25,0.2400,1,"{1020, 1080, 217, 206, 1034, 725}"
54,1,298,0,0.3758881,"\int \frac{\sqrt{a+c x^2}}{d+e x+f x^2} \, dx","Int[Sqrt[a + c*x^2]/(d + e*x + f*x^2),x]","-\frac{\sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f}}+\frac{\sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{f}","-\frac{\sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f}}+\frac{\sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{f}",1,"(Sqrt[c]*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/f - (Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f]) + (Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f])","A",8,5,24,0.2083,1,"{991, 217, 206, 1034, 725}"
55,1,358,0,1.3135063,"\int \frac{\sqrt{a+c x^2}}{x \left(d+e x+f x^2\right)} \, dx","Int[Sqrt[a + c*x^2]/(x*(d + e*x + f*x^2)),x]","\frac{\left(\left(e-\sqrt{e^2-4 d f}\right) (c d-a f)+2 a e f\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(\left(\sqrt{e^2-4 d f}+e\right) (c d-a f)+2 a e f\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d}","\frac{\left(\left(e-\sqrt{e^2-4 d f}\right) (c d-a f)+2 a e f\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(\left(\sqrt{e^2-4 d f}+e\right) (c d-a f)+2 a e f\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d}",1,"((2*a*e*f + (c*d - a*f)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((2*a*e*f + (c*d - a*f)*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) - (Sqrt[a]*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d","A",12,9,27,0.3333,1,"{6728, 266, 50, 63, 208, 1020, 1034, 725, 206}"
56,1,382,0,1.4170496,"\int \frac{\sqrt{a+c x^2}}{x^2 \left(d+e x+f x^2\right)} \, dx","Int[Sqrt[a + c*x^2]/(x^2*(d + e*x + f*x^2)),x]","-\frac{f \left(a \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)+2 c d^2\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(a \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)+2 c d^2\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\sqrt{a} e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^2}-\frac{\sqrt{a+c x^2}}{d x}","-\frac{f \left(a \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)+2 c d^2\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(a \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)+2 c d^2\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\sqrt{a} e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^2}-\frac{\sqrt{a+c x^2}}{d x}",1,"-(Sqrt[a + c*x^2]/(d*x)) - (f*(2*c*d^2 + a*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + (f*(2*c*d^2 + a*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) + (Sqrt[a]*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^2","A",18,12,27,0.4444,1,"{6728, 277, 217, 206, 266, 50, 63, 208, 1020, 1080, 1034, 725}"
57,1,507,0,1.8798161,"\int \frac{\sqrt{a+c x^2}}{x^3 \left(d+e x+f x^2\right)} \, dx","Int[Sqrt[a + c*x^2]/(x^3*(d + e*x + f*x^2)),x]","\frac{f \left(a \left(e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)+c d^2 \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(a \left(-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)+c d^2 \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{a} \left(e^2-d f\right) \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^3}+\frac{e \sqrt{a+c x^2}}{d^2 x}-\frac{\sqrt{a+c x^2}}{2 d x^2}-\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 \sqrt{a} d}","\frac{f \left(a \left(e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)+c d^2 \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(a \left(-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)+c d^2 \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{a} \left(e^2-d f\right) \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^3}+\frac{e \sqrt{a+c x^2}}{d^2 x}-\frac{\sqrt{a+c x^2}}{2 d x^2}-\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 \sqrt{a} d}",1,"-Sqrt[a + c*x^2]/(2*d*x^2) + (e*Sqrt[a + c*x^2])/(d^2*x) + (f*(c*d^2*(e + Sqrt[e^2 - 4*d*f]) + a*(e^3 - 3*d*e*f + e^2*Sqrt[e^2 - 4*d*f] - d*f*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - (f*(c*d^2*(e - Sqrt[e^2 - 4*d*f]) + a*(e^3 - 3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) - (c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*Sqrt[a]*d) - (Sqrt[a]*(e^2 - d*f)*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^3","A",22,13,27,0.4815,1,"{6728, 266, 47, 63, 208, 277, 217, 206, 50, 1020, 1080, 1034, 725}"
58,1,795,0,4.2640181,"\int \frac{x^2 \left(a+c x^2\right)^{3/2}}{d+e x+f x^2} \, dx","Int[(x^2*(a + c*x^2)^(3/2))/(d + e*x + f*x^2),x]","-\frac{(4 e-3 f x) \left(c x^2+a\right)^{3/2}}{12 f^2}-\frac{\left(8 e \left(a f^2+c \left(e^2-2 d f\right)\right)-f \left(3 a f^2+4 c \left(e^2-d f\right)\right) x\right) \sqrt{c x^2+a}}{8 f^4}+\frac{\left(3 a^2 f^4+12 a c \left(e^2-d f\right) f^2+8 c^2 \left(e^4-3 d f e^2+d^2 f^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right)}{8 \sqrt{c} f^5}-\frac{\left(a^2 \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right) f^4+2 a c \left(e^4-\sqrt{e^2-4 d f} e^3-4 d f e^2+2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right) f^2+c^2 \left(e^6-\sqrt{e^2-4 d f} e^5-6 d f e^4+4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2-3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right)\right) \tanh ^{-1}\left(\frac{2 a f-c \left(e-\sqrt{e^2-4 d f}\right) x}{\sqrt{2} \sqrt{2 a f^2+c \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right)} \sqrt{c x^2+a}}\right)}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right)}}+\frac{\left(a^2 \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right) f^4+2 a c \left(e^4+\sqrt{e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right) f^2+c^2 \left(e^6+\sqrt{e^2-4 d f} e^5-6 d f e^4-4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2+3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right)\right) \tanh ^{-1}\left(\frac{2 a f-c \left(e+\sqrt{e^2-4 d f}\right) x}{\sqrt{2} \sqrt{2 a f^2+c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \sqrt{c x^2+a}}\right)}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)}}","-\frac{(4 e-3 f x) \left(c x^2+a\right)^{3/2}}{12 f^2}-\frac{\left(8 e \left(a f^2+c \left(e^2-2 d f\right)\right)-f \left(3 a f^2+4 c \left(e^2-d f\right)\right) x\right) \sqrt{c x^2+a}}{8 f^4}+\frac{\left(3 a^2 f^4+12 a c \left(e^2-d f\right) f^2+8 c^2 \left(e^4-3 d f e^2+d^2 f^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right)}{8 \sqrt{c} f^5}-\frac{\left(a^2 \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right) f^4+2 a c \left(e^4-\sqrt{e^2-4 d f} e^3-4 d f e^2+2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right) f^2+c^2 \left(e^6-\sqrt{e^2-4 d f} e^5-6 d f e^4+4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2-3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right)\right) \tanh ^{-1}\left(\frac{2 a f-c \left(e-\sqrt{e^2-4 d f}\right) x}{\sqrt{2} \sqrt{2 a f^2+c \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right)} \sqrt{c x^2+a}}\right)}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right)}}+\frac{\left(a^2 \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right) f^4+2 a c \left(e^4+\sqrt{e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right) f^2+c^2 \left(e^6+\sqrt{e^2-4 d f} e^5-6 d f e^4-4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2+3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right)\right) \tanh ^{-1}\left(\frac{2 a f-c \left(e+\sqrt{e^2-4 d f}\right) x}{\sqrt{2} \sqrt{2 a f^2+c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \sqrt{c x^2+a}}\right)}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)}}",1,"-((8*e*(a*f^2 + c*(e^2 - 2*d*f)) - f*(3*a*f^2 + 4*c*(e^2 - d*f))*x)*Sqrt[a + c*x^2])/(8*f^4) - ((4*e - 3*f*x)*(a + c*x^2)^(3/2))/(12*f^2) + ((3*a^2*f^4 + 12*a*c*f^2*(e^2 - d*f) + 8*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*Sqrt[c]*f^5) - ((a^2*f^4*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + 2*a*c*f^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2 - e^3*Sqrt[e^2 - 4*d*f] + 2*d*e*f*Sqrt[e^2 - 4*d*f]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 - e^5*Sqrt[e^2 - 4*d*f] + 4*d*e^3*f*Sqrt[e^2 - 4*d*f] - 3*d^2*e*f^2*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((a^2*f^4*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + 2*a*c*f^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + e^3*Sqrt[e^2 - 4*d*f] - 2*d*e*f*Sqrt[e^2 - 4*d*f]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 + e^5*Sqrt[e^2 - 4*d*f] - 4*d*e^3*f*Sqrt[e^2 - 4*d*f] + 3*d^2*e*f^2*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",10,7,27,0.2593,1,"{1069, 1068, 1080, 217, 206, 1034, 725}"
59,1,553,0,2.4305936,"\int \frac{x \left(a+c x^2\right)^{3/2}}{d+e x+f x^2} \, dx","Int[(x*(a + c*x^2)^(3/2))/(d + e*x + f*x^2),x]","-\frac{\left(2 c d e f \left(2 a f^2+c \left(e^2-2 d f\right)\right)-\left(e-\sqrt{e^2-4 d f}\right) \left(a^2 f^4+2 a c f^2 \left(e^2-d f\right)+c^2 \left(d^2 f^2-3 d e^2 f+e^4\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(2 c d e f \left(2 a f^2+c \left(e^2-2 d f\right)\right)-\left(\sqrt{e^2-4 d f}+e\right) \left(a^2 f^4+2 a c f^2 \left(e^2-d f\right)+c^2 \left(d^2 f^2-3 d e^2 f+e^4\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\sqrt{a+c x^2} \left(2 \left(a f^2+c \left(e^2-d f\right)\right)-c e f x\right)}{2 f^3}-\frac{\sqrt{c} e \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a f^2+2 c \left(e^2-2 d f\right)\right)}{2 f^4}+\frac{\left(a+c x^2\right)^{3/2}}{3 f}","-\frac{\left(2 c d e f \left(2 a f^2+c \left(e^2-2 d f\right)\right)-\left(e-\sqrt{e^2-4 d f}\right) \left(a^2 f^4+2 a c f^2 \left(e^2-d f\right)+c^2 \left(d^2 f^2-3 d e^2 f+e^4\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(2 c d e f \left(2 a f^2+c \left(e^2-2 d f\right)\right)-\left(\sqrt{e^2-4 d f}+e\right) \left(a^2 f^4+2 a c f^2 \left(e^2-d f\right)+c^2 \left(d^2 f^2-3 d e^2 f+e^4\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\sqrt{a+c x^2} \left(2 \left(a f^2+c \left(e^2-d f\right)\right)-c e f x\right)}{2 f^3}-\frac{\sqrt{c} e \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a f^2+2 c \left(e^2-2 d f\right)\right)}{2 f^4}+\frac{\left(a+c x^2\right)^{3/2}}{3 f}",1,"((2*(a*f^2 + c*(e^2 - d*f)) - c*e*f*x)*Sqrt[a + c*x^2])/(2*f^3) + (a + c*x^2)^(3/2)/(3*f) - (Sqrt[c]*e*(3*a*f^2 + 2*c*(e^2 - 2*d*f))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*f^4) - ((2*c*d*e*f*(2*a*f^2 + c*(e^2 - 2*d*f)) - (e - Sqrt[e^2 - 4*d*f])*(a^2*f^4 + 2*a*c*f^2*(e^2 - d*f) + c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^4*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((2*c*d*e*f*(2*a*f^2 + c*(e^2 - 2*d*f)) - (e + Sqrt[e^2 - 4*d*f])*(a^2*f^4 + 2*a*c*f^2*(e^2 - d*f) + c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^4*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",10,7,25,0.2800,1,"{1020, 1068, 1080, 217, 206, 1034, 725}"
60,1,482,0,4.2398651,"\int \frac{\left(a+c x^2\right)^{3/2}}{d+e x+f x^2} \, dx","Int[(a + c*x^2)^(3/2)/(d + e*x + f*x^2),x]","\frac{\left(-2 a^2 f^4-c e \left(e-\sqrt{e^2-4 d f}\right) \left(2 a f^2+c \left(e^2-2 d f\right)\right)+4 a c d f^3+2 c^2 d f \left(e^2-d f\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(-2 a^2 f^4-c e \left(\sqrt{e^2-4 d f}+e\right) \left(2 a f^2+c \left(e^2-2 d f\right)\right)+4 a c d f^3+2 c^2 d f \left(e^2-d f\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a f^2+2 c \left(e^2-d f\right)\right)}{2 f^3}-\frac{c \sqrt{a+c x^2} (2 e-f x)}{2 f^2}","-\frac{\left(c e \left(e-\sqrt{e^2-4 d f}\right) \left(2 a f^2+c \left(e^2-2 d f\right)\right)-2 f \left(-a^2 f^3+2 a c d f^2+c^2 d \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(c e \left(\sqrt{e^2-4 d f}+e\right) \left(2 a f^2+c \left(e^2-2 d f\right)\right)-2 f \left(-a^2 f^3+2 a c d f^2+c^2 d \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a f^2+2 c \left(e^2-d f\right)\right)}{2 f^3}-\frac{c \sqrt{a+c x^2} (2 e-f x)}{2 f^2}",1,"-(c*(2*e - f*x)*Sqrt[a + c*x^2])/(2*f^2) + (Sqrt[c]*(3*a*f^2 + 2*c*(e^2 - d*f))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*f^3) + ((4*a*c*d*f^3 - 2*a^2*f^4 + 2*c^2*d*f*(e^2 - d*f) - c*e*(e - Sqrt[e^2 - 4*d*f])*(2*a*f^2 + c*(e^2 - 2*d*f)))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((4*a*c*d*f^3 - 2*a^2*f^4 + 2*c^2*d*f*(e^2 - d*f) - c*e*(e + Sqrt[e^2 - 4*d*f])*(2*a*f^2 + c*(e^2 - 2*d*f)))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",9,6,24,0.2500,1,"{979, 1080, 217, 206, 1034, 725}"
61,1,496,0,2.5688493,"\int \frac{\left(a+c x^2\right)^{3/2}}{x \left(d+e x+f x^2\right)} \, dx","Int[(a + c*x^2)^(3/2)/(x*(d + e*x + f*x^2)),x]","-\frac{\left(2 e f \left(c^2 d^2-a^2 f^2\right)-\left(e-\sqrt{e^2-4 d f}\right) \left(c^2 d e^2-f (c d-a f)^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(2 e f \left(c^2 d^2-a^2 f^2\right)-\left(\sqrt{e^2-4 d f}+e\right) \left(c^2 d e^2-f (c d-a f)^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d}-\frac{c^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{f^2}+\frac{\sqrt{a+c x^2} (c d-a f)}{d f}+\frac{a \sqrt{a+c x^2}}{d}","-\frac{\left(2 e f \left(c^2 d^2-a^2 f^2\right)-\left(e-\sqrt{e^2-4 d f}\right) \left(c^2 d e^2-f (c d-a f)^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(2 e f \left(c^2 d^2-a^2 f^2\right)-\left(\sqrt{e^2-4 d f}+e\right) \left(c^2 d e^2-f (c d-a f)^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d}-\frac{c^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{f^2}+\frac{\sqrt{a+c x^2} (c d-a f)}{d f}+\frac{a \sqrt{a+c x^2}}{d}",1,"(a*Sqrt[a + c*x^2])/d + ((c*d - a*f)*Sqrt[a + c*x^2])/(d*f) - (c^(3/2)*e*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/f^2 - ((2*e*f*(c^2*d^2 - a^2*f^2) - (c^2*d*e^2 - f*(c*d - a*f)^2)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((2*e*f*(c^2*d^2 - a^2*f^2) - (c^2*d*e^2 - f*(c*d - a*f)^2)*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) - (a^(3/2)*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d","A",17,11,27,0.4074,1,"{6728, 266, 50, 63, 208, 1020, 1080, 217, 206, 1034, 725}"
62,1,604,0,2.8093571,"\int \frac{\left(a+c x^2\right)^{3/2}}{x^2 \left(d+e x+f x^2\right)} \, dx","Int[(a + c*x^2)^(3/2)/(x^2*(d + e*x + f*x^2)),x]","-\frac{\left(a^2 f^2 \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)+4 a c d^2 f^2+c^2 d^2 \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(a^2 f^2 \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)+4 a c d^2 f^2+c^2 d^2 \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{a^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^2}-\frac{a e \sqrt{a+c x^2}}{d^2}+\frac{\sqrt{a+c x^2} (2 a e-c d x)}{2 d^2}+\frac{\sqrt{c} (2 c d-3 a f) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 d f}-\frac{\left(a+c x^2\right)^{3/2}}{d x}+\frac{3 c x \sqrt{a+c x^2}}{2 d}+\frac{3 a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 d}","-\frac{\left(a^2 f^2 \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)+4 a c d^2 f^2+c^2 d^2 \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(a^2 f^2 \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)+4 a c d^2 f^2+c^2 d^2 \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{a^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^2}-\frac{a e \sqrt{a+c x^2}}{d^2}+\frac{\sqrt{a+c x^2} (2 a e-c d x)}{2 d^2}+\frac{\sqrt{c} (2 c d-3 a f) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 d f}-\frac{\left(a+c x^2\right)^{3/2}}{d x}+\frac{3 c x \sqrt{a+c x^2}}{2 d}+\frac{3 a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 d}",1,"-((a*e*Sqrt[a + c*x^2])/d^2) + (3*c*x*Sqrt[a + c*x^2])/(2*d) + ((2*a*e - c*d*x)*Sqrt[a + c*x^2])/(2*d^2) - (a + c*x^2)^(3/2)/(d*x) + (3*a*Sqrt[c]*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*d) + (Sqrt[c]*(2*c*d - 3*a*f)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*d*f) - ((4*a*c*d^2*f^2 + c^2*d^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + a^2*f^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((4*a*c*d^2*f^2 + a^2*f^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + c^2*d^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) + (a^(3/2)*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^2","A",21,14,27,0.5185,1,"{6728, 277, 195, 217, 206, 266, 50, 63, 208, 1020, 1068, 1080, 1034, 725}"
63,1,668,0,3.4647326,"\int \frac{\left(a+c x^2\right)^{3/2}}{x^3 \left(d+e x+f x^2\right)} \, dx","Int[(a + c*x^2)^(3/2)/(x^3*(d + e*x + f*x^2)),x]","\frac{\left(a^2 f \left(e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)+2 a c d^2 f \left(\sqrt{e^2-4 d f}+e\right)+c^2 d^3 \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(a^2 f \left(-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)+2 a c d^2 f \left(e-\sqrt{e^2-4 d f}\right)+c^2 d^3 \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{a^{3/2} \left(e^2-d f\right) \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^3}+\frac{a \sqrt{a+c x^2} \left(e^2-d f\right)}{d^3}-\frac{\sqrt{a+c x^2} \left(2 \left(a \left(e^2-d f\right)+c d^2\right)-c d e x\right)}{2 d^3}+\frac{e \left(a+c x^2\right)^{3/2}}{d^2 x}-\frac{3 c e x \sqrt{a+c x^2}}{2 d^2}-\frac{\left(a+c x^2\right)^{3/2}}{2 d x^2}+\frac{3 c \sqrt{a+c x^2}}{2 d}-\frac{3 \sqrt{a} c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 d}","\frac{\left(a^2 f \left(e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)+2 a c d^2 f \left(\sqrt{e^2-4 d f}+e\right)+c^2 d^3 \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(a^2 f \left(-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)+2 a c d^2 f \left(e-\sqrt{e^2-4 d f}\right)+c^2 d^3 \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{a^{3/2} \left(e^2-d f\right) \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^3}+\frac{a \sqrt{a+c x^2} \left(e^2-d f\right)}{d^3}-\frac{\sqrt{a+c x^2} \left(2 \left(a \left(e^2-d f\right)+c d^2\right)-c d e x\right)}{2 d^3}+\frac{e \left(a+c x^2\right)^{3/2}}{d^2 x}-\frac{3 c e x \sqrt{a+c x^2}}{2 d^2}-\frac{\left(a+c x^2\right)^{3/2}}{2 d x^2}+\frac{3 c \sqrt{a+c x^2}}{2 d}-\frac{3 \sqrt{a} c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 d}",1,"(3*c*Sqrt[a + c*x^2])/(2*d) + (a*(e^2 - d*f)*Sqrt[a + c*x^2])/d^3 - (3*c*e*x*Sqrt[a + c*x^2])/(2*d^2) - ((2*(c*d^2 + a*(e^2 - d*f)) - c*d*e*x)*Sqrt[a + c*x^2])/(2*d^3) - (a + c*x^2)^(3/2)/(2*d*x^2) + (e*(a + c*x^2)^(3/2))/(d^2*x) + ((c^2*d^3*(e - Sqrt[e^2 - 4*d*f]) + 2*a*c*d^2*f*(e + Sqrt[e^2 - 4*d*f]) + a^2*f*(e^3 - 3*d*e*f + e^2*Sqrt[e^2 - 4*d*f] - d*f*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((2*a*c*d^2*f*(e - Sqrt[e^2 - 4*d*f]) + c^2*d^3*(e + Sqrt[e^2 - 4*d*f]) + a^2*f*(e^3 - 3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) - (3*Sqrt[a]*c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*d) - (a^(3/2)*(e^2 - d*f)*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^3","A",26,15,27,0.5556,1,"{6728, 266, 47, 50, 63, 208, 277, 195, 217, 206, 1020, 1068, 1080, 1034, 725}"
64,1,380,0,1.1695106,"\int \frac{x^3}{\sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[x^3/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{\left(2 d e f-\left(e^2-d f\right) \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(2 d e f-\left(e^2-d f\right) \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{e \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c} f^2}+\frac{\sqrt{a+c x^2}}{c f}","-\frac{\left(2 d e f-\left(e^2-d f\right) \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(2 d e f-\left(e^2-d f\right) \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{e \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c} f^2}+\frac{\sqrt{a+c x^2}}{c f}",1,"Sqrt[a + c*x^2]/(c*f) - (e*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(Sqrt[c]*f^2) - ((2*d*e*f - (e^2 - d*f)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((2*d*e*f - (e^2 - d*f)*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",10,6,27,0.2222,1,"{6728, 217, 206, 261, 1034, 725}"
65,1,344,0,0.5409442,"\int \frac{x^2}{\sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[x^2/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{\left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(2 d f-e \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c} f}","-\frac{\left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(2 d f-e \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c} f}",1,"ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]]/(Sqrt[c]*f) - ((e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((2*d*f - e*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",8,5,27,0.1852,1,"{1081, 217, 206, 1034, 725}"
66,1,294,0,0.2352707,"\int \frac{x}{\sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[x/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\left(e-\sqrt{e^2-4 d f}\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}","\frac{\left(e-\sqrt{e^2-4 d f}\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}",1,"((e - Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((e + Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",5,3,25,0.1200,1,"{1034, 725, 206}"
67,1,266,0,0.1506515,"\int \frac{1}{\sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[1/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\sqrt{2} f \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{2} f \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}","\frac{\sqrt{2} f \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{2} f \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}",1,"-((Sqrt[2]*f*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])])) + (Sqrt[2]*f*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",5,3,24,0.1250,1,"{985, 725, 206}"
68,1,330,0,0.8245966,"\int \frac{1}{x \sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[1/(x*Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","\frac{f \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(e-\sqrt{e^2-4 d f}\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d}","\frac{f \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(e-\sqrt{e^2-4 d f}\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(f*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - (f*(e - Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) - ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]/(Sqrt[a]*d)","A",10,7,27,0.2593,1,"{6728, 266, 63, 208, 1034, 725, 206}"
69,1,367,0,1.1992369,"\int \frac{1}{x^2 \sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[1/(x^2*Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{f \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d^2}-\frac{\sqrt{a+c x^2}}{a d x}","-\frac{f \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d^2}-\frac{\sqrt{a+c x^2}}{a d x}",1,"-(Sqrt[a + c*x^2]/(a*d*x)) - (f*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + (f*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) + (e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(Sqrt[a]*d^2)","A",11,8,27,0.2963,1,"{6728, 264, 266, 63, 208, 1034, 725, 206}"
70,1,457,0,1.8633264,"\int \frac{1}{x^3 \sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[1/(x^3*Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 a^{3/2} d}+\frac{f \left(-\left(e^2-d f\right) \left(e-\sqrt{e^2-4 d f}\right)-4 d e f+2 e^3\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(-\left(e^2-d f\right) \left(\sqrt{e^2-4 d f}+e\right)-4 d e f+2 e^3\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(e^2-d f\right) \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d^3}+\frac{e \sqrt{a+c x^2}}{a d^2 x}-\frac{\sqrt{a+c x^2}}{2 a d x^2}","\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 a^{3/2} d}+\frac{f \left(-\left(e^2-d f\right) \left(e-\sqrt{e^2-4 d f}\right)-4 d e f+2 e^3\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(-\left(e^2-d f\right) \left(\sqrt{e^2-4 d f}+e\right)-4 d e f+2 e^3\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(e^2-d f\right) \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d^3}+\frac{e \sqrt{a+c x^2}}{a d^2 x}-\frac{\sqrt{a+c x^2}}{2 a d x^2}",1,"-Sqrt[a + c*x^2]/(2*a*d*x^2) + (e*Sqrt[a + c*x^2])/(a*d^2*x) + (f*(2*e^3 - 4*d*e*f - (e^2 - d*f)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - (f*(2*e^3 - 4*d*e*f - (e^2 - d*f)*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) + (c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(3/2)*d) - ((e^2 - d*f)*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(Sqrt[a]*d^3)","A",15,9,27,0.3333,1,"{6728, 266, 51, 63, 208, 264, 1034, 725, 206}"
71,1,499,0,2.1068004,"\int \frac{x^3}{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[x^3/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{c e x \left(a \left(e^2-2 d f\right)+c d^2\right)+a f \left(a \left(e^2-d f\right)+c d^2\right)}{a f^2 \sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}-\frac{\left(2 a d e f-\left(e-\sqrt{e^2-4 d f}\right) \left(a \left(e^2-d f\right)+c d^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(2 a d e f-\left(\sqrt{e^2-4 d f}+e\right) \left(a \left(e^2-d f\right)+c d^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{e x}{a f^2 \sqrt{a+c x^2}}-\frac{1}{c f \sqrt{a+c x^2}}","\frac{c e x \left(a \left(e^2-2 d f\right)+c d^2\right)+a f \left(a \left(e^2-d f\right)+c d^2\right)}{a f^2 \sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}-\frac{\left(2 a d e f-\left(e-\sqrt{e^2-4 d f}\right) \left(a \left(e^2-d f\right)+c d^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(2 a d e f-\left(\sqrt{e^2-4 d f}+e\right) \left(a \left(e^2-d f\right)+c d^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{e x}{a f^2 \sqrt{a+c x^2}}-\frac{1}{c f \sqrt{a+c x^2}}",1,"-(1/(c*f*Sqrt[a + c*x^2])) - (e*x)/(a*f^2*Sqrt[a + c*x^2]) + (a*f*(c*d^2 + a*(e^2 - d*f)) + c*e*(c*d^2 + a*(e^2 - 2*d*f))*x)/(a*f^2*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2]) - ((2*a*d*e*f - (e - Sqrt[e^2 - 4*d*f])*(c*d^2 + a*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((2*a*d*e*f - (e + Sqrt[e^2 - 4*d*f])*(c*d^2 + a*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",10,7,27,0.2593,1,"{6728, 191, 261, 1017, 1034, 725, 206}"
72,1,410,0,0.7089326,"\int \frac{x^2}{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[x^2/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","-\frac{f \left(2 d (c d-a f)+a e \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(2 d (c d-a f)+a e \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{x (c d-a f)+a e}{\sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}","-\frac{f \left(2 d (c d-a f)+a e \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(2 d (c d-a f)+a e \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{x (c d-a f)+a e}{\sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}",1,"-((a*e + (c*d - a*f)*x)/((a*c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2])) - (f*(2*d*(c*d - a*f) + a*e*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + (f*(2*d*(c*d - a*f) + a*e*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",6,4,27,0.1481,1,"{1063, 1034, 725, 206}"
73,1,411,0,0.8255142,"\int \frac{x}{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[x/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{f \left(2 c d e-\left(e-\sqrt{e^2-4 d f}\right) (c d-a f)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(2 c d e-\left(\sqrt{e^2-4 d f}+e\right) (c d-a f)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{-a f+c d-c e x}{\sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}","\frac{f \left(2 c d e-\left(e-\sqrt{e^2-4 d f}\right) (c d-a f)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(2 c d e-\left(\sqrt{e^2-4 d f}+e\right) (c d-a f)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{-a f+c d-c e x}{\sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}",1,"-((c*d - a*f - c*e*x)/((a*c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2])) + (f*(2*c*d*e - (c*d - a*f)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - (f*(2*c*d*e - (c*d - a*f)*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",6,4,25,0.1600,1,"{1017, 1034, 725, 206}"
74,1,416,0,0.6170498,"\int \frac{1}{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[1/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","-\frac{f \left(2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{c (x (c d-a f)+a e)}{a \sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}","-\frac{f \left(2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{c (x (c d-a f)+a e)}{a \sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}",1,"(c*(a*e + (c*d - a*f)*x))/(a*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2]) - (f*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + (f*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])","A",6,4,24,0.1667,1,"{976, 1034, 725, 206}"
75,1,526,0,2.1830682,"\int \frac{1}{x \left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[1/(x*(a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{a \left(a f^2+c \left(e^2-d f\right)\right)+c^2 d e x}{a d \sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}+\frac{f \left(2 e \left(a f^2+c \left(e^2-2 d f\right)\right)-\left(e-\sqrt{e^2-4 d f}\right) \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(2 e \left(a f^2+c \left(e^2-2 d f\right)\right)-\left(\sqrt{e^2-4 d f}+e\right) \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{1}{a d \sqrt{a+c x^2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{a \left(a f^2+c \left(e^2-d f\right)\right)+c^2 d e x}{a d \sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}+\frac{f \left(2 e \left(a f^2+c \left(e^2-2 d f\right)\right)-\left(e-\sqrt{e^2-4 d f}\right) \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(2 e \left(a f^2+c \left(e^2-2 d f\right)\right)-\left(\sqrt{e^2-4 d f}+e\right) \left(a f^2+c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{1}{a d \sqrt{a+c x^2}}",1,"1/(a*d*Sqrt[a + c*x^2]) - (a*(a*f^2 + c*(e^2 - d*f)) + c^2*d*e*x)/(a*d*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2]) + (f*(2*e*(a*f^2 + c*(e^2 - 2*d*f)) - (e - Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - (f*(2*e*(a*f^2 + c*(e^2 - 2*d*f)) - (e + Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) - ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]/(a^(3/2)*d)","A",12,9,27,0.3333,1,"{6728, 266, 51, 63, 208, 1017, 1034, 725, 206}"
76,1,618,0,2.2802966,"\int \frac{1}{x^2 \left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[1/(x^2*(a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{a^{3/2} d^2}-\frac{2 c x}{a^2 d \sqrt{a+c x^2}}+\frac{c d x \left(a f^2+c \left(e^2-d f\right)\right)+a e \left(a f^2+c \left(e^2-2 d f\right)\right)}{a d^2 \sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}+\frac{f \left(e \left(e-\sqrt{e^2-4 d f}\right) \left(a f^2+c \left(e^2-2 d f\right)\right)-2 \left(a f^2 \left(e^2-d f\right)+c \left(d^2 f^2-3 d e^2 f+e^4\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(e \left(\sqrt{e^2-4 d f}+e\right) \left(a f^2+c \left(e^2-2 d f\right)\right)-2 \left(a f^2 \left(e^2-d f\right)+c \left(d^2 f^2-3 d e^2 f+e^4\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{e}{a d^2 \sqrt{a+c x^2}}-\frac{1}{a d x \sqrt{a+c x^2}}","\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{a^{3/2} d^2}-\frac{2 c x}{a^2 d \sqrt{a+c x^2}}+\frac{c d x \left(a f^2+c \left(e^2-d f\right)\right)+a e \left(a f^2+c \left(e^2-2 d f\right)\right)}{a d^2 \sqrt{a+c x^2} \left((c d-a f)^2+a c e^2\right)}+\frac{f \left(e \left(e-\sqrt{e^2-4 d f}\right) \left(a f^2+c \left(e^2-2 d f\right)\right)-2 \left(a f^2 \left(e^2-d f\right)+c \left(d^2 f^2-3 d e^2 f+e^4\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(e-\sqrt{e^2-4 d f}\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{f \left(e \left(\sqrt{e^2-4 d f}+e\right) \left(a f^2+c \left(e^2-2 d f\right)\right)-2 \left(a f^2 \left(e^2-d f\right)+c \left(d^2 f^2-3 d e^2 f+e^4\right)\right)\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \left((c d-a f)^2+a c e^2\right) \sqrt{2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{e}{a d^2 \sqrt{a+c x^2}}-\frac{1}{a d x \sqrt{a+c x^2}}",1,"-(e/(a*d^2*Sqrt[a + c*x^2])) - 1/(a*d*x*Sqrt[a + c*x^2]) - (2*c*x)/(a^2*d*Sqrt[a + c*x^2]) + (a*e*(a*f^2 + c*(e^2 - 2*d*f)) + c*d*(a*f^2 + c*(e^2 - d*f))*x)/(a*d^2*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2]) + (f*(e*(e - Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - 2*d*f)) - 2*(a*f^2*(e^2 - d*f) + c*(e^4 - 3*d*e^2*f + d^2*f^2)))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - (f*(e*(e + Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - 2*d*f)) - 2*(a*f^2*(e^2 - d*f) + c*(e^4 - 3*d*e^2*f + d^2*f^2)))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) + (e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(a^(3/2)*d^2)","A",14,11,27,0.4074,1,"{6728, 271, 191, 266, 51, 63, 208, 1017, 1034, 725, 206}"
77,1,392,0,0.952051,"\int \frac{x^3 \sqrt{a+b x+c x^2}}{d-f x^2} \, dx","Int[(x^3*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]","-\frac{b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{5/2} f}+\frac{b (b+2 c x) \sqrt{a+b x+c x^2}}{8 c^2 f}-\frac{d \sqrt{a+b x+c x^2}}{f^2}-\frac{b d \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} f^2}-\frac{d \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^{5/2}}+\frac{d \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^{5/2}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{3 c f}","-\frac{b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{5/2} f}+\frac{b (b+2 c x) \sqrt{a+b x+c x^2}}{8 c^2 f}-\frac{d \sqrt{a+b x+c x^2}}{f^2}-\frac{b d \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} f^2}-\frac{d \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^{5/2}}+\frac{d \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^{5/2}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{3 c f}",1,"-((d*Sqrt[a + b*x + c*x^2])/f^2) + (b*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8*c^2*f) - (a + b*x + c*x^2)^(3/2)/(3*c*f) - (b*d*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f^2) - (b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(5/2)*f) - (d*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(5/2)) + (d*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(5/2))","A",15,9,28,0.3214,1,"{6725, 640, 612, 621, 206, 1021, 1078, 1033, 724}"
78,1,316,0,0.4886276,"\int \frac{x^2 \sqrt{a+b x+c x^2}}{d-f x^2} \, dx","Int[(x^2*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]","-\frac{\left(4 a c f+b^2 (-f)+8 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 c^{3/2} f^2}+\frac{\sqrt{d} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^2}+\frac{\sqrt{d} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^2}-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{4 c f}","-\frac{\left(4 a c f+b^2 (-f)+8 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 c^{3/2} f^2}+\frac{\sqrt{d} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^2}+\frac{\sqrt{d} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^2}-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{4 c f}",1,"-((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c*f) - ((8*c^2*d - b^2*f + 4*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2)*f^2) + (Sqrt[d]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^2) + (Sqrt[d]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^2)","A",9,6,28,0.2143,1,"{1071, 1078, 621, 206, 1033, 724}"
79,1,282,0,0.2949891,"\int \frac{x \sqrt{a+b x+c x^2}}{d-f x^2} \, dx","Int[(x*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]","-\frac{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^{3/2}}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{f}-\frac{b \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} f}","-\frac{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^{3/2}}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{f}-\frac{b \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} f}",1,"-(Sqrt[a + b*x + c*x^2]/f) - (b*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f) - (Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(3/2)) + (Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(3/2))","A",9,6,26,0.2308,1,"{1021, 1078, 621, 206, 1033, 724}"
80,1,266,0,0.2264483,"\int \frac{\sqrt{a+b x+c x^2}}{d-f x^2} \, dx","Int[Sqrt[a + b*x + c*x^2]/(d - f*x^2),x]","\frac{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} f}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} f}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{f}","\frac{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} f}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} f}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{f}",1,"-((Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/f) + (Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*f) + (Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*f)","A",8,5,25,0.2000,1,"{990, 621, 206, 1033, 724}"
81,1,267,0,0.7780775,"\int \frac{\sqrt{a+b x+c x^2}}{x \left(d-f x^2\right)} \, dx","Int[Sqrt[a + b*x + c*x^2]/(x*(d - f*x^2)),x]","-\frac{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d \sqrt{f}}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d \sqrt{f}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d}","-\frac{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d \sqrt{f}}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d \sqrt{f}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d}",1,"-((Sqrt[a]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d) - (Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*Sqrt[f]) + (Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*Sqrt[f])","A",17,9,28,0.3214,1,"{6725, 734, 843, 621, 206, 724, 1021, 1078, 1033}"
82,1,286,0,0.7057134,"\int \frac{\sqrt{a+b x+c x^2}}{x^2 \left(d-f x^2\right)} \, dx","Int[Sqrt[a + b*x + c*x^2]/(x^2*(d - f*x^2)),x]","\frac{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^{3/2}}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{a} d}","\frac{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^{3/2}}+\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{a} d}",1,"-(Sqrt[a + b*x + c*x^2]/(d*x)) - (b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[a]*d) + (Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2)) + (Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2))","A",16,8,28,0.2857,1,"{6725, 732, 843, 621, 206, 724, 990, 1033}"
83,1,353,0,0.8788149,"\int \frac{\sqrt{a+b x+c x^2}}{x^3 \left(d-f x^2\right)} \, dx","Int[Sqrt[a + b*x + c*x^2]/(x^3*(d - f*x^2)),x]","\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{8 a^{3/2} d}-\frac{\sqrt{a} f \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d^2}-\frac{\sqrt{f} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^2}+\frac{\sqrt{f} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^2}-\frac{(2 a+b x) \sqrt{a+b x+c x^2}}{4 a d x^2}","\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{8 a^{3/2} d}-\frac{\sqrt{a} f \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d^2}-\frac{\sqrt{f} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^2}+\frac{\sqrt{f} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^2}-\frac{(2 a+b x) \sqrt{a+b x+c x^2}}{4 a d x^2}",1,"-((2*a + b*x)*Sqrt[a + b*x + c*x^2])/(4*a*d*x^2) + ((b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(3/2)*d) - (Sqrt[a]*f*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d^2 - (Sqrt[f]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2) + (Sqrt[f]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2)","A",20,10,28,0.3571,1,"{6725, 720, 724, 206, 734, 843, 621, 1021, 1078, 1033}"
84,1,501,0,1.4121523,"\int \frac{x^3 \left(a+b x+c x^2\right)^{3/2}}{d-f x^2} \, dx","Int[(x^3*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x]","-\frac{d \sqrt{a+b x+c x^2} \left(8 a c f+b^2 f+2 b c f x+8 c^2 d\right)}{8 c f^3}-\frac{b d \left(12 a c f+b^2 (-f)+24 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} f^3}-\frac{3 b \left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^3 f}+\frac{3 b \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{256 c^{7/2} f}+\frac{b (b+2 c x) \left(a+b x+c x^2\right)^{3/2}}{16 c^2 f}-\frac{d \left(a+b x+c x^2\right)^{3/2}}{3 f^2}-\frac{d \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^{7/2}}+\frac{d \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^{7/2}}-\frac{\left(a+b x+c x^2\right)^{5/2}}{5 c f}","-\frac{d \sqrt{a+b x+c x^2} \left(8 a c f+b^2 f+2 b c f x+8 c^2 d\right)}{8 c f^3}-\frac{b d \left(12 a c f+b^2 (-f)+24 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} f^3}-\frac{3 b \left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^3 f}+\frac{3 b \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{256 c^{7/2} f}+\frac{b (b+2 c x) \left(a+b x+c x^2\right)^{3/2}}{16 c^2 f}-\frac{d \left(a+b x+c x^2\right)^{3/2}}{3 f^2}-\frac{d \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^{7/2}}+\frac{d \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^{7/2}}-\frac{\left(a+b x+c x^2\right)^{5/2}}{5 c f}",1,"(-3*b*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^3*f) - (d*(8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^3) - (d*(a + b*x + c*x^2)^(3/2))/(3*f^2) + (b*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(16*c^2*f) - (a + b*x + c*x^2)^(5/2)/(5*c*f) + (3*b*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(7/2)*f) - (b*d*(24*c^2*d - b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*f^3) - (d*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(7/2)) + (d*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(7/2))","A",17,10,28,0.3571,1,"{6725, 640, 612, 621, 206, 1021, 1070, 1078, 1033, 724}"
85,1,417,0,1.0152028,"\int \frac{x^2 \left(a+b x+c x^2\right)^{3/2}}{d-f x^2} \, dx","Int[(x^2*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x]","-\frac{\left(48 c^2 f \left(a^2 f+b^2 d\right)-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{128 c^{5/2} f^3}-\frac{\sqrt{a+b x+c x^2} \left(2 c x \left(12 a c f-3 b^2 f+16 c^2 d\right)+b \left(12 a c f-3 b^2 f+80 c^2 d\right)\right)}{64 c^2 f^2}+\frac{\sqrt{d} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^3}+\frac{\sqrt{d} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^3}-\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/2}}{8 c f}","-\frac{\left(48 c^2 f \left(a^2 f+b^2 d\right)-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{128 c^{5/2} f^3}-\frac{\sqrt{a+b x+c x^2} \left(2 c x \left(12 a c f-3 b^2 f+16 c^2 d\right)+b \left(12 a c f-3 b^2 f+80 c^2 d\right)\right)}{64 c^2 f^2}+\frac{\sqrt{d} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^3}+\frac{\sqrt{d} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^3}-\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/2}}{8 c f}",1,"-((b*(80*c^2*d - 3*b^2*f + 12*a*c*f) + 2*c*(16*c^2*d - 3*b^2*f + 12*a*c*f)*x)*Sqrt[a + b*x + c*x^2])/(64*c^2*f^2) - ((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(8*c*f) - ((128*c^4*d^2 + 192*a*c^3*d*f + 3*b^4*f^2 - 24*a*b^2*c*f^2 + 48*c^2*f*(b^2*d + a^2*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(5/2)*f^3) + (Sqrt[d]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^3) + (Sqrt[d]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^3)","A",10,7,28,0.2500,1,"{1071, 1070, 1078, 621, 206, 1033, 724}"
86,1,349,0,0.5215441,"\int \frac{x \left(a+b x+c x^2\right)^{3/2}}{d-f x^2} \, dx","Int[(x*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x]","-\frac{\sqrt{a+b x+c x^2} \left(8 a c f+b^2 f+2 b c f x+8 c^2 d\right)}{8 c f^2}-\frac{b \left(12 a c f+b^2 (-f)+24 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} f^2}-\frac{\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^{5/2}}+\frac{\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^{5/2}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{3 f}","-\frac{\sqrt{a+b x+c x^2} \left(8 a c f+b^2 f+2 b c f x+8 c^2 d\right)}{8 c f^2}-\frac{b \left(12 a c f+b^2 (-f)+24 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} f^2}-\frac{\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^{5/2}}+\frac{\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^{5/2}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{3 f}",1,"-((8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - (a + b*x + c*x^2)^(3/2)/(3*f) - (b*(24*c^2*d - b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*f^2) - ((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(5/2)) + ((c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(5/2))","A",10,7,26,0.2692,1,"{1021, 1070, 1078, 621, 206, 1033, 724}"
87,1,315,0,0.5170155,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{d-f x^2} \, dx","Int[(a + b*x + c*x^2)^(3/2)/(d - f*x^2),x]","-\frac{\left(12 a c f+3 b^2 f+8 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c} f^2}+\frac{\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} f^2}+\frac{\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} f^2}-\frac{(5 b+2 c x) \sqrt{a+b x+c x^2}}{4 f}","-\frac{\left(12 a c f+3 b^2 f+8 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c} f^2}+\frac{\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} f^2}+\frac{\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} f^2}-\frac{(5 b+2 c x) \sqrt{a+b x+c x^2}}{4 f}",1,"-((5*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*f) - ((8*c^2*d + 3*b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*f^2) + ((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*f^2) + ((c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*f^2)","A",9,6,25,0.2400,1,"{978, 1078, 621, 206, 1033, 724}"
88,1,469,0,1.2741459,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{x \left(d-f x^2\right)} \, dx","Int[(a + b*x + c*x^2)^(3/2)/(x*(d - f*x^2)),x]","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d}-\frac{\sqrt{a+b x+c x^2} \left(8 a c f+b^2 f+2 b c f x+8 c^2 d\right)}{8 c d f}-\frac{b \left(12 a c f+b^2 (-f)+24 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} d f}-\frac{b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} d}+\frac{\left(8 a c+b^2+2 b c x\right) \sqrt{a+b x+c x^2}}{8 c d}-\frac{\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d f^{3/2}}+\frac{\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d f^{3/2}}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d}-\frac{\sqrt{a+b x+c x^2} \left(8 a c f+b^2 f+2 b c f x+8 c^2 d\right)}{8 c d f}-\frac{b \left(12 a c f+b^2 (-f)+24 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} d f}-\frac{b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} d}+\frac{\left(8 a c+b^2+2 b c x\right) \sqrt{a+b x+c x^2}}{8 c d}-\frac{\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d f^{3/2}}+\frac{\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d f^{3/2}}",1,"((b^2 + 8*a*c + 2*b*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*d) - ((8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f*x)*Sqrt[a + b*x + c*x^2])/(8*c*d*f) - (a^(3/2)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d - (b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d) - (b*(24*c^2*d - b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d*f) - ((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*f^(3/2)) + ((c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*f^(3/2))","A",19,11,28,0.3929,1,"{6725, 734, 814, 843, 621, 206, 724, 1021, 1070, 1078, 1033}"
89,1,463,0,1.2011395,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{x^2 \left(d-f x^2\right)} \, dx","Int[(a + b*x + c*x^2)^(3/2)/(x^2*(d - f*x^2)),x]","-\frac{\left(12 a c f+3 b^2 f+8 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c} d f}+\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c} d}+\frac{\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^{3/2} f}+\frac{\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^{3/2} f}-\frac{\left(a+b x+c x^2\right)^{3/2}}{d x}+\frac{3 (3 b+2 c x) \sqrt{a+b x+c x^2}}{4 d}-\frac{(5 b+2 c x) \sqrt{a+b x+c x^2}}{4 d}-\frac{3 \sqrt{a} b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 d}","-\frac{\left(12 a c f+3 b^2 f+8 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c} d f}+\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c} d}+\frac{\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^{3/2} f}+\frac{\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^{3/2} f}-\frac{\left(a+b x+c x^2\right)^{3/2}}{d x}+\frac{3 (3 b+2 c x) \sqrt{a+b x+c x^2}}{4 d}-\frac{(5 b+2 c x) \sqrt{a+b x+c x^2}}{4 d}-\frac{3 \sqrt{a} b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 d}",1,"(3*(3*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d) - ((5*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d) - (a + b*x + c*x^2)^(3/2)/(d*x) - (3*Sqrt[a]*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*d) + (3*(b^2 + 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*d) - ((8*c^2*d + 3*b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*d*f) + ((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2)*f) + ((c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2)*f)","A",18,10,28,0.3571,1,"{6725, 732, 814, 843, 621, 206, 724, 978, 1078, 1033}"
90,1,614,0,1.4360005,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{x^3 \left(d-f x^2\right)} \, dx","Int[(a + b*x + c*x^2)^(3/2)/(x^3*(d - f*x^2)),x]","-\frac{a^{3/2} f \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d^2}-\frac{\sqrt{a+b x+c x^2} \left(8 a c f+b^2 f+2 b c f x+8 c^2 d\right)}{8 c d^2}-\frac{b f \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} d^2}-\frac{b \left(12 a c f+b^2 (-f)+24 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} d^2}+\frac{f \left(8 a c+b^2+2 b c x\right) \sqrt{a+b x+c x^2}}{8 c d^2}-\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{a} d}-\frac{\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^2 \sqrt{f}}+\frac{\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^2 \sqrt{f}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{2 d x^2}-\frac{3 (b-2 c x) \sqrt{a+b x+c x^2}}{4 d x}+\frac{3 b \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 d}","-\frac{a^{3/2} f \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d^2}-\frac{\sqrt{a+b x+c x^2} \left(8 a c f+b^2 f+2 b c f x+8 c^2 d\right)}{8 c d^2}-\frac{b f \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} d^2}-\frac{b \left(12 a c f+b^2 (-f)+24 c^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} d^2}+\frac{f \left(8 a c+b^2+2 b c x\right) \sqrt{a+b x+c x^2}}{8 c d^2}-\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{a} d}-\frac{\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^2 \sqrt{f}}+\frac{\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^2 \sqrt{f}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{2 d x^2}-\frac{3 (b-2 c x) \sqrt{a+b x+c x^2}}{4 d x}+\frac{3 b \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 d}",1,"(-3*(b - 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d*x) + (f*(b^2 + 8*a*c + 2*b*c*x)*Sqrt[a + b*x + c*x^2])/(8*c*d^2) - ((8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f*x)*Sqrt[a + b*x + c*x^2])/(8*c*d^2) - (a + b*x + c*x^2)^(3/2)/(2*d*x^2) - (3*(b^2 + 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[a]*d) - (a^(3/2)*f*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d^2 + (3*b*Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*d) - (b*(b^2 - 12*a*c)*f*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^2) - (b*(24*c^2*d - b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^2) - ((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2*Sqrt[f]) + ((c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2*Sqrt[f])","A",26,13,28,0.4643,1,"{6725, 732, 812, 843, 621, 206, 724, 734, 814, 1021, 1070, 1078, 1033}"
91,1,189,0,0.284912,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{1-x^2} \, dx","Int[(a + b*x + c*x^2)^(3/2)/(1 - x^2),x]","-\frac{\left(12 a c+3 b^2+8 c^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c}}-\frac{1}{4} (5 b+2 c x) \sqrt{a+b x+c x^2}-\frac{1}{2} (a-b+c)^{3/2} \tanh ^{-1}\left(\frac{2 a+x (b-2 c)-b}{2 \sqrt{a-b+c} \sqrt{a+b x+c x^2}}\right)+\frac{1}{2} (a+b+c)^{3/2} \tanh ^{-1}\left(\frac{2 a+x (b+2 c)+b}{2 \sqrt{a+b+c} \sqrt{a+b x+c x^2}}\right)","-\frac{\left(12 a c+3 b^2+8 c^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c}}-\frac{1}{4} (5 b+2 c x) \sqrt{a+b x+c x^2}-\frac{1}{2} (a-b+c)^{3/2} \tanh ^{-1}\left(\frac{2 a+x (b-2 c)-b}{2 \sqrt{a-b+c} \sqrt{a+b x+c x^2}}\right)+\frac{1}{2} (a+b+c)^{3/2} \tanh ^{-1}\left(\frac{2 a+x (b+2 c)+b}{2 \sqrt{a+b+c} \sqrt{a+b x+c x^2}}\right)",1,"-((5*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/4 - ((a - b + c)^(3/2)*ArcTanh[(2*a - b + (b - 2*c)*x)/(2*Sqrt[a - b + c]*Sqrt[a + b*x + c*x^2])])/2 - ((3*b^2 + 12*a*c + 8*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]) + ((a + b + c)^(3/2)*ArcTanh[(2*a + b + (b + 2*c)*x)/(2*Sqrt[a + b + c]*Sqrt[a + b*x + c*x^2])])/2","A",9,6,24,0.2500,1,"{978, 1078, 621, 206, 1033, 724}"
92,1,75,0,0.0582723,"\int \frac{\sqrt{-1-x+x^2}}{1-x^2} \, dx","Int[Sqrt[-1 - x + x^2]/(1 - x^2),x]","-\frac{1}{2} \tan ^{-1}\left(\frac{3-x}{2 \sqrt{x^2-x-1}}\right)+\tanh ^{-1}\left(\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{3 x+1}{2 \sqrt{x^2-x-1}}\right)","-\frac{1}{2} \tan ^{-1}\left(\frac{3-x}{2 \sqrt{x^2-x-1}}\right)+\tanh ^{-1}\left(\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{3 x+1}{2 \sqrt{x^2-x-1}}\right)",1,"-ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])]/2 + ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])] + ArcTanh[(1 + 3*x)/(2*Sqrt[-1 - x + x^2])]/2","A",8,6,22,0.2727,1,"{990, 621, 206, 1033, 724, 204}"
93,1,130,0,0.1600663,"\int \frac{\left(x+x^2\right)^{3/2}}{1+x^2} \, dx","Int[(x + x^2)^(3/2)/(1 + x^2),x]","\frac{1}{4} \sqrt{x^2+x} (2 x+5)+\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{-x+\sqrt{2}+1}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{x^2+x}}\right)-\sqrt{\sqrt{2}-1} \tanh ^{-1}\left(\frac{-x-\sqrt{2}+1}{\sqrt{2 \left(\sqrt{2}-1\right)} \sqrt{x^2+x}}\right)-\frac{5}{4} \tanh ^{-1}\left(\frac{x}{\sqrt{x^2+x}}\right)","\frac{1}{4} \sqrt{x^2+x} (2 x+5)+\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{-x+\sqrt{2}+1}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{x^2+x}}\right)-\sqrt{\sqrt{2}-1} \tanh ^{-1}\left(\frac{-x-\sqrt{2}+1}{\sqrt{2 \left(\sqrt{2}-1\right)} \sqrt{x^2+x}}\right)-\frac{5}{4} \tanh ^{-1}\left(\frac{x}{\sqrt{x^2+x}}\right)",1,"((5 + 2*x)*Sqrt[x + x^2])/4 + Sqrt[1 + Sqrt[2]]*ArcTan[(1 + Sqrt[2] - x)/(Sqrt[2*(1 + Sqrt[2])]*Sqrt[x + x^2])] - Sqrt[-1 + Sqrt[2]]*ArcTanh[(1 - Sqrt[2] - x)/(Sqrt[2*(-1 + Sqrt[2])]*Sqrt[x + x^2])] - (5*ArcTanh[x/Sqrt[x + x^2]])/4","A",10,9,17,0.5294,1,"{978, 1078, 620, 206, 12, 1036, 1030, 207, 203}"
94,1,369,0,0.8094075,"\int \frac{x^4}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Int[x^4/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","-\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 c^{5/2} f}+\frac{3 b \sqrt{a+b x+c x^2}}{4 c^2 f}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^2 \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^2 \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{d \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} f^2}-\frac{x \sqrt{a+b x+c x^2}}{2 c f}","-\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 c^{5/2} f}+\frac{3 b \sqrt{a+b x+c x^2}}{4 c^2 f}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^2 \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^2 \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{d \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} f^2}-\frac{x \sqrt{a+b x+c x^2}}{2 c f}",1,"(3*b*Sqrt[a + b*x + c*x^2])/(4*c^2*f) - (x*Sqrt[a + b*x + c*x^2])/(2*c*f) - (d*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f^2) - ((3*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2)*f) + (d^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (d^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])","A",13,7,28,0.2500,1,"{6725, 621, 206, 742, 640, 984, 724}"
95,1,287,0,0.6325427,"\int \frac{x^3}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Int[x^3/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{b \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 c^{3/2} f}-\frac{d \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^{3/2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{d \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^{3/2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\sqrt{a+b x+c x^2}}{c f}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 c^{3/2} f}-\frac{d \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f^{3/2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{d \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f^{3/2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\sqrt{a+b x+c x^2}}{c f}",1,"-(Sqrt[a + b*x + c*x^2]/(c*f)) + (b*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(3/2)*f) - (d*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(3/2)*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (d*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(3/2)*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])","A",10,6,28,0.2143,1,"{6725, 640, 621, 206, 1033, 724}"
96,1,266,0,0.2164247,"\int \frac{x^2}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Int[x^2/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{\sqrt{d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\sqrt{d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} f}","\frac{\sqrt{d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\sqrt{d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} f}",1,"-(ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(Sqrt[c]*f)) + (Sqrt[d]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (Sqrt[d]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])","A",8,5,28,0.1786,1,"{1079, 621, 206, 984, 724}"
97,1,220,0,0.1296195,"\int \frac{x}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Int[x/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{\tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{f} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{f} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}","\frac{\tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{f} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{f} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}",1,"-ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])]/(2*Sqrt[f]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])]/(2*Sqrt[f]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])","A",5,3,26,0.1154,1,"{1033, 724, 206}"
98,1,220,0,0.117588,"\int \frac{1}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Int[1/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{\tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}","\frac{\tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}",1,"ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])]/(2*Sqrt[d]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])]/(2*Sqrt[d]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])","A",5,3,25,0.1200,1,"{984, 724, 206}"
99,1,267,0,0.6634893,"\int \frac{1}{x \sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Int[1/(x*Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","-\frac{\sqrt{f} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\sqrt{f} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} d}","-\frac{\sqrt{f} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\sqrt{f} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} d}",1,"-(ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])]/(Sqrt[a]*d)) - (Sqrt[f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (Sqrt[f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])","A",9,4,28,0.1429,1,"{6725, 724, 206, 1033}"
100,1,291,0,0.6567503,"\int \frac{1}{x^2 \sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Int[1/(x^2*Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 a^{3/2} d}+\frac{f \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^{3/2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{f \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^{3/2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\sqrt{a+b x+c x^2}}{a d x}","\frac{b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 a^{3/2} d}+\frac{f \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^{3/2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{f \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^{3/2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\sqrt{a+b x+c x^2}}{a d x}",1,"-(Sqrt[a + b*x + c*x^2]/(a*d*x)) + (b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)*d) + (f*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2)*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (f*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2)*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])","A",10,5,28,0.1786,1,"{6725, 730, 724, 206, 984}"
101,1,376,0,0.7341048,"\int \frac{1}{x^3 \sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Int[1/(x^3*Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","-\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{8 a^{5/2} d}+\frac{3 b \sqrt{a+b x+c x^2}}{4 a^2 d x}-\frac{f^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^2 \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{f^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^2 \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{f \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} d^2}-\frac{\sqrt{a+b x+c x^2}}{2 a d x^2}","-\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{8 a^{5/2} d}+\frac{3 b \sqrt{a+b x+c x^2}}{4 a^2 d x}-\frac{f^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^2 \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{f^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^2 \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{f \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} d^2}-\frac{\sqrt{a+b x+c x^2}}{2 a d x^2}",1,"-Sqrt[a + b*x + c*x^2]/(2*a*d*x^2) + (3*b*Sqrt[a + b*x + c*x^2])/(4*a^2*d*x) - ((3*b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(5/2)*d) - (f*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a]*d^2) - (f^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + (f^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])","A",13,6,28,0.2143,1,"{6725, 744, 806, 724, 206, 1033}"
102,1,466,0,1.3465556,"\int \frac{x^4}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Int[x^4/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","-\frac{2 d^2 \left(b \left(b^2 f-c (3 a f+c d)\right)-c x \left(2 a c f+b^2 (-f)+2 c^2 d\right)\right)}{f^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}+\frac{2 d (b+2 c x)}{f^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{2 b \sqrt{a+b x+c x^2}}{c f \left(b^2-4 a c\right)}-\frac{2 x (2 a+b x)}{f \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{c^{3/2} f}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}","-\frac{2 d^2 \left(b \left(b^2 f-c (3 a f+c d)\right)-c x \left(2 a c f+b^2 (-f)+2 c^2 d\right)\right)}{f^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}+\frac{2 d (b+2 c x)}{f^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{2 b \sqrt{a+b x+c x^2}}{c f \left(b^2-4 a c\right)}-\frac{2 x (2 a+b x)}{f \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{c^{3/2} f}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 f \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 f \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}",1,"(-2*x*(2*a + b*x))/((b^2 - 4*a*c)*f*Sqrt[a + b*x + c*x^2]) + (2*d*(b + 2*c*x))/((b^2 - 4*a*c)*f^2*Sqrt[a + b*x + c*x^2]) - (2*d^2*(b*(b^2*f - c*(c*d + 3*a*f)) - c*(2*c^2*d - b^2*f + 2*a*c*f)*x))/((b^2 - 4*a*c)*f^2*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) + (2*b*Sqrt[a + b*x + c*x^2])/(c*(b^2 - 4*a*c)*f) - ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(c^(3/2)*f) + (d^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (d^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))","A",13,9,28,0.3214,1,"{6725, 613, 738, 640, 621, 206, 975, 1033, 724}"
103,1,341,0,1.0411511,"\int \frac{x^3}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Int[x^3/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","-\frac{2 d \left(a \left(2 a c f+b^2 (-f)+2 c^2 d\right)+b c x (c d-a f)\right)}{f \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}-\frac{2 (2 a+b x)}{f \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{d \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{f} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{d \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{f} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}","-\frac{2 d \left(a \left(2 a c f+b^2 (-f)+2 c^2 d\right)+b c x (c d-a f)\right)}{f \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}-\frac{2 (2 a+b x)}{f \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{d \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{f} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{d \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{f} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}",1,"(-2*(2*a + b*x))/((b^2 - 4*a*c)*f*Sqrt[a + b*x + c*x^2]) - (2*d*(a*(2*c^2*d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*f*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) - (d*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[f]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (d*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[f]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))","A",9,6,28,0.2143,1,"{6725, 636, 1018, 1033, 724, 206}"
104,1,297,0,0.4538713,"\int \frac{x^2}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Int[x^2/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","\frac{2 \left(c x \left(b^2 d-2 a (a f+c d)\right)+a b (c d-a f)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}+\frac{\sqrt{d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{\sqrt{d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}","\frac{2 \left(c x \left(b^2 d-2 a (a f+c d)\right)+a b (c d-a f)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}+\frac{\sqrt{d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{\sqrt{d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}",1,"(2*(a*b*(c*d - a*f) + c*(b^2*d - 2*a*(c*d + a*f))*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) + (Sqrt[d]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (Sqrt[d]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))","A",6,4,28,0.1429,1,"{1065, 1033, 724, 206}"
105,1,299,0,0.4003567,"\int \frac{x}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Int[x/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","-\frac{2 \left(a \left(2 a c f+b^2 (-f)+2 c^2 d\right)+b c x (c d-a f)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}-\frac{\sqrt{f} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{\sqrt{f} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}","-\frac{2 \left(a \left(2 a c f+b^2 (-f)+2 c^2 d\right)+b c x (c d-a f)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}-\frac{\sqrt{f} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{\sqrt{f} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}",1,"(-2*(a*(2*c^2*d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) - (Sqrt[f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (Sqrt[f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))","A",6,4,26,0.1538,1,"{1018, 1033, 724, 206}"
106,1,310,0,0.4106104,"\int \frac{1}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Int[1/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","-\frac{2 \left(b \left(b^2 f-c (3 a f+c d)\right)-c x \left(2 a c f+b^2 (-f)+2 c^2 d\right)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}+\frac{f \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{f \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}","-\frac{2 \left(b \left(b^2 f-c (3 a f+c d)\right)-c x \left(2 a c f+b^2 (-f)+2 c^2 d\right)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}+\frac{f \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 \sqrt{d} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{f \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \sqrt{d} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}",1,"(-2*(b*(b^2*f - c*(c*d + 3*a*f)) - c*(2*c^2*d - b^2*f + 2*a*c*f)*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) + (f*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (f*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))","A",6,4,25,0.1600,1,"{975, 1033, 724, 206}"
107,1,394,0,1.1818092,"\int \frac{1}{x \left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Int[1/(x*(a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{a^{3/2} d}-\frac{2 f \left(a \left(2 a c f+b^2 (-f)+2 c^2 d\right)+b c x (c d-a f)\right)}{d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}+\frac{2 \left(-2 a c+b^2+b c x\right)}{a d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{f^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{f^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{a^{3/2} d}-\frac{2 f \left(a \left(2 a c f+b^2 (-f)+2 c^2 d\right)+b c x (c d-a f)\right)}{d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}+\frac{2 \left(-2 a c+b^2+b c x\right)}{a d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{f^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{f^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}",1,"(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) - (2*f*(a*(2*c^2*d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*d*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) - ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])]/(a^(3/2)*d) - (f^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (f^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))","A",12,7,28,0.2500,1,"{6725, 740, 12, 724, 206, 1018, 1033}"
108,1,454,0,1.1933543,"\int \frac{1}{x^2 \left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Int[1/(x^2*(a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","-\frac{\left(3 b^2-8 a c\right) \sqrt{a+b x+c x^2}}{a^2 d x \left(b^2-4 a c\right)}+\frac{3 b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 a^{5/2} d}-\frac{2 f \left(b \left(b^2 f-c (3 a f+c d)\right)-c x \left(2 a c f+b^2 (-f)+2 c^2 d\right)\right)}{d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}+\frac{2 \left(-2 a c+b^2+b c x\right)}{a d x \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{f^2 \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^{3/2} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{f^2 \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^{3/2} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}","-\frac{\left(3 b^2-8 a c\right) \sqrt{a+b x+c x^2}}{a^2 d x \left(b^2-4 a c\right)}+\frac{3 b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 a^{5/2} d}-\frac{2 f \left(b \left(b^2 f-c (3 a f+c d)\right)-c x \left(2 a c f+b^2 (-f)+2 c^2 d\right)\right)}{d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(b^2 d f-(a f+c d)^2\right)}+\frac{2 \left(-2 a c+b^2+b c x\right)}{a d x \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{f^2 \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+x \left(2 c \sqrt{d}-b \sqrt{f}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{2 d^{3/2} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{f^2 \tanh ^{-1}\left(\frac{2 a \sqrt{f}+x \left(b \sqrt{f}+2 c \sqrt{d}\right)+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^{3/2} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}",1,"(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*x*Sqrt[a + b*x + c*x^2]) - (2*f*(b*(b^2*f - c*(c*d + 3*a*f)) - c*(2*c^2*d - b^2*f + 2*a*c*f)*x))/((b^2 - 4*a*c)*d*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) - ((3*b^2 - 8*a*c)*Sqrt[a + b*x + c*x^2])/(a^2*(b^2 - 4*a*c)*d*x) + (3*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(5/2)*d) + (f^2*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2)*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (f^2*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2)*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2))","A",12,7,28,0.2500,1,"{6725, 740, 806, 724, 206, 975, 1033}"
109,1,761,0,3.1353465,"\int \frac{x^2 \sqrt{a+b x+c x^2}}{d+e x+f x^2} \, dx","Int[(x^2*Sqrt[a + b*x + c*x^2])/(d + e*x + f*x^2),x]","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(4 c f (b e-a f)+b^2 f^2-8 c^2 \left(e^2-d f\right)\right)}{8 c^{3/2} f^3}-\frac{\left(f \left(a f \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)-b \left(-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)\right)+c \left(2 d^2 f^2-e^3 \sqrt{e^2-4 d f}-4 d e^2 f+2 d e f \sqrt{e^2-4 d f}+e^4\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(f \left(a f \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)-b \left(e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)\right)+c \left(2 d^2 f^2+e^3 \sqrt{e^2-4 d f}-4 d e^2 f-2 d e f \sqrt{e^2-4 d f}+e^4\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{a+b x+c x^2} (-b f+4 c e-2 c f x)}{4 c f^2}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(4 c f (b e-a f)+b^2 f^2-8 c^2 \left(e^2-d f\right)\right)}{8 c^{3/2} f^3}-\frac{\left(f \left(a f \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)-b \left(-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)\right)+c \left(2 d^2 f^2-e^3 \sqrt{e^2-4 d f}-4 d e^2 f+2 d e f \sqrt{e^2-4 d f}+e^4\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(f \left(a f \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)-b \left(e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right)\right)+c \left(2 d^2 f^2+e^3 \sqrt{e^2-4 d f}-4 d e^2 f-2 d e f \sqrt{e^2-4 d f}+e^4\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{a+b x+c x^2} (-b f+4 c e-2 c f x)}{4 c f^2}",1,"-((4*c*e - b*f - 2*c*f*x)*Sqrt[a + b*x + c*x^2])/(4*c*f^2) - ((b^2*f^2 + 4*c*f*(b*e - a*f) - 8*c^2*(e^2 - d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2)*f^3) - ((c*(e^4 - 4*d*e^2*f + 2*d^2*f^2 - e^3*Sqrt[e^2 - 4*d*f] + 2*d*e*f*Sqrt[e^2 - 4*d*f]) + f*(a*f*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - b*(e^3 - 3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4*d*f])))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))]) + ((c*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + e^3*Sqrt[e^2 - 4*d*f] - 2*d*e*f*Sqrt[e^2 - 4*d*f]) + f*(a*f*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) - b*(e^3 - 3*d*e*f + e^2*Sqrt[e^2 - 4*d*f] - d*f*Sqrt[e^2 - 4*d*f])))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))])","A",9,6,30,0.2000,1,"{1067, 1076, 621, 206, 1032, 724}"
110,1,549,0,7.0278336,"\int \frac{x \sqrt{a+b x+c x^2}}{d+e x+f x^2} \, dx","Int[(x*Sqrt[a + b*x + c*x^2])/(d + e*x + f*x^2),x]","-\frac{\left(\left(e-\sqrt{e^2-4 d f}\right) \left(f (b e-a f)-c \left(e^2-d f\right)\right)+2 d f (c e-b f)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\left(\left(\sqrt{e^2-4 d f}+e\right) \left(f (b e-a f)-c \left(e^2-d f\right)\right)+2 d f (c e-b f)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{(2 c e-b f) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} f^2}+\frac{\sqrt{a+b x+c x^2}}{f}","-\frac{\left(\left(e-\sqrt{e^2-4 d f}\right) \left(f (b e-a f)-c \left(e^2-d f\right)\right)+2 d f (c e-b f)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\left(\left(\sqrt{e^2-4 d f}+e\right) \left(f (b e-a f)-c \left(e^2-d f\right)\right)+2 d f (c e-b f)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{(2 c e-b f) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} f^2}+\frac{\sqrt{a+b x+c x^2}}{f}",1,"Sqrt[a + b*x + c*x^2]/f - ((2*c*e - b*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f^2) - ((2*d*f*(c*e - b*f) + (e - Sqrt[e^2 - 4*d*f])*(f*(b*e - a*f) - c*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) + ((2*d*f*(c*e - b*f) + (e + Sqrt[e^2 - 4*d*f])*(f*(b*e - a*f) - c*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",9,6,28,0.2143,1,"{1019, 1076, 621, 206, 1032, 724}"
111,1,431,0,0.6496436,"\int \frac{\sqrt{a+b x+c x^2}}{d+e x+f x^2} \, dx","Int[Sqrt[a + b*x + c*x^2]/(d + e*x + f*x^2),x]","-\frac{\sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f}}+\frac{\sqrt{f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{f}","-\frac{\sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f}}+\frac{\sqrt{f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{f}",1,"(Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/f - (Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))]*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f]) + (Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))]*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f])","A",8,5,27,0.1852,1,"{989, 621, 206, 1032, 724}"
112,1,521,0,3.7014379,"\int \frac{\sqrt{a+b x+c x^2}}{x \left(d+e x+f x^2\right)} \, dx","Int[Sqrt[a + b*x + c*x^2]/(x*(d + e*x + f*x^2)),x]","-\frac{\left(-a f \left(\sqrt{e^2-4 d f}+e\right)+2 b d f-c d \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\left(-a f \left(e-\sqrt{e^2-4 d f}\right)+2 b d f-c d \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d}","\frac{\left(c d \left(e-\sqrt{e^2-4 d f}\right)-f \left(2 b d-a \left(\sqrt{e^2-4 d f}+e\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\left(c d \left(\sqrt{e^2-4 d f}+e\right)-f \left(2 b d-a \left(e-\sqrt{e^2-4 d f}\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d}",1,"-((Sqrt[a]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d) - ((2*b*d*f - c*d*(e - Sqrt[e^2 - 4*d*f]) - a*f*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))]) + ((2*b*d*f - a*f*(e - Sqrt[e^2 - 4*d*f]) - c*d*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))])","A",17,9,30,0.3000,1,"{6728, 734, 843, 621, 206, 724, 1019, 1076, 1032}"
113,1,736,0,3.4755407,"\int \frac{\sqrt{a+b x+c x^2}}{x^2 \left(d+e x+f x^2\right)} \, dx","Int[Sqrt[a + b*x + c*x^2]/(x^2*(d + e*x + f*x^2)),x]","-\frac{f \left(a \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)-b d \left(\sqrt{e^2-4 d f}+e\right)+2 c d^2\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(a \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)-b d \left(e-\sqrt{e^2-4 d f}\right)+2 c d^2\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\sqrt{a} e \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d^2}-\frac{b e \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} d^2}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} d^2}-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{a} d}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{d}","-\frac{f \left(a \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)-b d \left(\sqrt{e^2-4 d f}+e\right)+2 c d^2\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(a \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)-b d \left(e-\sqrt{e^2-4 d f}\right)+2 c d^2\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{\sqrt{a} e \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{d^2}-\frac{b e \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} d^2}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} d^2}-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{a} d}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{d}",1,"-(Sqrt[a + b*x + c*x^2]/(d*x)) - (b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[a]*d) + (Sqrt[a]*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d^2 + (Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/d - (b*e*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*d^2) - ((2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*d^2) - (f*(2*c*d^2 - b*d*(e + Sqrt[e^2 - 4*d*f]) + a*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))]) + (f*(2*c*d^2 - b*d*(e - Sqrt[e^2 - 4*d*f]) + a*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))])","A",23,10,30,0.3333,1,"{6728, 732, 843, 621, 206, 724, 734, 1019, 1076, 1032}"
114,1,545,0,3.7221266,"\int \frac{x^3}{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[x^3/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{b \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 c^{3/2} f}-\frac{\left(2 d e f-\left(e^2-d f\right) \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\left(2 d e f-\left(e^2-d f\right) \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{e \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} f^2}+\frac{\sqrt{a+b x+c x^2}}{c f}","-\frac{b \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 c^{3/2} f}-\frac{\left(2 d e f-\left(e^2-d f\right) \left(e-\sqrt{e^2-4 d f}\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\left(2 d e f-\left(e^2-d f\right) \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{e \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} f^2}+\frac{\sqrt{a+b x+c x^2}}{c f}",1,"Sqrt[a + b*x + c*x^2]/(c*f) - (e*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f^2) - (b*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(3/2)*f) - ((2*d*e*f - (e^2 - d*f)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) + ((2*d*e*f - (e^2 - d*f)*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",12,6,30,0.2000,1,"{6728, 621, 206, 640, 1032, 724}"
115,1,463,0,3.4356442,"\int \frac{x^2}{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[x^2/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{\left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left(2 d f-e \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} f}","-\frac{\left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left(2 d f-e \left(\sqrt{e^2-4 d f}+e\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} f \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} f}",1,"ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(Sqrt[c]*f) - ((e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - ((2*d*f - e*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",8,5,30,0.1667,1,"{1077, 621, 206, 1032, 724}"
116,1,402,0,0.9628642,"\int \frac{x}{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[x/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\left(e-\sqrt{e^2-4 d f}\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}","\frac{\left(e-\sqrt{e^2-4 d f}\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}",1,"((e - Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - ((e + Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",5,3,28,0.1071,1,"{1032, 724, 206}"
117,1,374,0,0.313594,"\int \frac{1}{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[1/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\sqrt{2} f \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\sqrt{2} f \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}","\frac{\sqrt{2} f \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\sqrt{2} f \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}",1,"-((Sqrt[2]*f*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]])) + (Sqrt[2]*f*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",5,3,27,0.1111,1,"{983, 724, 206}"
118,1,451,0,2.6316409,"\int \frac{1}{x \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[1/(x*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","\frac{f \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{f \left(e-\sqrt{e^2-4 d f}\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} d}","\frac{f \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{f \left(e-\sqrt{e^2-4 d f}\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} d}",1,"-(ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])]/(Sqrt[a]*d)) + (f*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - (f*(e - Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",9,4,30,0.1333,1,"{6728, 724, 206, 1032}"
119,1,543,0,4.5917632,"\int \frac{1}{x^2 \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[1/(x^2*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","\frac{b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 a^{3/2} d}-\frac{f \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{e \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} d^2}-\frac{\sqrt{a+b x+c x^2}}{a d x}","\frac{b \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 a^{3/2} d}-\frac{f \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{e \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} d^2}-\frac{\sqrt{a+b x+c x^2}}{a d x}",1,"-(Sqrt[a + b*x + c*x^2]/(a*d*x)) + (b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)*d) + (e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a]*d^2) - (f*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))]) + (f*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",12,5,30,0.1667,1,"{6728, 730, 724, 206, 1032}"
120,1,679,0,11.2259121,"\int \frac{1}{x^3 \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Int[1/(x^3*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{8 a^{5/2} d}-\frac{b e \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 a^{3/2} d^2}+\frac{3 b \sqrt{a+b x+c x^2}}{4 a^2 d x}+\frac{f \left(-\left(e^2-d f\right) \left(e-\sqrt{e^2-4 d f}\right)-4 d e f+2 e^3\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{f \left(-\left(e^2-d f\right) \left(\sqrt{e^2-4 d f}+e\right)-4 d e f+2 e^3\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left(e^2-d f\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} d^3}+\frac{e \sqrt{a+b x+c x^2}}{a d^2 x}-\frac{\sqrt{a+b x+c x^2}}{2 a d x^2}","-\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{8 a^{5/2} d}-\frac{b e \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 a^{3/2} d^2}+\frac{3 b \sqrt{a+b x+c x^2}}{4 a^2 d x}+\frac{f \left(-\left(e^2-d f\right) \left(e-\sqrt{e^2-4 d f}\right)-4 d e f+2 e^3\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{f \left(-\left(e^2-d f\right) \left(\sqrt{e^2-4 d f}+e\right)-4 d e f+2 e^3\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left(e^2-d f\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a} d^3}+\frac{e \sqrt{a+b x+c x^2}}{a d^2 x}-\frac{\sqrt{a+b x+c x^2}}{2 a d x^2}",1,"-Sqrt[a + b*x + c*x^2]/(2*a*d*x^2) + (3*b*Sqrt[a + b*x + c*x^2])/(4*a^2*d*x) + (e*Sqrt[a + b*x + c*x^2])/(a*d^2*x) - ((3*b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(5/2)*d) - (b*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)*d^2) - ((e^2 - d*f)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a]*d^3) + (f*(2*e^3 - 4*d*e*f - (e^2 - d*f)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - (f*(2*e^3 - 4*d*e*f - (e^2 - d*f)*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",16,7,30,0.2333,1,"{6728, 744, 806, 724, 206, 730, 1032}"
121,1,779,0,14.1698151,"\int \frac{x^3}{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[x^3/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{2 \left(c x \left(\left(e^2-d f\right) (a b f-2 a c e+b c d)-d e \left(-c (2 a f+b e)+b^2 f+2 c^2 d\right)\right)-\left(a d f-a e^2+b d e\right) \left(-c (2 a f+b e)+b^2 f+2 c^2 d\right)+c d e (a b f-2 a c e+b c d)\right)}{f^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left((c d-a f)^2-(b d-a e) (c e-b f)\right)}+\frac{2 e (b+2 c x)}{f^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{2 (2 a+b x)}{f \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{\left(\left(e-\sqrt{e^2-4 d f}\right) \left(a \left(e^2-d f\right)-b d e+c d^2\right)+2 d f (b d-a e)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left(\left(\sqrt{e^2-4 d f}+e\right) \left(a \left(e^2-d f\right)-b d e+c d^2\right)+2 d f (b d-a e)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}","\frac{2 \left(c x \left(\left(e^2-d f\right) (a b f-2 a c e+b c d)-d e \left(-c (2 a f+b e)+b^2 f+2 c^2 d\right)\right)-\left(a d f-a e^2+b d e\right) \left(-c (2 a f+b e)+b^2 f+2 c^2 d\right)+c d e (a b f-2 a c e+b c d)\right)}{f^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left((c d-a f)^2-(b d-a e) (c e-b f)\right)}+\frac{2 e (b+2 c x)}{f^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{2 (2 a+b x)}{f \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{\left(\left(e-\sqrt{e^2-4 d f}\right) \left(a \left(e^2-d f\right)-b d e+c d^2\right)+2 d f (b d-a e)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left(\left(\sqrt{e^2-4 d f}+e\right) \left(a \left(e^2-d f\right)-b d e+c d^2\right)+2 d f (b d-a e)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}",1,"(2*(2*a + b*x))/((b^2 - 4*a*c)*f*Sqrt[a + b*x + c*x^2]) + (2*e*(b + 2*c*x))/((b^2 - 4*a*c)*f^2*Sqrt[a + b*x + c*x^2]) + (2*(c*d*e*(b*c*d - 2*a*c*e + a*b*f) - (b*d*e - a*e^2 + a*d*f)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*((b*c*d - 2*a*c*e + a*b*f)*(e^2 - d*f) - d*e*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*x))/((b^2 - 4*a*c)*f^2*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[a + b*x + c*x^2]) + ((2*d*(b*d - a*e)*f + (e - Sqrt[e^2 - 4*d*f])*(c*d^2 - b*d*e + a*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - ((2*d*(b*d - a*e)*f + (e + Sqrt[e^2 - 4*d*f])*(c*d^2 - b*d*e + a*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",10,7,30,0.2333,1,"{6728, 613, 636, 1016, 1032, 724, 206}"
122,1,609,0,5.8416833,"\int \frac{x^2}{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[x^2/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","-\frac{2 \left(c x \left(-a b e-2 a (c d-a f)+b^2 d\right)+a (a b f-2 a c e+b c d)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left((c d-a f)^2-(b d-a e) (c e-b f)\right)}-\frac{f \left(2 d (c d-a f)-\left(e-\sqrt{e^2-4 d f}\right) (b d-a e)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{f \left(2 d (c d-a f)-\left(\sqrt{e^2-4 d f}+e\right) (b d-a e)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}","-\frac{2 \left(c x \left(-a b e-2 a (c d-a f)+b^2 d\right)+a (a b f-2 a c e+b c d)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left((c d-a f)^2-(b d-a e) (c e-b f)\right)}-\frac{f \left(2 d (c d-a f)-\left(e-\sqrt{e^2-4 d f}\right) (b d-a e)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{f \left(2 d (c d-a f)-\left(\sqrt{e^2-4 d f}+e\right) (b d-a e)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}",1,"(-2*(a*(b*c*d - 2*a*c*e + a*b*f) + c*(b^2*d - a*b*e - 2*a*(c*d - a*f))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[a + b*x + c*x^2]) - (f*(2*d*(c*d - a*f) - (b*d - a*e)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) + (f*(2*d*(c*d - a*f) - (b*d - a*e)*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",6,4,30,0.1333,1,"{1061, 1032, 724, 206}"
123,1,609,0,5.6425884,"\int \frac{x}{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[x/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{2 \left(a \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)+c x (a b f-2 a c e+b c d)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left((c d-a f)^2-(b d-a e) (c e-b f)\right)}+\frac{f \left(2 d (c e-b f)-\left(e-\sqrt{e^2-4 d f}\right) (c d-a f)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{f \left(2 d (c e-b f)-\left(\sqrt{e^2-4 d f}+e\right) (c d-a f)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}","\frac{2 \left(a \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)+c x (a b f-2 a c e+b c d)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left((c d-a f)^2-(b d-a e) (c e-b f)\right)}+\frac{f \left(2 d (c e-b f)-\left(e-\sqrt{e^2-4 d f}\right) (c d-a f)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{f \left(2 d (c e-b f)-\left(\sqrt{e^2-4 d f}+e\right) (c d-a f)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}",1,"(2*(a*(2*c^2*d - b*c*e + b^2*f - 2*a*c*f) + c*(b*c*d - 2*a*c*e + a*b*f)*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[a + b*x + c*x^2]) + (f*(2*d*(c*e - b*f) - (c*d - a*f)*(e - Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - (f*(2*d*(c*e - b*f) - (c*d - a*f)*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",6,4,28,0.1429,1,"{1016, 1032, 724, 206}"
124,1,666,0,1.7459374,"\int \frac{1}{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[1/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{2 \left(-c x \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)-b c (c d-3 a f)-2 a c^2 e+b^2 c e+b^3 (-f)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left((c d-a f)^2-(b d-a e) (c e-b f)\right)}-\frac{f \left(f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}","\frac{2 \left(-c x \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)-b c (c d-3 a f)-2 a c^2 e+b^2 c e+b^3 (-f)\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left((c d-a f)^2-(b d-a e) (c e-b f)\right)}-\frac{f \left(f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right)-b \left(e-\sqrt{e^2-4 d f}\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}+\frac{f \left(f \left(2 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \tanh ^{-1}\left(\frac{4 a f+2 x \left(b f-c \left(\sqrt{e^2-4 d f}+e\right)\right)-b \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right)}{\sqrt{2} \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{f \left(2 a f-b \left(\sqrt{e^2-4 d f}+e\right)\right)+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}",1,"(2*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f) - c*(2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[a + b*x + c*x^2]) - (f*(c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f])))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))]) + (f*(c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f])))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))])","A",6,4,27,0.1481,1,"{974, 1032, 724, 206}"
125,1,814,0,15.9158428,"\int \frac{1}{x \left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Int[1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{2 \left(b^2+c x b-2 a c\right)}{a \left(b^2-4 a c\right) d \sqrt{c x^2+b x+a}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{c x^2+b x+a}}\right)}{a^{3/2} d}-\frac{f \left(2 f \left(b e^2-a f e-b d f\right)-2 c \left(e^3-2 d e f\right)-\left(e-\sqrt{e^2-4 d f}\right) \left(f (b e-a f)-c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e-\sqrt{e^2-4 d f}\right)+2 \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt{e^2-4 d f}}}+\frac{f \left(2 f \left(b e^2-a f e-b d f\right)-2 c \left(e^3-2 d e f\right)-\left(e+\sqrt{e^2-4 d f}\right) \left(f (b e-a f)-c \left(e^2-d f\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e+\sqrt{e^2-4 d f}\right)+2 \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt{e^2-4 d f}}}+\frac{2 \left(c e (2 a c e-b (c d+a f))+(b e-a f) \left(f b^2+2 c^2 d-c (b e+2 a f)\right)+c \left(2 d e c^2-b \left(e^2+d f\right) c+b f (b e-a f)\right) x\right)}{\left(b^2-4 a c\right) d \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{c x^2+b x+a}}","\frac{2 \left(b^2+c x b-2 a c\right)}{a \left(b^2-4 a c\right) d \sqrt{c x^2+b x+a}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{c x^2+b x+a}}\right)}{a^{3/2} d}+\frac{f \left(\left(e-\sqrt{e^2-4 d f}\right) \left(f (b e-a f)-c \left(e^2-d f\right)\right)-2 \left(f \left(b e^2-a f e-b d f\right)-c \left(e^3-2 d e f\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e-\sqrt{e^2-4 d f}\right)+2 \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt{e^2-4 d f}}}-\frac{f \left(\left(e+\sqrt{e^2-4 d f}\right) \left(f (b e-a f)-c \left(e^2-d f\right)\right)-2 \left(f \left(b e^2-a f e-b d f\right)-c \left(e^3-2 d e f\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e+\sqrt{e^2-4 d f}\right)+2 \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right)}{\sqrt{2} d \sqrt{e^2-4 d f} \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt{e^2-4 d f}}}+\frac{2 \left(c e (2 a c e-b (c d+a f))+(b e-a f) \left(f b^2+2 c^2 d-c (b e+2 a f)\right)+c \left(2 d e c^2-b \left(e^2+d f\right) c+b f (b e-a f)\right) x\right)}{\left(b^2-4 a c\right) d \left((c d-a f)^2-(b d-a e) (c e-b f)\right) \sqrt{c x^2+b x+a}}",1,"(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) + (2*(c*e*(2*a*c*e - b*(c*d + a*f)) + (b*e - a*f)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(2*c^2*d*e + b*f*(b*e - a*f) - b*c*(e^2 + d*f))*x))/((b^2 - 4*a*c)*d*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[a + b*x + c*x^2]) - ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])]/(a^(3/2)*d) - (f*(2*f*(b*e^2 - b*d*f - a*e*f) - 2*c*(e^3 - 2*d*e*f) - (e - Sqrt[e^2 - 4*d*f])*(f*(b*e - a*f) - c*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) + (f*(2*f*(b*e^2 - b*d*f - a*e*f) - 2*c*(e^3 - 2*d*e*f) - (e + Sqrt[e^2 - 4*d*f])*(f*(b*e - a*f) - c*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]])","A",12,7,30,0.2333,1,"{6728, 740, 12, 724, 206, 1016, 1032}"
126,1,140,0,0.5006272,"\int \frac{x^4}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Int[x^4/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","-\frac{1}{4} \sqrt{-x^2-4 x-3} x+\frac{5}{2} \sqrt{-x^2-4 x-3}+\frac{\tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)}{2 \sqrt{2}}-\frac{5}{4} \tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)+\frac{11}{2} \sin ^{-1}(x+2)","-\frac{1}{4} \sqrt{-x^2-4 x-3} x+\frac{5}{2} \sqrt{-x^2-4 x-3}+\frac{\tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)}{2 \sqrt{2}}-\frac{5}{4} \tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)+\frac{11}{2} \sin ^{-1}(x+2)",1,"(5*Sqrt[-3 - 4*x - x^2])/2 - (x*Sqrt[-3 - 4*x - x^2])/4 + (11*ArcSin[2 + x])/2 + ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/(2*Sqrt[2]) - ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/(2*Sqrt[2]) - (5*ArcTanh[x/Sqrt[-3 - 4*x - x^2]])/4","A",24,14,30,0.4667,1,"{6728, 619, 216, 640, 742, 1028, 986, 12, 1026, 1161, 618, 204, 1027, 206}"
127,1,115,0,0.4201025,"\int \frac{x^3}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Int[x^3/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","-\frac{1}{2} \sqrt{-x^2-4 x-3}+\frac{\tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)}{2 \sqrt{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)-2 \sin ^{-1}(x+2)","-\frac{1}{2} \sqrt{-x^2-4 x-3}+\frac{\tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)}{2 \sqrt{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)-2 \sin ^{-1}(x+2)",1,"-Sqrt[-3 - 4*x - x^2]/2 - 2*ArcSin[2 + x] + ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/(2*Sqrt[2]) - ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/(2*Sqrt[2]) + ArcTanh[x/Sqrt[-3 - 4*x - x^2]]","A",20,13,30,0.4333,1,"{6728, 619, 216, 640, 1028, 986, 12, 1026, 1161, 618, 204, 1027, 206}"
128,1,98,0,0.1981545,"\int \frac{x^2}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Int[x^2/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)}{\sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)+\frac{1}{2} \sin ^{-1}(x+2)","-\frac{\tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)}{\sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)+\frac{1}{2} \sin ^{-1}(x+2)",1,"ArcSin[2 + x]/2 - ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/Sqrt[2] + ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/Sqrt[2] - ArcTanh[x/Sqrt[-3 - 4*x - x^2]]/2","A",16,12,30,0.4000,1,"{1077, 619, 216, 1028, 986, 12, 1026, 1161, 618, 204, 1027, 206}"
129,1,69,0,0.0615909,"\int \frac{x}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Int[x/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\frac{\tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)}{\sqrt{2}}","\frac{\tan ^{-1}\left(\frac{\frac{3 \sqrt{-x-1}}{\sqrt{x+3}}+1}{\sqrt{2}}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{3 \sqrt{-x-1}}{\sqrt{x+3}}}{\sqrt{2}}\right)}{\sqrt{2}}",1,"ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/Sqrt[2] - ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/Sqrt[2]","A",6,4,28,0.1429,1,"{1026, 1161, 618, 204}"
130,1,95,0,0.1111734,"\int \frac{1}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Int[1/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","-\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)+\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)+\frac{1}{3} \tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)","-\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)+\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)+\frac{1}{3} \tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)",1,"-(Sqrt[2]*ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/3 + (Sqrt[2]*ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/3 + ArcTanh[x/Sqrt[-3 - 4*x - x^2]]/3","A",10,8,27,0.2963,1,"{986, 12, 1026, 1161, 618, 204, 1027, 206}"
131,1,130,0,0.4174218,"\int \frac{1}{x \sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Int[1/(x*Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{2 x+3}{\sqrt{3} \sqrt{-x^2-4 x-3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \sqrt{2} \tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)-\frac{1}{9} \sqrt{2} \tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)-\frac{4}{9} \tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)","-\frac{\tan ^{-1}\left(\frac{2 x+3}{\sqrt{3} \sqrt{-x^2-4 x-3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \sqrt{2} \tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)-\frac{1}{9} \sqrt{2} \tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)-\frac{4}{9} \tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)",1,"-ArcTan[(3 + 2*x)/(Sqrt[3]*Sqrt[-3 - 4*x - x^2])]/(3*Sqrt[3]) + (Sqrt[2]*ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/9 - (Sqrt[2]*ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/9 - (4*ArcTanh[x/Sqrt[-3 - 4*x - x^2]])/9","A",17,11,30,0.3667,1,"{6728, 724, 204, 1028, 986, 12, 1026, 1161, 618, 1027, 206}"
132,1,151,0,0.4536253,"\int \frac{1}{x^2 \sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Int[1/(x^2*Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\frac{\sqrt{-x^2-4 x-3}}{9 x}+\frac{2 \tan ^{-1}\left(\frac{2 x+3}{\sqrt{3} \sqrt{-x^2-4 x-3}}\right)}{3 \sqrt{3}}+\frac{2}{27} \sqrt{2} \tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)-\frac{2}{27} \sqrt{2} \tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)+\frac{10}{27} \tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)","\frac{\sqrt{-x^2-4 x-3}}{9 x}+\frac{2 \tan ^{-1}\left(\frac{2 x+3}{\sqrt{3} \sqrt{-x^2-4 x-3}}\right)}{3 \sqrt{3}}+\frac{2}{27} \sqrt{2} \tan ^{-1}\left(\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right)-\frac{2}{27} \sqrt{2} \tan ^{-1}\left(\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right)+\frac{10}{27} \tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)",1,"Sqrt[-3 - 4*x - x^2]/(9*x) + (2*ArcTan[(3 + 2*x)/(Sqrt[3]*Sqrt[-3 - 4*x - x^2])])/(3*Sqrt[3]) + (2*Sqrt[2]*ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/27 - (2*Sqrt[2]*ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/27 + (10*ArcTanh[x/Sqrt[-3 - 4*x - x^2]])/27","A",20,12,30,0.4000,1,"{6728, 730, 724, 204, 1028, 986, 12, 1026, 1161, 618, 1027, 206}"
133,1,149,0,0.0933228,"\int (2+3 x)^2 \left(30+31 x-12 x^2\right)^2 \sqrt{6+17 x+12 x^2} \, dx","Int[(2 + 3*x)^2*(30 + 31*x - 12*x^2)^2*Sqrt[6 + 17*x + 12*x^2],x]","-\frac{1}{32} (10-3 x) \left(12 x^2+17 x+6\right)^{7/2}-\frac{873 \left(12 x^2+17 x+6\right)^{7/2}}{1792}+\frac{25091 (24 x+17) \left(12 x^2+17 x+6\right)^{5/2}}{24576}-\frac{125455 (24 x+17) \left(12 x^2+17 x+6\right)^{3/2}}{4718592}+\frac{125455 (24 x+17) \sqrt{12 x^2+17 x+6}}{150994944}-\frac{125455 \tanh ^{-1}\left(\frac{24 x+17}{4 \sqrt{3} \sqrt{12 x^2+17 x+6}}\right)}{603979776 \sqrt{3}}","-\frac{1}{32} (10-3 x) \left(12 x^2+17 x+6\right)^{7/2}-\frac{873 \left(12 x^2+17 x+6\right)^{7/2}}{1792}+\frac{25091 (24 x+17) \left(12 x^2+17 x+6\right)^{5/2}}{24576}-\frac{125455 (24 x+17) \left(12 x^2+17 x+6\right)^{3/2}}{4718592}+\frac{125455 (24 x+17) \sqrt{12 x^2+17 x+6}}{150994944}-\frac{125455 \tanh ^{-1}\left(\frac{24 x+17}{4 \sqrt{3} \sqrt{12 x^2+17 x+6}}\right)}{603979776 \sqrt{3}}",1,"(125455*(17 + 24*x)*Sqrt[6 + 17*x + 12*x^2])/150994944 - (125455*(17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2))/4718592 + (25091*(17 + 24*x)*(6 + 17*x + 12*x^2)^(5/2))/24576 - (873*(6 + 17*x + 12*x^2)^(7/2))/1792 - ((10 - 3*x)*(6 + 17*x + 12*x^2)^(7/2))/32 - (125455*ArcTanh[(17 + 24*x)/(4*Sqrt[3]*Sqrt[6 + 17*x + 12*x^2])])/(603979776*Sqrt[3])","A",8,6,34,0.1765,1,"{1002, 742, 640, 612, 621, 206}"
134,1,103,0,0.0417328,"\int (2+3 x) \left(30+31 x-12 x^2\right) \sqrt{6+17 x+12 x^2} \, dx","Int[(2 + 3*x)*(30 + 31*x - 12*x^2)*Sqrt[6 + 17*x + 12*x^2],x]","-\frac{1}{20} \left(12 x^2+17 x+6\right)^{5/2}+\frac{97}{768} (24 x+17) \left(12 x^2+17 x+6\right)^{3/2}-\frac{97 (24 x+17) \sqrt{12 x^2+17 x+6}}{24576}+\frac{97 \tanh ^{-1}\left(\frac{24 x+17}{4 \sqrt{3} \sqrt{12 x^2+17 x+6}}\right)}{98304 \sqrt{3}}","-\frac{1}{20} \left(12 x^2+17 x+6\right)^{5/2}+\frac{97}{768} (24 x+17) \left(12 x^2+17 x+6\right)^{3/2}-\frac{97 (24 x+17) \sqrt{12 x^2+17 x+6}}{24576}+\frac{97 \tanh ^{-1}\left(\frac{24 x+17}{4 \sqrt{3} \sqrt{12 x^2+17 x+6}}\right)}{98304 \sqrt{3}}",1,"(-97*(17 + 24*x)*Sqrt[6 + 17*x + 12*x^2])/24576 + (97*(17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2))/768 - (6 + 17*x + 12*x^2)^(5/2)/20 + (97*ArcTanh[(17 + 24*x)/(4*Sqrt[3]*Sqrt[6 + 17*x + 12*x^2])])/(98304*Sqrt[3])","A",6,5,30,0.1667,1,"{1002, 640, 612, 621, 206}"
135,1,28,0,0.0482939,"\int \frac{\sqrt{6+17 x+12 x^2}}{(2+3 x) \left(30+31 x-12 x^2\right)} \, dx","Int[Sqrt[6 + 17*x + 12*x^2]/((2 + 3*x)*(30 + 31*x - 12*x^2)),x]","\frac{1}{42} \tanh ^{-1}\left(\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right)","\frac{1}{42} \tanh ^{-1}\left(\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right)",1,"ArcTanh[(206 + 291*x)/(84*Sqrt[6 + 17*x + 12*x^2])]/42","A",3,3,34,0.08824,1,"{1002, 724, 206}"
136,1,84,0,0.0777517,"\int \frac{\sqrt{6+17 x+12 x^2}}{(2+3 x)^2 \left(30+31 x-12 x^2\right)^2} \, dx","Int[Sqrt[6 + 17*x + 12*x^2]/((2 + 3*x)^2*(30 + 31*x - 12*x^2)^2),x]","-\frac{388 x+275}{98 (10-3 x) \sqrt{12 x^2+17 x+6}}+\frac{3137 \sqrt{12 x^2+17 x+6}}{38416 (10-3 x)}+\frac{97 \tanh ^{-1}\left(\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right)}{3226944}","-\frac{388 x+275}{98 (10-3 x) \sqrt{12 x^2+17 x+6}}+\frac{3137 \sqrt{12 x^2+17 x+6}}{38416 (10-3 x)}+\frac{97 \tanh ^{-1}\left(\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right)}{3226944}",1,"-(275 + 388*x)/(98*(10 - 3*x)*Sqrt[6 + 17*x + 12*x^2]) + (3137*Sqrt[6 + 17*x + 12*x^2])/(38416*(10 - 3*x)) + (97*ArcTanh[(206 + 291*x)/(84*Sqrt[6 + 17*x + 12*x^2])])/3226944","A",5,5,34,0.1471,1,"{1002, 740, 806, 724, 206}"
137,1,139,0,0.1174852,"\int \frac{\sqrt{6+17 x+12 x^2}}{(2+3 x)^3 \left(30+31 x-12 x^2\right)^3} \, dx","Int[Sqrt[6 + 17*x + 12*x^2]/((2 + 3*x)^3*(30 + 31*x - 12*x^2)^3),x]","-\frac{388 x+275}{294 (10-3 x)^2 \left(12 x^2+17 x+6\right)^{3/2}}-\frac{1634466587 \sqrt{12 x^2+17 x+6}}{7589772288 (10-3 x)}-\frac{50555899 \sqrt{12 x^2+17 x+6}}{19361664 (10-3 x)^2}+\frac{1042556 x+738029}{8232 (10-3 x)^2 \sqrt{12 x^2+17 x+6}}+\frac{40325 \tanh ^{-1}\left(\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right)}{637540872192}","-\frac{388 x+275}{294 (10-3 x)^2 \left(12 x^2+17 x+6\right)^{3/2}}-\frac{1634466587 \sqrt{12 x^2+17 x+6}}{7589772288 (10-3 x)}-\frac{50555899 \sqrt{12 x^2+17 x+6}}{19361664 (10-3 x)^2}+\frac{1042556 x+738029}{8232 (10-3 x)^2 \sqrt{12 x^2+17 x+6}}+\frac{40325 \tanh ^{-1}\left(\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right)}{637540872192}",1,"-(275 + 388*x)/(294*(10 - 3*x)^2*(6 + 17*x + 12*x^2)^(3/2)) + (738029 + 1042556*x)/(8232*(10 - 3*x)^2*Sqrt[6 + 17*x + 12*x^2]) - (50555899*Sqrt[6 + 17*x + 12*x^2])/(19361664*(10 - 3*x)^2) - (1634466587*Sqrt[6 + 17*x + 12*x^2])/(7589772288*(10 - 3*x)) + (40325*ArcTanh[(206 + 291*x)/(84*Sqrt[6 + 17*x + 12*x^2])])/637540872192","A",7,7,34,0.2059,1,"{1002, 740, 822, 834, 806, 724, 206}"
138,1,15,0,0.0039147,"\int (-3+2 x) \left(-3 x+x^2\right)^{2/3} \, dx","Int[(-3 + 2*x)*(-3*x + x^2)^(2/3),x]","\frac{3}{5} \left(x^2-3 x\right)^{5/3}","\frac{3}{5} \left(x^2-3 x\right)^{5/3}",1,"(3*(-3*x + x^2)^(5/3))/5","A",1,1,17,0.05882,1,"{629}"
139,1,16,0,0.006378,"\int ((-3+x) x)^{2/3} (-3+2 x) \, dx","Int[((-3 + x)*x)^(2/3)*(-3 + 2*x),x]","\frac{3}{5} (-(3-x) x)^{5/3}","\frac{3}{5} (-(3-x) x)^{5/3}",1,"(3*(-((3 - x)*x))^(5/3))/5","A",1,1,15,0.06667,1,"{1588}"
140,1,15,0,0.0282344,"\int \frac{x \left(9-9 x+2 x^2\right)}{\sqrt[3]{-3 x+x^2}} \, dx","Int[(x*(9 - 9*x + 2*x^2))/(-3*x + x^2)^(1/3),x]","\frac{3}{5} \left(x^2-3 x\right)^{5/3}","\frac{3}{5} \left(x^2-3 x\right)^{5/3}",1,"(3*(-3*x + x^2)^(5/3))/5","A",2,2,23,0.08696,1,"{1631, 629}"
141,1,15,0,0.0575034,"\int \frac{x \left(9-9 x+2 x^2\right)}{\sqrt[3]{(-3+x) x}} \, dx","Int[(x*(9 - 9*x + 2*x^2))/((-3 + x)*x)^(1/3),x]","\frac{3}{5} \left(x^2-3 x\right)^{5/3}","\frac{3}{5} \left(x^2-3 x\right)^{5/3}",1,"(3*(-3*x + x^2)^(5/3))/5","A",3,3,21,0.1429,1,"{1985, 1631, 629}"
142,1,242,0,0.0927643,"\int \frac{g+h x}{\sqrt[3]{-\frac{c g^2}{h^2}+9 c x^2} \left(g^2+3 h^2 x^2\right)} \, dx","Int[(g + h*x)/((-((c*g^2)/h^2) + 9*c*x^2)^(1/3)*(g^2 + 3*h^2*x^2)),x]","\frac{\sqrt[3]{1-\frac{9 h^2 x^2}{g^2}} \log \left(g^2+3 h^2 x^2\right)}{6\ 2^{2/3} h \sqrt[3]{9 c x^2-\frac{c g^2}{h^2}}}-\frac{\sqrt[3]{1-\frac{9 h^2 x^2}{g^2}} \log \left(\left(1-\frac{3 h x}{g}\right)^{2/3}+\sqrt[3]{2} \sqrt[3]{\frac{3 h x}{g}+1}\right)}{2\ 2^{2/3} h \sqrt[3]{9 c x^2-\frac{c g^2}{h^2}}}+\frac{\sqrt[3]{1-\frac{9 h^2 x^2}{g^2}} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left(1-\frac{3 h x}{g}\right)^{2/3}}{\sqrt{3} \sqrt[3]{\frac{3 h x}{g}+1}}\right)}{2^{2/3} \sqrt{3} h \sqrt[3]{9 c x^2-\frac{c g^2}{h^2}}}","\frac{\sqrt[3]{1-\frac{9 h^2 x^2}{g^2}} \log \left(g^2+3 h^2 x^2\right)}{6\ 2^{2/3} h \sqrt[3]{9 c x^2-\frac{c g^2}{h^2}}}-\frac{\sqrt[3]{1-\frac{9 h^2 x^2}{g^2}} \log \left(\left(1-\frac{3 h x}{g}\right)^{2/3}+\sqrt[3]{2} \sqrt[3]{\frac{3 h x}{g}+1}\right)}{2\ 2^{2/3} h \sqrt[3]{9 c x^2-\frac{c g^2}{h^2}}}+\frac{\sqrt[3]{1-\frac{9 h^2 x^2}{g^2}} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left(1-\frac{3 h x}{g}\right)^{2/3}}{\sqrt{3} \sqrt[3]{\frac{3 h x}{g}+1}}\right)}{2^{2/3} \sqrt{3} h \sqrt[3]{9 c x^2-\frac{c g^2}{h^2}}}",1,"((1 - (9*h^2*x^2)/g^2)^(1/3)*ArcTan[1/Sqrt[3] - (2^(2/3)*(1 - (3*h*x)/g)^(2/3))/(Sqrt[3]*(1 + (3*h*x)/g)^(1/3))])/(2^(2/3)*Sqrt[3]*h*(-((c*g^2)/h^2) + 9*c*x^2)^(1/3)) + ((1 - (9*h^2*x^2)/g^2)^(1/3)*Log[g^2 + 3*h^2*x^2])/(6*2^(2/3)*h*(-((c*g^2)/h^2) + 9*c*x^2)^(1/3)) - ((1 - (9*h^2*x^2)/g^2)^(1/3)*Log[(1 - (3*h*x)/g)^(2/3) + 2^(1/3)*(1 + (3*h*x)/g)^(1/3)])/(2*2^(2/3)*h*(-((c*g^2)/h^2) + 9*c*x^2)^(1/3))","A",2,2,40,0.05000,1,"{1009, 1008}"
143,1,488,0,0.3604191,"\int \frac{g+h x}{\sqrt[3]{\frac{-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left(\frac{f \left(b^2-\frac{-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right)}{c^2}+\frac{b f x}{c}+f x^2\right)} \, dx","Int[(g + h*x)/(((-(c^2*g^2) + b*c*g*h + 2*b^2*h^2)/(9*c*h^2) + b*x + c*x^2)^(1/3)*((f*(b^2 - (-(c^2*g^2) + b*c*g*h + 2*b^2*h^2)/(3*h^2)))/c^2 + (b*f*x)/c + f*x^2)),x]","\frac{3^{2/3} h \sqrt[3]{\frac{c h^2 \left(\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right)}{(2 c g-b h)^2}} \log \left(\frac{f \left(b^2 h^2-b c g h+c^2 g^2\right)}{3 c^2 h^2}+\frac{b f x}{c}+f x^2\right)}{2 f \sqrt[3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}-\frac{3\ 3^{2/3} h \sqrt[3]{\frac{c h^2 \left(\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right)}{(2 c g-b h)^2}} \log \left(\left(1-\frac{3 h (b+2 c x)}{2 c g-b h}\right)^{2/3}+\sqrt[3]{2} \sqrt[3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}\right)}{2 f \sqrt[3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}+\frac{3 \sqrt[6]{3} h \sqrt[3]{\frac{c h^2 \left(\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right)}{(2 c g-b h)^2}} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left(1-\frac{3 h (b+2 c x)}{2 c g-b h}\right)^{2/3}}{\sqrt{3} \sqrt[3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}}\right)}{f \sqrt[3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}","\frac{3^{2/3} h \sqrt[3]{\frac{c h^2 \left(\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right)}{(2 c g-b h)^2}} \log \left(\frac{f \left(b^2 h^2-b c g h+c^2 g^2\right)}{3 c^2 h^2}+\frac{b f x}{c}+f x^2\right)}{2 f \sqrt[3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}-\frac{3\ 3^{2/3} h \sqrt[3]{\frac{c h^2 \left(\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right)}{(2 c g-b h)^2}} \log \left(\left(1-\frac{3 h (b+2 c x)}{2 c g-b h}\right)^{2/3}+\sqrt[3]{2} \sqrt[3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}\right)}{2 f \sqrt[3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}+\frac{3 \sqrt[6]{3} h \sqrt[3]{\frac{c h^2 \left(\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right)}{(2 c g-b h)^2}} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left(1-\frac{3 h (b+2 c x)}{2 c g-b h}\right)^{2/3}}{\sqrt{3} \sqrt[3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}}\right)}{f \sqrt[3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}",1,"(3*3^(1/6)*h*((c*h^2*(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2) - 9*b*x - 9*c*x^2))/(2*c*g - b*h)^2)^(1/3)*ArcTan[1/Sqrt[3] - (2^(2/3)*(1 - (3*h*(b + 2*c*x))/(2*c*g - b*h))^(2/3))/(Sqrt[3]*(1 + (3*h*(b + 2*c*x))/(2*c*g - b*h))^(1/3))])/(f*(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3)) + (3^(2/3)*h*((c*h^2*(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2) - 9*b*x - 9*c*x^2))/(2*c*g - b*h)^2)^(1/3)*Log[(f*(c^2*g^2 - b*c*g*h + b^2*h^2))/(3*c^2*h^2) + (b*f*x)/c + f*x^2])/(2*f*(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3)) - (3*3^(2/3)*h*((c*h^2*(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2) - 9*b*x - 9*c*x^2))/(2*c*g - b*h)^2)^(1/3)*Log[(1 - (3*h*(b + 2*c*x))/(2*c*g - b*h))^(2/3) + 2^(1/3)*(1 + (3*h*(b + 2*c*x))/(2*c*g - b*h))^(1/3)])/(2*f*(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3))","A",2,2,104,0.01923,1,"{1041, 1040}"