1,1,86,94,0.0715971,"\int \frac{(A+B x) \left(a+b x+c x^2\right)}{d+f x^2} \, dx","Integrate[((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)-\frac{2 \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) (-a A f+A c d+b B d)}{\sqrt{d}}+f x (2 A c+2 b B+B c x)}{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{\log \left(d+f x^2\right) (-a B f-A b f+B c d)}{2 f^2}+\frac{x (A c+b B)}{f}+\frac{B c x^2}{2 f}",1,"(f*x*(2*b*B + 2*A*c + B*c*x) - (2*Sqrt[f]*(b*B*d + A*c*d - a*A*f)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/Sqrt[d] + (-(B*c*d) + A*b*f + a*B*f)*Log[d + f*x^2])/(2*f^2)","A",1
2,1,204,228,0.2042649,"\int \frac{(A+B x) \left(a+b x+c x^2\right)^2}{d+f x^2} \, dx","Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/(d + f*x^2),x]","\frac{6 \log \left(d+f x^2\right) \left(B \left(a^2 f^2-2 a c d f+b^2 (-d) f+c^2 d^2\right)+2 A b f (a f-c d)\right)+f x \left(4 A c \left(6 a f-3 c d+c f x^2\right)+4 b B \left(6 a f-6 c d+2 c f x^2\right)+3 B c x \left(4 a f-2 c d+c f x^2\right)+6 b^2 f (2 A+B x)+12 A b c f x\right)}{12 f^3}+\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{\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{\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{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{c x^3 (A c+2 b B)}{3 f}+\frac{B c^2 x^4}{4 f}",1,"((-(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)) + (f*x*(12*A*b*c*f*x + 6*b^2*f*(2*A + B*x) + 3*B*c*x*(-2*c*d + 4*a*f + c*f*x^2) + 4*A*c*(-3*c*d + 6*a*f + c*f*x^2) + 4*b*B*(-6*c*d + 6*a*f + 2*c*f*x^2)) + 6*(2*A*b*f*(-(c*d) + a*f) + B*(c^2*d^2 - b^2*d*f - 2*a*c*d*f + a^2*f^2))*Log[d + f*x^2])/(12*f^3)","A",1
3,1,422,441,0.4593077,"\int \frac{(A+B x) \left(a+b x+c x^2\right)^3}{d+f x^2} \, dx","Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + f*x^2),x]","\frac{f x \left(3 b \left(4 B \left(15 a^2 f^2+10 a c f \left(f x^2-3 d\right)+c^2 \left(15 d^2-5 d f x^2+3 f^2 x^4\right)\right)+15 A c f x \left(4 a f-2 c d+c f x^2\right)\right)+c \left(4 A \left(45 a^2 f^2+15 a c f \left(f x^2-3 d\right)+c^2 \left(15 d^2-5 d f x^2+3 f^2 x^4\right)\right)+5 B x \left(18 a^2 f^2+9 a c f \left(f x^2-2 d\right)+c^2 \left(6 d^2-3 d f x^2+2 f^2 x^4\right)\right)\right)+15 b^2 f \left(4 A \left(3 a f-3 c d+c f x^2\right)+3 B x \left(2 a f-2 c d+c f x^2\right)\right)+10 b^3 f \left(3 A f x-6 B d+2 B f x^2\right)\right)-30 \log \left(d+f x^2\right) \left(A b f \left(-3 a^2 f^2+6 a c d f+b^2 d f-3 c^2 d^2\right)+B (c d-a f) \left(a^2 f^2-2 a c d f-3 b^2 d f+c^2 d^2\right)\right)}{60 f^4}+\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{\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^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{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^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)) + (f*x*(10*b^3*f*(-6*B*d + 3*A*f*x + 2*B*f*x^2) + 15*b^2*f*(3*B*x*(-2*c*d + 2*a*f + c*f*x^2) + 4*A*(-3*c*d + 3*a*f + c*f*x^2)) + 3*b*(15*A*c*f*x*(-2*c*d + 4*a*f + c*f*x^2) + 4*B*(15*a^2*f^2 + 10*a*c*f*(-3*d + f*x^2) + c^2*(15*d^2 - 5*d*f*x^2 + 3*f^2*x^4))) + c*(5*B*x*(18*a^2*f^2 + 9*a*c*f*(-2*d + f*x^2) + c^2*(6*d^2 - 3*d*f*x^2 + 2*f^2*x^4)) + 4*A*(45*a^2*f^2 + 15*a*c*f*(-3*d + f*x^2) + c^2*(15*d^2 - 5*d*f*x^2 + 3*f^2*x^4)))) - 30*(A*b*f*(-3*c^2*d^2 + b^2*d*f + 6*a*c*d*f - 3*a^2*f^2) + B*(c*d - a*f)*(c^2*d^2 - 3*b^2*d*f - 2*a*c*d*f + a^2*f^2))*Log[d + f*x^2])/(60*f^4)","A",1
4,1,212,274,0.3675794,"\int \frac{A+B x}{\left(a+b x+c x^2\right) \left(d+f x^2\right)} \, dx","Integrate[(A + B*x)/((a + b*x + c*x^2)*(d + f*x^2)),x]","\frac{\sqrt{d} \left(\sqrt{4 a c-b^2} (-a B f+A b f+B c d) \left(\log (a+x (b+c x))-\log \left(d+f x^2\right)\right)+2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right)\right)+2 \sqrt{f} \sqrt{4 a c-b^2} \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) (a A f-A c d+b B d)}{2 \sqrt{d} \sqrt{4 a c-b^2} \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,"(2*Sqrt[-b^2 + 4*a*c]*Sqrt[f]*(b*B*d - A*c*d + a*A*f)*ArcTan[(Sqrt[f]*x)/Sqrt[d]] + Sqrt[d]*(2*(A*b^2*f + 2*A*c*(c*d - a*f) - b*B*(c*d + a*f))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*(B*c*d + A*b*f - a*B*f)*(-Log[d + f*x^2] + Log[a + x*(b + c*x)])))/(2*Sqrt[-b^2 + 4*a*c]*Sqrt[d]*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f)))","A",1
5,1,523,596,1.8423323,"\int \frac{A+B x}{\left(a+b x+c x^2\right)^2 \left(d+f x^2\right)} \, dx","Integrate[(A + B*x)/((a + b*x + c*x^2)^2*(d + f*x^2)),x]","\frac{f \log \left(d+f x^2\right) \left(B \left(f \left(a^2 f-b^2 d\right)-2 a c d f+c^2 d^2\right)+2 A b f (c d-a f)\right)+f \log (a+x (b+c x)) \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 (a f-c d)\right)-\frac{2 \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right) \left(B \left(2 a^2 c f-a \left(b^2 f+b c f x+2 c^2 d\right)-b c^2 d x\right)+A \left(b c (c d-3 a f)+2 c^2 x (c d-a f)+b^3 f+b^2 c f x\right)\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}-\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\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^3 f^3+3 a^2 c d f^2-7 a c^2 d^2 f+c^3 d^3\right)+2 A b^4 f^2 (a f-c d)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{2 f^{3/2} \tan ^{-1}\left(\frac{\sqrt{f} x}{\sqrt{d}}\right) \left(A (c d-a f)^2+2 b B d (a f-c d)-A b^2 d f\right)}{\sqrt{d}}}{2 \left(f \left(a^2 f+b^2 d\right)-2 a c d f+c^2 d^2\right)^2}","-\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{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{\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^3 f^3+3 a^2 c d f^2-7 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{-(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,"((-2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))*(A*(b^3*f + b*c*(c*d - 3*a*f) + b^2*c*f*x + 2*c^2*(c*d - a*f)*x) + B*(2*a^2*c*f - b*c^2*d*x - a*(2*c^2*d + b^2*f + b*c*f*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*f^(3/2)*(-(A*b^2*d*f) + A*(c*d - a*f)^2 + 2*b*B*d*(-(c*d) + a*f))*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/Sqrt[d] - (2*(b^5*B*d*f^2 - 4*A*c^2*(c*d - 3*a*f)*(c*d - a*f)^2 + 2*A*b^4*f^2*(-(c*d) + a*f) - 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))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 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] + 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 + x*(b + c*x)])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2)","A",1
6,1,322,331,0.8010602,"\int \frac{(A+B x) \sqrt{a+b x+c x^2}}{d-f x^2} \, dx","Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]","\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}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)-\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}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)-2 B \sqrt{d} \sqrt{f} \sqrt{a+x (b+c x)}}{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+x (b+c x)}}\right)}{2 \sqrt{c} 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,"-1/2*((b*B + 2*A*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(Sqrt[c]*f) + (-2*B*Sqrt[d]*Sqrt[f]*Sqrt[a + x*(b + c*x)] + (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]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] - (B*Sqrt[d] - A*Sqrt[f])*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(2*Sqrt[d]*f^(3/2))","A",1
7,1,249,249,0.2398324,"\int \frac{A+B x}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Integrate[(A + B*x)/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{-\frac{\left(B \sqrt{d}-A \sqrt{f}\right) \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}-\frac{\left(A \sqrt{f}+B \sqrt{d}\right) \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}}{2 \sqrt{d} \sqrt{f}}","\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*Sqrt[d] - A*Sqrt[f])*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) - ((B*Sqrt[d] + A*Sqrt[f])*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])/(2*Sqrt[d]*Sqrt[f])","A",1
8,1,440,381,0.649575,"\int \frac{A+B x}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Integrate[(A + B*x)/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","\frac{2 \left(\frac{B \left(2 a^2 c f+a \left(b^2 (-f)-b c f x+2 c^2 d\right)+b c^2 d x\right)+A \left(-b c (3 a f+c d)-2 c^2 x (a f+c d)+b^3 f+b^2 c f x\right)}{\sqrt{a+x (b+c x)}}+\frac{\sqrt{f} \left(b^2-4 a c\right) \left(A \sqrt{f}-B \sqrt{d}\right) \left(a f+b \sqrt{d} \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{4 \sqrt{d} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\sqrt{f} \left(4 a c-b^2\right) \left(A \sqrt{f}+B \sqrt{d}\right) \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{4 \sqrt{d} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\left(b^2-4 a c\right) \left((a f+c d)^2-b^2 d f\right)}","-\frac{2 \left(A \left(b^3 f-b c (3 a f+c d)\right)+c x \left(-2 A c (a f+c d)+b B (c d-a f)+A b^2 f\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 - b*c*(c*d + 3*a*f) + b^2*c*f*x - 2*c^2*(c*d + a*f)*x) + B*(2*a^2*c*f + b*c^2*d*x + a*(2*c^2*d - b^2*f - b*c*f*x)))/Sqrt[a + x*(b + c*x)] + ((b^2 - 4*a*c)*(-(B*Sqrt[d]) + A*Sqrt[f])*Sqrt[f]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(4*Sqrt[d]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ((-b^2 + 4*a*c)*(B*Sqrt[d] + A*Sqrt[f])*Sqrt[f]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(4*Sqrt[d]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])))/((b^2 - 4*a*c)*(-(b^2*d*f) + (c*d + a*f)^2))","A",1
9,1,674,797,4.0599346,"\int \frac{A+B x}{\left(a+b x+c x^2\right)^{5/2} \left(d-f x^2\right)} \, dx","Integrate[(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^2)),x]","\frac{2 \left(\frac{B \left(2 a^2 c f+a \left(b^2 (-f)-b c f x+2 c^2 d\right)+b c^2 d x\right)+A \left(-b c (3 a f+c d)-2 c^2 x (a f+c d)+b^3 f+b^2 c f x\right)}{(a+x (b+c x))^{3/2}}-\frac{3 f \left(-b^2 \left(B \left(-a^2 f^2+2 a c d f+c^2 d^2\right)+2 a A c f^2 x\right)+b^3 f (B c d x-A (2 a f+c d))+b c (a f+c d) (5 a A f+a B f x+A c d-3 B c d x)+2 c (a f+c d)^2 (A c x-a B)+b^4 B d f\right)}{\sqrt{a+x (b+c x)} \left(f \left(a^2 f-b^2 d\right)+2 a c d f+c^2 d^2\right)}+\frac{3 f^{3/2} \left(b^2-4 a c\right) \left(\frac{\left(A \sqrt{f}-B \sqrt{d}\right) \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^2 \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}-\frac{\left(A \sqrt{f}+B \sqrt{d}\right) \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^2 \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{4 \sqrt{d} \left((a f+c d)^2-b^2 d f\right)}+\frac{4 c (b+2 c x) \left(2 A c (a f+c d)+b B (a f-c d)-A b^2 f\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}\right)}{3 \left(b^2-4 a c\right) \left((a f+c d)^2-b^2 d f\right)}","-\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*((4*c*(-(A*b^2*f) + b*B*(-(c*d) + a*f) + 2*A*c*(c*d + a*f))*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]) - (3*f*(b^4*B*d*f + 2*c*(c*d + a*f)^2*(-(a*B) + A*c*x) + b^3*f*(-(A*(c*d + 2*a*f)) + B*c*d*x) + b*c*(c*d + a*f)*(A*c*d + 5*a*A*f - 3*B*c*d*x + a*B*f*x) - b^2*(B*(c^2*d^2 + 2*a*c*d*f - a^2*f^2) + 2*a*A*c*f^2*x)))/((c^2*d^2 + 2*a*c*d*f + f*(-(b^2*d) + a^2*f))*Sqrt[a + x*(b + c*x)]) + (A*(b^3*f - b*c*(c*d + 3*a*f) + b^2*c*f*x - 2*c^2*(c*d + a*f)*x) + B*(2*a^2*c*f + b*c^2*d*x + a*(2*c^2*d - b^2*f - b*c*f*x)))/(a + x*(b + c*x))^(3/2) + (3*(b^2 - 4*a*c)*f^(3/2)*(((-(B*Sqrt[d]) + A*Sqrt[f])*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^2*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f] - ((B*Sqrt[d] + A*Sqrt[f])*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^2*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]))/(4*Sqrt[d]*(-(b^2*d*f) + (c*d + a*f)^2))))/(3*(b^2 - 4*a*c)*(-(b^2*d*f) + (c*d + a*f)^2))","A",1
10,1,49,47,0.0077106,"\int \frac{1+2 x}{\left(-1+x^2\right) \sqrt{-1+x+x^2}} \, dx","Integrate[(1 + 2*x)/((-1 + x^2)*Sqrt[-1 + x + x^2]),x]","\frac{1}{2} \tan ^{-1}\left(\frac{-x-3}{2 \sqrt{x^2+x-1}}\right)-\frac{3}{2} \tanh ^{-1}\left(\frac{3 x-1}{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",1
11,1,78,117,0.0330073,"\int \frac{1+2 x}{\left(1+x^2\right) \sqrt{-1+x+x^2}} \, dx","Integrate[(1 + 2*x)/((1 + x^2)*Sqrt[-1 + x + x^2]),x]","-\frac{1}{2} i \left(\sqrt{2+i} \tanh ^{-1}\left(\frac{\sqrt{2+i} (x-i)}{2 \sqrt{x^2+x-1}}\right)-\sqrt{2-i} \tanh ^{-1}\left(\frac{\sqrt{2-i} (x+i)}{2 \sqrt{x^2+x-1}}\right)\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,"(-1/2*I)*(Sqrt[2 + I]*ArcTanh[(Sqrt[2 + I]*(-I + x))/(2*Sqrt[-1 + x + x^2])] - Sqrt[2 - I]*ArcTanh[(Sqrt[2 - I]*(I + x))/(2*Sqrt[-1 + x + x^2])])","C",1
12,1,136,484,0.0828157,"\int \frac{a-c+b x}{\left(1+x^2\right) \sqrt{a+b x+c x^2}} \, dx","Integrate[(a - c + b*x)/((1 + x^2)*Sqrt[a + b*x + c*x^2]),x]","\frac{1}{2} i \left(\sqrt{a+i b-c} \tanh ^{-1}\left(\frac{2 a+b (x+i)+2 i c x}{2 \sqrt{a+i b-c} \sqrt{a+x (b+c x)}}\right)-\sqrt{a-i b-c} \tanh ^{-1}\left(\frac{2 a+b (x-i)-2 i c x}{2 \sqrt{a-i b-c} \sqrt{a+x (b+c x)}}\right)\right)","-\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,"(I/2)*(-(Sqrt[a - I*b - c]*ArcTanh[(2*a - (2*I)*c*x + b*(-I + x))/(2*Sqrt[a - I*b - c]*Sqrt[a + x*(b + c*x)])]) + Sqrt[a + I*b - c]*ArcTanh[(2*a + (2*I)*c*x + b*(I + x))/(2*Sqrt[a + I*b - c]*Sqrt[a + x*(b + c*x)])])","C",1
13,1,175,184,0.2112268,"\int \frac{(A+B x) \left(a+b x+c x^2\right)}{d+e x+f x^2} \, dx","Integrate[((A + B*x)*(a + b*x + c*x^2))/(d + e*x + f*x^2),x]","\frac{\log (d+x (e+f x)) \left(B f (a f-b e)+A f (b f-c e)+B c \left(e^2-d f\right)\right)-\frac{2 \tan ^{-1}\left(\frac{e+2 f x}{\sqrt{4 d f-e^2}}\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)}{\sqrt{4 d f-e^2}}+2 f x (A c f+b B f-B c e)+B c f^2 x^2}{2 f^3}","-\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,"(2*f*(-(B*c*e) + b*B*f + A*c*f)*x + B*c*f^2*x^2 - (2*(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))*ArcTan[(e + 2*f*x)/Sqrt[-e^2 + 4*d*f]])/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 + x*(e + f*x)])/(2*f^3)","A",1
14,1,535,542,0.6042326,"\int \frac{(A+B x) \left(a+b x+c x^2\right)^2}{d+e x+f x^2} \, dx","Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x + f*x^2),x]","\frac{6 \log (d+x (e+f x)) \left(B \left(f^2 \left(a^2 f^2-2 a b e f+b^2 \left(e^2-d f\right)\right)-2 c f \left(a f \left(d f-e^2\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 (b f-c e) \left(f (2 a f-b e)+c \left(e^2-2 d f\right)\right)\right)-\frac{12 \tan ^{-1}\left(\frac{e+2 f x}{\sqrt{4 d f-e^2}}\right) \left(B \left(f^2 \left(a^2 e f^2+2 a b f \left(2 d f-e^2\right)+b^2 \left(e^3-3 d e f\right)\right)-2 c f \left(b \left(2 d^2 f^2-4 d e^2 f+e^4\right)-a e f \left(e^2-3 d f\right)\right)+c^2 \left(5 d^2 e f^2-5 d e^3 f+e^5\right)\right)-A f \left(f^2 \left(2 a^2 f^2-2 a b e f+b^2 \left(e^2-2 d f\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)\right)}{\sqrt{4 d f-e^2}}+6 f^2 x^2 \left(B \left(2 c f (a f-b e)+b^2 f^2+c^2 \left(e^2-d f\right)\right)+A c f (2 b f-c e)\right)+12 f x \left(A f \left(2 c f (a f-b e)+b^2 f^2+c^2 \left(e^2-d f\right)\right)-B (c e-b f) \left(f (2 a f-b e)+c \left(e^2-2 d f\right)\right)\right)+4 c f^3 x^3 (A c f+2 b B f-B c e)+3 B c^2 f^4 x^4}{12 f^5}","\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-\left(b^2 \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-\left(b^2 \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 \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{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{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,"(12*f*(-(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 + 6*f^2*(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 + 4*c*f^3*(-(B*c*e) + 2*b*B*f + A*c*f)*x^3 + 3*B*c^2*f^4*x^4 - (12*(-(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))))*ArcTan[(e + 2*f*x)/Sqrt[-e^2 + 4*d*f]])/Sqrt[-e^2 + 4*d*f] + 6*(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 + x*(e + f*x)])/(12*f^5)","A",1
15,1,267,406,0.4617067,"\int \frac{A+B x}{\left(a+b x+c x^2\right) \left(d+e x+f x^2\right)} \, dx","Integrate[(A + B*x)/((a + b*x + c*x^2)*(d + e*x + f*x^2)),x]","\frac{\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\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{4 a c-b^2}}-\frac{2 \tan ^{-1}\left(\frac{e+2 f x}{\sqrt{4 d f-e^2}}\right) \left(A \left(-2 a f^2+b e f+2 c d f-c e^2\right)+B (a e f-2 b d f+c d e)\right)}{\sqrt{4 d f-e^2}}+\log (a+x (b+c x)) (-a B f+A b f-A c e+B c d)+\log (d+x (e+f x)) (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{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,"((2*(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))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - (2*(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))*ArcTan[(e + 2*f*x)/Sqrt[-e^2 + 4*d*f]])/Sqrt[-e^2 + 4*d*f] + (B*c*d - A*c*e + A*b*f - a*B*f)*Log[a + x*(b + c*x)] + (-(B*c*d) + A*c*e - A*b*f + a*B*f)*Log[d + x*(e + f*x)])/(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",1
16,1,952,1075,6.6995417,"\int \frac{A+B x}{\left(a+b x+c x^2\right)^2 \left(d+e x+f x^2\right)} \, dx","Integrate[(A + B*x)/((a + b*x + c*x^2)^2*(d + e*x + f*x^2)),x]","\frac{-\frac{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) \left(A \left(f b^3+c (f x-e) b^2+c (c (d-e x)-3 a f) b+2 c^2 (c d x+a (e-f x))\right)+B \left(2 c f a^2-\left(f b^2+c (f x-e) b+2 c^2 (d-e x)\right) a-b c^2 d x\right)\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}-\frac{2 \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 B e \left(c^2 d^2-3 a^2 f^2-a c \left(e^2-2 d f\right)\right)+A \left(-c^3 d^3+a c^2 \left(5 d f-3 e^2\right) d+3 a^3 f^3+a^2 c f \left(e^2-7 d f\right)\right)\right)\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{2 \left(B \left(d e \left(3 d f-e^2\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) \tan ^{-1}\left(\frac{e+2 f x}{\sqrt{4 d f-e^2}}\right)}{\sqrt{4 d f-e^2}}-\left(A (c e-b f) \left(f (2 a f-b e)+c \left(e^2-2 d f\right)\right)+B \left(d \left(d f-e^2\right) c^2+2 d f (b e-a f) c+f^2 \left(a^2 f-b^2 d\right)\right)\right) \log (a+x (b+c x))+\left(A (c e-b f) \left(f (2 a f-b e)+c \left(e^2-2 d f\right)\right)+B \left(d \left(d f-e^2\right) c^2+2 d f (b e-a f) c+f^2 \left(a^2 f-b^2 d\right)\right)\right) \log (d+x (e+f x))}{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(\left(e^2-2 d f\right) b^2\right)+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,"((-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*(b^3*f + b^2*c*(-e + f*x) + b*c*(-3*a*f + c*(d - e*x)) + 2*c^2*(c*d*x + a*(e - f*x))) + B*(2*a^2*c*f - b*c^2*d*x - a*(b^2*f + 2*c^2*(d - e*x) + b*c*(-e + f*x)))))/((b^2 - 4*a*c)*(a + x*(b + 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*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))) + 4*c^2*(a*B*e*(c^2*d^2 - 3*a^2*f^2 - a*c*(e^2 - 2*d*f)) + 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))) + 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))))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (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))))*ArcTan[(e + 2*f*x)/Sqrt[-e^2 + 4*d*f]])/Sqrt[-e^2 + 4*d*f] - (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 + x*(b + c*x)] + (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 + x*(e + f*x)])/(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",1
17,1,131,140,0.1432499,"\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","Integrate[(g + h*x)/((a + b*x + c*x^2)*(a*d + b*d*x + c*d*x^2)^2),x]","\frac{\frac{\left(b^2-4 a c\right) (2 a h-b g+b h x-2 c g x)}{(a+x (b+c x))^2}-\frac{12 c (b h-2 c g) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{3 (b+2 c x) (2 c g-b h)}{a+x (b+c x)}}{2 d^2 \left(b^2-4 a c\right)^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^2 - 4*a*c)*(-(b*g) + 2*a*h - 2*c*g*x + b*h*x))/(a + x*(b + c*x))^2 + (3*(2*c*g - b*h)*(b + 2*c*x))/(a + x*(b + c*x)) - (12*c*(-2*c*g + b*h)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c)^2*d^2)","A",1
18,1,131,140,0.0271669,"\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","Integrate[(g + h*x)/((a + b*x + c*x^2)^2*(a*d + b*d*x + c*d*x^2)),x]","\frac{\frac{\left(b^2-4 a c\right) (2 a h-b g+b h x-2 c g x)}{(a+x (b+c x))^2}-\frac{12 c (b h-2 c g) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{3 (b+2 c x) (2 c g-b h)}{a+x (b+c x)}}{2 d \left(b^2-4 a c\right)^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^2 - 4*a*c)*(-(b*g) + 2*a*h - 2*c*g*x + b*h*x))/(a + x*(b + c*x))^2 + (3*(2*c*g - b*h)*(b + 2*c*x))/(a + x*(b + c*x)) - (12*c*(-2*c*g + b*h)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c)^2*d)","A",1
19,1,517,617,2.0860031,"\int \frac{(A+B x) \sqrt{a+b x+c x^2}}{d+e x+f x^2} \, dx","Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/(d + e*x + f*x^2),x]","\frac{-\sqrt{2} \left(B \left(\sqrt{e^2-4 d f}+e\right)-2 A 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)} \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)-\sqrt{2} \left(2 A f+B \left(\sqrt{e^2-4 d f}-e\right)\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)} \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)+4 B f \sqrt{e^2-4 d f} \sqrt{a+x (b+c x)}}{4 f^2 \sqrt{e^2-4 d f}}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) (2 A c f+b B f-2 B c e)}{2 \sqrt{c} f^2}","\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,"((-2*B*c*e + b*B*f + 2*A*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(2*Sqrt[c]*f^2) + (4*B*f*Sqrt[e^2 - 4*d*f]*Sqrt[a + x*(b + c*x)] - Sqrt[2]*(-2*A*f + B*(e + 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 - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])] - Sqrt[2]*(2*A*f + B*(-e + 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 + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(4*f^2*Sqrt[e^2 - 4*d*f])","A",1
20,1,1627,1092,6.5643379,"\int \frac{(A+B x) \left(a+b x+c x^2\right)^{3/2}}{d+e x+f x^2} \, dx","Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x + f*x^2),x]","\frac{\left(B-\frac{B e-2 A f}{\sqrt{e^2-4 d f}}\right) (a+x (b+c x))^{3/2}}{6 f}+\frac{\left(B+\frac{B e-2 A f}{\sqrt{e^2-4 d f}}\right) (a+x (b+c x))^{3/2}}{6 f}-\frac{\left(B+\frac{2 A f-B e}{\sqrt{e^2-4 d f}}\right) \left(\frac{\left(4 c f \left(b \left(e-\sqrt{e^2-4 d f}\right)-4 a f\right)+2 \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) \left(2 c \left(e-\sqrt{e^2-4 d f}\right)-b f\right)-4 c f \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) x\right) \sqrt{c x^2+b x+a}}{8 c f^2}-\frac{-\frac{2 \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) \left(-4 \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right) c^2-4 f \left(3 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right) c+b^2 f^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{\sqrt{c} f}-\frac{2 \sqrt{2} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \left(4 \left(e-\sqrt{e^2-4 d f}\right) \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) \left(-4 \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right) c^2-4 f \left(3 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right) c+b^2 f^2\right)+4 f \left(2 c f \left(4 a f-b \left(e-\sqrt{e^2-4 d f}\right)\right)^2-\left(e-\sqrt{e^2-4 d f}\right) \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) \left(f b^2-2 c \left(e-\sqrt{e^2-4 d f}\right) b+4 a c f\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e-\sqrt{e^2-4 d f}\right)-\left(2 c \left(e-\sqrt{e^2-4 d f}\right)-2 b f\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right)}{f \left(16 a f^2-8 b \left(e-\sqrt{e^2-4 d f}\right) f+4 c \left(e-\sqrt{e^2-4 d f}\right)^2\right)}}{16 c f^2}\right) (a+x (b+c x))^{3/2}}{4 f \left(c x^2+b x+a\right)^{3/2}}-\frac{\left(B-\frac{2 A f-B e}{\sqrt{e^2-4 d f}}\right) \left(\frac{\left(4 c f \left(b \left(e+\sqrt{e^2-4 d f}\right)-4 a f\right)+2 \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) \left(2 c \left(e+\sqrt{e^2-4 d f}\right)-b f\right)-4 c f \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) x\right) \sqrt{c x^2+b x+a}}{8 c f^2}-\frac{-\frac{2 \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) \left(-4 \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right) c^2-4 f \left(3 a f-b \left(e+\sqrt{e^2-4 d f}\right)\right) c+b^2 f^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{\sqrt{c} f}-\frac{2 \sqrt{2} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \left(4 \left(e+\sqrt{e^2-4 d f}\right) \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) \left(-4 \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right) c^2-4 f \left(3 a f-b \left(e+\sqrt{e^2-4 d f}\right)\right) c+b^2 f^2\right)+4 f \left(2 c f \left(4 a f-b \left(e+\sqrt{e^2-4 d f}\right)\right)^2-\left(e+\sqrt{e^2-4 d f}\right) \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) \left(f b^2-2 c \left(e+\sqrt{e^2-4 d f}\right) b+4 a c f\right)\right)\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e+\sqrt{e^2-4 d f}\right)-\left(2 c \left(e+\sqrt{e^2-4 d f}\right)-2 b f\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right)}{f \left(16 a f^2-8 b \left(e+\sqrt{e^2-4 d f}\right) f+4 c \left(e+\sqrt{e^2-4 d f}\right)^2\right)}}{16 c f^2}\right) (a+x (b+c x))^{3/2}}{4 f \left(c x^2+b x+a\right)^{3/2}}","\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(\left(e^2-d f\right) b^2\right)+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(\left(e^2-d f\right) b^2\right)+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,"((B - (B*e - 2*A*f)/Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2))/(6*f) + ((B + (B*e - 2*A*f)/Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2))/(6*f) - ((B + (-(B*e) + 2*A*f)/Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2)*(((4*c*f*(-4*a*f + b*(e - Sqrt[e^2 - 4*d*f])) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(-(b*f) + 2*c*(e - Sqrt[e^2 - 4*d*f])) - 4*c*f*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - ((-2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e - Sqrt[e^2 - 4*d*f])))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f) - (2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*(4*(e - Sqrt[e^2 - 4*d*f])*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e - Sqrt[e^2 - 4*d*f]))) + 4*f*(2*c*f*(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]))^2 - (e - Sqrt[e^2 - 4*d*f])*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(b^2*f + 4*a*c*f - 2*b*c*(e - Sqrt[e^2 - 4*d*f]))))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) - (-2*b*f + 2*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*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f*(16*a*f^2 - 8*b*f*(e - Sqrt[e^2 - 4*d*f]) + 4*c*(e - Sqrt[e^2 - 4*d*f])^2)))/(16*c*f^2)))/(4*f*(a + b*x + c*x^2)^(3/2)) - ((B - (-(B*e) + 2*A*f)/Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2)*(((4*c*f*(-4*a*f + b*(e + Sqrt[e^2 - 4*d*f])) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*(-(b*f) + 2*c*(e + Sqrt[e^2 - 4*d*f])) - 4*c*f*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - ((-2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e + Sqrt[e^2 - 4*d*f])))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f) - (2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*(4*(e + Sqrt[e^2 - 4*d*f])*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e + Sqrt[e^2 - 4*d*f]))) + 4*f*(2*c*f*(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]))^2 - (e + Sqrt[e^2 - 4*d*f])*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*(b^2*f + 4*a*c*f - 2*b*c*(e + Sqrt[e^2 - 4*d*f]))))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) - (-2*b*f + 2*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*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f*(16*a*f^2 - 8*b*f*(e + Sqrt[e^2 - 4*d*f]) + 4*c*(e + Sqrt[e^2 - 4*d*f])^2)))/(16*c*f^2)))/(4*f*(a + b*x + c*x^2)^(3/2))","A",1
21,1,393,416,4.1753722,"\int \frac{A+B x}{\left(a+b x+c x^2\right) \sqrt{d+e x+f x^2}} \, dx","Integrate[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]),x]","\frac{-\frac{\left(B \sqrt{b^2-4 a c}+2 A c-b B\right) \tanh ^{-1}\left(\frac{\left(\sqrt{b^2-4 a c}-b\right) (e+2 f x)+2 c (2 d+e x)}{2 \sqrt{2} \sqrt{d+x (e+f x)} \sqrt{c \left(e \sqrt{b^2-4 a c}-2 a f-b e\right)+b f \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d}}\right)}{\sqrt{c \left(e \sqrt{b^2-4 a c}-2 a f-b e\right)+b f \left(b-\sqrt{b^2-4 a c}\right)+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 (2 d+e x)-\left(\sqrt{b^2-4 a c}+b\right) (e+2 f x)}{2 \sqrt{d+x (e+f x)} \sqrt{-2 c \left(e \sqrt{b^2-4 a c}+2 a f+b e\right)+2 b f \left(\sqrt{b^2-4 a c}+b\right)+4 c^2 d}}\right)}{\sqrt{-c \left(e \sqrt{b^2-4 a c}+2 a f+b e\right)+b f \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d}}}{\sqrt{2} \sqrt{b^2-4 a c}}","\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[(2*c*(2*d + e*x) + (-b + Sqrt[b^2 - 4*a*c])*(e + 2*f*x))/(2*Sqrt[2]*Sqrt[2*c^2*d + b*(b - Sqrt[b^2 - 4*a*c])*f + c*(-(b*e) + Sqrt[b^2 - 4*a*c]*e - 2*a*f)]*Sqrt[d + x*(e + f*x)])])/Sqrt[2*c^2*d + b*(b - Sqrt[b^2 - 4*a*c])*f + c*(-(b*e) + Sqrt[b^2 - 4*a*c]*e - 2*a*f)]) - ((b*B - 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*(2*d + e*x) - (b + Sqrt[b^2 - 4*a*c])*(e + 2*f*x))/(2*Sqrt[4*c^2*d + 2*b*(b + Sqrt[b^2 - 4*a*c])*f - 2*c*(b*e + Sqrt[b^2 - 4*a*c]*e + 2*a*f)]*Sqrt[d + x*(e + f*x)])])/Sqrt[2*c^2*d + b*(b + Sqrt[b^2 - 4*a*c])*f - c*(b*e + Sqrt[b^2 - 4*a*c]*e + 2*a*f)])/(Sqrt[2]*Sqrt[b^2 - 4*a*c])","A",1
22,1,254,780,0.4085005,"\int \frac{A+B x}{\left(a+c x^2\right) \sqrt{d+e x+f x^2}} \, dx","Integrate[(A + B*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]),x]","\frac{\frac{\left(A \sqrt{c}-\sqrt{-a} B\right) \tanh ^{-1}\left(\frac{\sqrt{c} (2 d+e x)-\sqrt{-a} (e+2 f x)}{2 \sqrt{d+x (e+f x)} \sqrt{-\sqrt{-a} \sqrt{c} e-a f+c d}}\right)}{\sqrt{-\sqrt{-a} \sqrt{c} e-a f+c d}}-\frac{\left(\sqrt{-a} B+A \sqrt{c}\right) \tanh ^{-1}\left(\frac{\sqrt{-a} (e+2 f x)+\sqrt{c} (2 d+e x)}{2 \sqrt{d+x (e+f x)} \sqrt{\sqrt{-a} \sqrt{c} e-a f+c d}}\right)}{\sqrt{\sqrt{-a} \sqrt{c} e-a f+c d}}}{2 \sqrt{-a} \sqrt{c}}","\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) + A*Sqrt[c])*ArcTanh[(Sqrt[c]*(2*d + e*x) - Sqrt[-a]*(e + 2*f*x))/(2*Sqrt[c*d - Sqrt[-a]*Sqrt[c]*e - a*f]*Sqrt[d + x*(e + f*x)])])/Sqrt[c*d - Sqrt[-a]*Sqrt[c]*e - a*f] - ((Sqrt[-a]*B + A*Sqrt[c])*ArcTanh[(Sqrt[c]*(2*d + e*x) + Sqrt[-a]*(e + 2*f*x))/(2*Sqrt[c*d + Sqrt[-a]*Sqrt[c]*e - a*f]*Sqrt[d + x*(e + f*x)])])/Sqrt[c*d + Sqrt[-a]*Sqrt[c]*e - a*f])/(2*Sqrt[-a]*Sqrt[c])","A",1
23,1,283,302,0.4302557,"\int \frac{A+B x}{\left(a+b x+c x^2\right) \sqrt{d+f x^2}} \, dx","Integrate[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]),x]","\frac{\sqrt{2} \left(-\frac{\left(B \sqrt{b^2-4 a c}+2 A c-b B\right) \tanh ^{-1}\left(\frac{f x \left(\sqrt{b^2-4 a c}-b\right)+2 c d}{\sqrt{d+f x^2} \sqrt{2 b f \left(b-\sqrt{b^2-4 a c}\right)-4 a c f+4 c^2 d}}\right)}{2 \sqrt{b f \left(b-\sqrt{b^2-4 a c}\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(\sqrt{b^2-4 a c}+b\right)}{\sqrt{d+f x^2} \sqrt{2 b f \left(\sqrt{b^2-4 a c}+b\right)-4 a c f+4 c^2 d}}\right)}{2 \sqrt{b f \left(\sqrt{b^2-4 a c}+b\right)-2 a c f+2 c^2 d}}\right)}{\sqrt{b^2-4 a c}}","\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,"(Sqrt[2]*(-1/2*((-(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[4*c^2*d - 4*a*c*f + 2*b*(b - Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/Sqrt[2*c^2*d - 2*a*c*f + b*(b - Sqrt[b^2 - 4*a*c])*f] - ((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[4*c^2*d - 4*a*c*f + 2*b*(b + Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(2*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f])))/Sqrt[b^2 - 4*a*c]","A",1
24,1,154,101,0.1737836,"\int \frac{A+B x}{\left(a+c x^2\right) \sqrt{d+f x^2}} \, dx","Integrate[(A + B*x)/((a + c*x^2)*Sqrt[d + f*x^2]),x]","\frac{\left(A \sqrt{c}-\sqrt{-a} B\right) \tanh ^{-1}\left(\frac{\sqrt{c} d-\sqrt{-a} f x}{\sqrt{d+f x^2} \sqrt{c d-a f}}\right)-\left(\sqrt{-a} B+A \sqrt{c}\right) \tanh ^{-1}\left(\frac{\sqrt{-a} f x+\sqrt{c} d}{\sqrt{d+f x^2} \sqrt{c d-a f}}\right)}{2 \sqrt{-a} \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,"((-(Sqrt[-a]*B) + A*Sqrt[c])*ArcTanh[(Sqrt[c]*d - Sqrt[-a]*f*x)/(Sqrt[c*d - a*f]*Sqrt[d + f*x^2])] - (Sqrt[-a]*B + A*Sqrt[c])*ArcTanh[(Sqrt[c]*d + Sqrt[-a]*f*x)/(Sqrt[c*d - a*f]*Sqrt[d + f*x^2])])/(2*Sqrt[-a]*Sqrt[c]*Sqrt[c*d - a*f])","A",1
25,1,140,139,0.3011884,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \sqrt{1+3 x-2 x^2}} \, dx","Integrate[(2 + x)/((2 + 4*x - 3*x^2)*Sqrt[1 + 3*x - 2*x^2]),x]","\frac{\left(4 \sqrt{10}-5\right) \tan ^{-1}\left(\frac{4 \sqrt{10} x+x-3 \sqrt{10}+12}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right)+3 \sqrt{5 \left(7+2 \sqrt{10}\right)} \tanh ^{-1}\left(\frac{-4 \sqrt{10} x+x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right)}{10 \sqrt{1+\sqrt{10}}}","\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,"((-5 + 4*Sqrt[10])*ArcTan[(12 - 3*Sqrt[10] + x + 4*Sqrt[10]*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])] + 3*Sqrt[5*(7 + 2*Sqrt[10])]*ArcTanh[(3*(4 + Sqrt[10]) + x - 4*Sqrt[10]*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/(10*Sqrt[1 + Sqrt[10]])","A",0
26,1,167,166,0.377637,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \left(1+3 x-2 x^2\right)^{3/2}} \, dx","Integrate[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x - 2*x^2)^(3/2)),x]","\frac{1}{170} \left(153 \sqrt{5 \left(3+\sqrt{10}\right)} \tanh ^{-1}\left(\frac{-4 \sqrt{10} x+x+3 \left(4+\sqrt{10}\right)}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right)-\frac{153 \sqrt{5 \left(\sqrt{10}-3\right)} \sqrt{-2 x^2+3 x+1} \tan ^{-1}\left(\frac{4 \sqrt{10} x+x-3 \sqrt{10}+12}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right)+280 x+300}{\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,"(-((300 + 280*x + 153*Sqrt[5*(-3 + Sqrt[10])]*Sqrt[1 + 3*x - 2*x^2]*ArcTan[(12 - 3*Sqrt[10] + x + 4*Sqrt[10]*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/Sqrt[1 + 3*x - 2*x^2]) + 153*Sqrt[5*(3 + Sqrt[10])]*ArcTanh[(3*(4 + Sqrt[10]) + x - 4*Sqrt[10]*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/170","A",1
27,1,185,193,0.5955289,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \left(1+3 x-2 x^2\right)^{5/2}} \, dx","Integrate[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x - 2*x^2)^(5/2)),x]","\frac{3}{10} \sqrt{1+\sqrt{10}} \left(7 \sqrt{10}-25\right) \tan ^{-1}\left(\frac{3 \left(\sqrt{10}-4\right)-\left(1+4 \sqrt{10}\right) x}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right)-\frac{3}{10} \sqrt{\sqrt{10}-1} \left(25+7 \sqrt{10}\right) \tanh ^{-1}\left(\frac{\left(4 \sqrt{10}-1\right) x-3 \left(4+\sqrt{10}\right)}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right)-\frac{2 \left(-9628 x^3+13860 x^2+5925 x+546\right)}{867 \left(-2 x^2+3 x+1\right)^{3/2}}","-\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*(546 + 5925*x + 13860*x^2 - 9628*x^3))/(867*(1 + 3*x - 2*x^2)^(3/2)) + (3*Sqrt[1 + Sqrt[10]]*(-25 + 7*Sqrt[10])*ArcTan[(3*(-4 + Sqrt[10]) - (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/10 - (3*Sqrt[-1 + Sqrt[10]]*(25 + 7*Sqrt[10])*ArcTanh[(-3*(4 + Sqrt[10]) + (-1 + 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/10","A",1
28,1,148,151,0.3555588,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \sqrt{1+3 x+2 x^2}} \, dx","Integrate[(2 + x)/((2 + 4*x - 3*x^2)*Sqrt[1 + 3*x + 2*x^2]),x]","\frac{\left(5-4 \sqrt{10}\right) \tanh ^{-1}\left(\frac{-4 \sqrt{10} x+17 x-3 \sqrt{10}+12}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)+3 \sqrt{285-90 \sqrt{10}} \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)}{10 \sqrt{55-17 \sqrt{10}}}","\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,"((5 - 4*Sqrt[10])*ArcTanh[(12 - 3*Sqrt[10] + 17*x - 4*Sqrt[10]*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])] + 3*Sqrt[285 - 90*Sqrt[10]]*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[55 - 17*Sqrt[10]])","A",0
29,1,172,174,0.5699045,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \left(1+3 x+2 x^2\right)^{3/2}} \, dx","Integrate[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x + 2*x^2)^(3/2)),x]","\frac{1}{50} \left(\frac{\sqrt{30975-9795 \sqrt{10}} \sqrt{2 x^2+3 x+1} \tanh ^{-1}\left(\frac{4 \sqrt{10} x+17 x+3 \sqrt{10}+12}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)+440 x+420}{\sqrt{2 x^2+3 x+1}}-\sqrt{30975+9795 \sqrt{10}} \tanh ^{-1}\left(\frac{-4 \sqrt{10} x+17 x-3 \sqrt{10}+12}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)\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,"(-(Sqrt[30975 + 9795*Sqrt[10]]*ArcTanh[(12 - 3*Sqrt[10] + 17*x - 4*Sqrt[10]*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])]) + (420 + 440*x + Sqrt[30975 - 9795*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2]*ArcTanh[(12 + 3*Sqrt[10] + 17*x + 4*Sqrt[10]*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/Sqrt[1 + 3*x + 2*x^2])/50","A",1
30,1,190,197,0.7095502,"\int \frac{2+x}{\left(2+4 x-3 x^2\right) \left(1+3 x+2 x^2\right)^{5/2}} \, dx","Integrate[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x + 2*x^2)^(5/2)),x]","\frac{1}{450} \left(\sqrt{55-17 \sqrt{10}} \left(7289+2305 \sqrt{10}\right) \tanh ^{-1}\left(\frac{\left(4 \sqrt{10}-17\right) x+3 \left(\sqrt{10}-4\right)}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)-\sqrt{55+17 \sqrt{10}} \left(2305 \sqrt{10}-7289\right) \tanh ^{-1}\left(\frac{-\left(\left(17+4 \sqrt{10}\right) x\right)-3 \left(4+\sqrt{10}\right)}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right)+\frac{60 \left(460 x^3+1236 x^2+1071 x+294\right)}{\left(2 x^2+3 x+1\right)^{3/2}}\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,"((60*(294 + 1071*x + 1236*x^2 + 460*x^3))/(1 + 3*x + 2*x^2)^(3/2) + Sqrt[55 - 17*Sqrt[10]]*(7289 + 2305*Sqrt[10])*ArcTanh[(3*(-4 + Sqrt[10]) + (-17 + 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])] - Sqrt[55 + 17*Sqrt[10]]*(-7289 + 2305*Sqrt[10])*ArcTanh[(-3*(4 + Sqrt[10]) - (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/450","A",1
31,1,79,15,0.0447355,"\int \frac{1+x}{\left(4+2 x+x^2\right) \sqrt{5+2 x+x^2}} \, dx","Integrate[(1 + x)/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]","\frac{1}{2} \left(-\tanh ^{-1}\left(\frac{-i \sqrt{3} x-i \sqrt{3}+4}{\sqrt{x^2+2 x+5}}\right)-\tanh ^{-1}\left(\frac{i \sqrt{3} x+i \sqrt{3}+4}{\sqrt{x^2+2 x+5}}\right)\right)","-\tanh ^{-1}\left(\sqrt{x^2+2 x+5}\right)",1,"(-ArcTanh[(4 - I*Sqrt[3] - I*Sqrt[3]*x)/Sqrt[5 + 2*x + x^2]] - ArcTanh[(4 + I*Sqrt[3] + I*Sqrt[3]*x)/Sqrt[5 + 2*x + x^2]])/2","C",1
32,1,101,44,0.0547138,"\int \frac{4+x}{\left(4+2 x+x^2\right) \sqrt{5+2 x+x^2}} \, dx","Integrate[(4 + x)/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]","-\frac{1}{2} \left(1+i \sqrt{3}\right) \tanh ^{-1}\left(\frac{-i \sqrt{3} x-i \sqrt{3}+4}{\sqrt{x^2+2 x+5}}\right)-\frac{1}{2} \left(1-i \sqrt{3}\right) \tanh ^{-1}\left(\frac{i \sqrt{3} x+i \sqrt{3}+4}{\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,"-1/2*((1 + I*Sqrt[3])*ArcTanh[(4 - I*Sqrt[3] - I*Sqrt[3]*x)/Sqrt[5 + 2*x + x^2]]) - ((1 - I*Sqrt[3])*ArcTanh[(4 + I*Sqrt[3] + I*Sqrt[3]*x)/Sqrt[5 + 2*x + x^2]])/2","C",1
33,1,90,24,0.0633911,"\int \frac{1+2 x}{\left(3+x+x^2\right) \sqrt{5+x+x^2}} \, dx","Integrate[(1 + 2*x)/((3 + x + x^2)*Sqrt[5 + x + x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{-2 i \sqrt{11} x-i \sqrt{11}+19}{4 \sqrt{2} \sqrt{x^2+x+5}}\right)+\tanh ^{-1}\left(\frac{2 i \sqrt{11} x+i \sqrt{11}+19}{4 \sqrt{2} \sqrt{x^2+x+5}}\right)}{\sqrt{2}}","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{x^2+x+5}}{\sqrt{2}}\right)",1,"-((ArcTanh[(19 - I*Sqrt[11] - (2*I)*Sqrt[11]*x)/(4*Sqrt[2]*Sqrt[5 + x + x^2])] + ArcTanh[(19 + I*Sqrt[11] + (2*I)*Sqrt[11]*x)/(4*Sqrt[2]*Sqrt[5 + x + x^2])])/Sqrt[2])","C",1
34,1,114,56,0.0608212,"\int \frac{x}{\left(3+x+x^2\right) \sqrt{5+x+x^2}} \, dx","Integrate[x/((3 + x + x^2)*Sqrt[5 + x + x^2]),x]","\frac{-\left(\left(\sqrt{11}-i\right) \tanh ^{-1}\left(\frac{-2 i \sqrt{11} x-i \sqrt{11}+19}{4 \sqrt{2} \sqrt{x^2+x+5}}\right)\right)-\left(\sqrt{11}+i\right) \tanh ^{-1}\left(\frac{2 i \sqrt{11} x+i \sqrt{11}+19}{4 \sqrt{2} \sqrt{x^2+x+5}}\right)}{2 \sqrt{22}}","-\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,"(-((-I + Sqrt[11])*ArcTanh[(19 - I*Sqrt[11] - (2*I)*Sqrt[11]*x)/(4*Sqrt[2]*Sqrt[5 + x + x^2])]) - (I + Sqrt[11])*ArcTanh[(19 + I*Sqrt[11] + (2*I)*Sqrt[11]*x)/(4*Sqrt[2]*Sqrt[5 + x + x^2])])/(2*Sqrt[22])","C",1
35,1,767,249,1.4886014,"\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","Integrate[(A + B*x)/(Sqrt[d + e*x + f*x^2]*(a*e + b*e*x + b*f*x^2)^2),x]","-\frac{-(a e+b x (e+f x)) \log \left(b (e+2 f x)-\sqrt{b} \sqrt{e} \sqrt{b e-4 a f}\right) \left(-8 a b e f (B e-2 A f)-b^{3/2} B e^{5/2} \sqrt{b e-4 a f}+4 a \sqrt{b} B e^{3/2} f \sqrt{b e-4 a f}+b^2 \left(4 d f+e^2\right) (B e-2 A f)\right)+(a e+b x (e+f x)) \log \left(\sqrt{b} \sqrt{e} \sqrt{b e-4 a f}+b (e+2 f x)\right) \left(-8 a b e f (B e-2 A f)+b^{3/2} B e^{5/2} \sqrt{b e-4 a f}-4 a \sqrt{b} B e^{3/2} f \sqrt{b e-4 a f}+b^2 \left(4 d f+e^2\right) (B e-2 A f)\right)-(a e+b x (e+f x)) \left(-8 a b e f (B e-2 A f)+b^{3/2} B e^{5/2} \sqrt{b e-4 a f}-4 a \sqrt{b} B e^{3/2} f \sqrt{b e-4 a f}+b^2 \left(4 d f+e^2\right) (B e-2 A f)\right) \log \left(\sqrt{b} \left(-4 f \sqrt{b d-a e} \sqrt{d+x (e+f x)}+e^{3/2} \sqrt{b e-4 a f}+2 \sqrt{e} f x \sqrt{b e-4 a f}+\sqrt{b} \left(e^2-4 d f\right)\right)\right)+(a e+b x (e+f x)) \left(-8 a b e f (B e-2 A f)-b^{3/2} B e^{5/2} \sqrt{b e-4 a f}+4 a \sqrt{b} B e^{3/2} f \sqrt{b e-4 a f}+b^2 \left(4 d f+e^2\right) (B e-2 A f)\right) \log \left(\sqrt{b} \left(4 f \sqrt{b d-a e} \sqrt{d+x (e+f x)}+e^{3/2} \sqrt{b e-4 a f}+2 \sqrt{e} f x \sqrt{b e-4 a f}-\sqrt{b} \left(e^2-4 d f\right)\right)\right)+4 b \sqrt{e} f \sqrt{b d-a e} \sqrt{b e-4 a f} \sqrt{d+x (e+f x)} (A b (e+2 f x)-B e (2 a+b x))}{4 b e^{3/2} f (b d-a e)^{3/2} (b e-4 a f)^{3/2} (a e+b x (e+f 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}}",1,"-1/4*(4*b*Sqrt[e]*Sqrt[b*d - a*e]*f*Sqrt[b*e - 4*a*f]*Sqrt[d + x*(e + f*x)]*(-(B*e*(2*a + b*x)) + A*b*(e + 2*f*x)) - (-(b^(3/2)*B*e^(5/2)*Sqrt[b*e - 4*a*f]) + 4*a*Sqrt[b]*B*e^(3/2)*f*Sqrt[b*e - 4*a*f] - 8*a*b*e*f*(B*e - 2*A*f) + b^2*(B*e - 2*A*f)*(e^2 + 4*d*f))*(a*e + b*x*(e + f*x))*Log[-(Sqrt[b]*Sqrt[e]*Sqrt[b*e - 4*a*f]) + b*(e + 2*f*x)] + (b^(3/2)*B*e^(5/2)*Sqrt[b*e - 4*a*f] - 4*a*Sqrt[b]*B*e^(3/2)*f*Sqrt[b*e - 4*a*f] - 8*a*b*e*f*(B*e - 2*A*f) + b^2*(B*e - 2*A*f)*(e^2 + 4*d*f))*(a*e + b*x*(e + f*x))*Log[Sqrt[b]*Sqrt[e]*Sqrt[b*e - 4*a*f] + b*(e + 2*f*x)] - (b^(3/2)*B*e^(5/2)*Sqrt[b*e - 4*a*f] - 4*a*Sqrt[b]*B*e^(3/2)*f*Sqrt[b*e - 4*a*f] - 8*a*b*e*f*(B*e - 2*A*f) + b^2*(B*e - 2*A*f)*(e^2 + 4*d*f))*(a*e + b*x*(e + f*x))*Log[Sqrt[b]*(e^(3/2)*Sqrt[b*e - 4*a*f] + Sqrt[b]*(e^2 - 4*d*f) + 2*Sqrt[e]*f*Sqrt[b*e - 4*a*f]*x - 4*Sqrt[b*d - a*e]*f*Sqrt[d + x*(e + f*x)])] + (-(b^(3/2)*B*e^(5/2)*Sqrt[b*e - 4*a*f]) + 4*a*Sqrt[b]*B*e^(3/2)*f*Sqrt[b*e - 4*a*f] - 8*a*b*e*f*(B*e - 2*A*f) + b^2*(B*e - 2*A*f)*(e^2 + 4*d*f))*(a*e + b*x*(e + f*x))*Log[Sqrt[b]*(e^(3/2)*Sqrt[b*e - 4*a*f] - Sqrt[b]*(e^2 - 4*d*f) + 2*Sqrt[e]*f*Sqrt[b*e - 4*a*f]*x + 4*Sqrt[b*d - a*e]*f*Sqrt[d + x*(e + f*x)])])/(b*e^(3/2)*(b*d - a*e)^(3/2)*f*(b*e - 4*a*f)^(3/2)*(a*e + b*x*(e + f*x)))","B",1
36,1,46,48,0.1851591,"\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","Integrate[((g + h*x)*Sqrt[a + b*x + c*x^2])/(a*d + b*d*x + c*d*x^2)^2,x]","\frac{4 a h-2 b g+2 b h x-4 c g x}{d^2 \left(b^2-4 a c\right) \sqrt{a+x (b+c 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}}",1,"(-2*b*g + 4*a*h - 4*c*g*x + 2*b*h*x)/((b^2 - 4*a*c)*d^2*Sqrt[a + x*(b + c*x)])","A",1
37,1,165,17,0.2878683,"\int \frac{3+2 x}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Integrate[(3 + 2*x)/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\frac{1}{6} \left(\sqrt{1-2 i \sqrt{2}} \left(\sqrt{2}+i\right) \tanh ^{-1}\left(\frac{\left(2-i \sqrt{2}\right) x-2 i \sqrt{2}+2}{\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)+\sqrt{1+2 i \sqrt{2}} \left(\sqrt{2}-i\right) \tanh ^{-1}\left(\frac{\left(2+i \sqrt{2}\right) x+2 i \sqrt{2}+2}{\sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)\right)","\tanh ^{-1}\left(\frac{x}{\sqrt{-x^2-4 x-3}}\right)",1,"(Sqrt[1 - (2*I)*Sqrt[2]]*(I + Sqrt[2])*ArcTanh[(2 - (2*I)*Sqrt[2] + (2 - I*Sqrt[2])*x)/(Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])] + Sqrt[1 + (2*I)*Sqrt[2]]*(-I + Sqrt[2])*ArcTanh[(2 + (2*I)*Sqrt[2] + (2 + I*Sqrt[2])*x)/(Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])])/6","C",1
38,1,150,86,0.1094095,"\int \frac{3+4 x}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Integrate[(3 + 4*x)/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","-\frac{1}{2} i \left(\sqrt{1+2 i \sqrt{2}} \tanh ^{-1}\left(\frac{\left(2-i \sqrt{2}\right) x-2 i \sqrt{2}+2}{\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)-\sqrt{1-2 i \sqrt{2}} \tanh ^{-1}\left(\frac{\left(2+i \sqrt{2}\right) x+2 i \sqrt{2}+2}{\sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)\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,"(-1/2*I)*(Sqrt[1 + (2*I)*Sqrt[2]]*ArcTanh[(2 - (2*I)*Sqrt[2] + (2 - I*Sqrt[2])*x)/(Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])] - Sqrt[1 - (2*I)*Sqrt[2]]*ArcTanh[(2 + (2*I)*Sqrt[2] + (2 + I*Sqrt[2])*x)/(Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])])","C",1
39,1,108,136,0.09104,"\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","Integrate[((g + h*x)*Sqrt[a + b*x + c*x^2])/(a*d + b*d*x + c*d*x^2)^(3/2),x]","\frac{(a+x (b+c x))^{3/2} \left((4 c g-2 b h) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)+h \sqrt{4 a c-b^2} \log (a+x (b+c x))\right)}{2 c \sqrt{4 a c-b^2} (d (a+x (b+c x)))^{3/2}}","\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,"((a + x*(b + c*x))^(3/2)*((4*c*g - 2*b*h)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*h*Log[a + x*(b + c*x)]))/(2*c*Sqrt[-b^2 + 4*a*c]*(d*(a + x*(b + c*x)))^(3/2))","A",1
40,1,129,212,0.1511919,"\int x^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2} \, dx","Integrate[x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2],x]","\frac{\sqrt{(a+b x)^2} \sqrt{c+d x^2} \left(\sqrt{\frac{d x^2}{c}+1} \left(15 a d x \left(c+2 d x^2\right)+8 b \left(-2 c^2+c d x^2+3 d^2 x^4\right)\right)-15 a c^{3/2} \sqrt{d} \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)\right)}{120 d^2 (a+b x) \sqrt{\frac{d x^2}{c}+1}}","-\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,"(Sqrt[(a + b*x)^2]*Sqrt[c + d*x^2]*(Sqrt[1 + (d*x^2)/c]*(15*a*d*x*(c + 2*d*x^2) + 8*b*(-2*c^2 + c*d*x^2 + 3*d^2*x^4)) - 15*a*c^(3/2)*Sqrt[d]*ArcSinh[(Sqrt[d]*x)/Sqrt[c]]))/(120*d^2*(a + b*x)*Sqrt[1 + (d*x^2)/c])","A",1
41,1,117,161,0.1156945,"\int x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2} \, dx","Integrate[x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2],x]","\frac{\sqrt{(a+b x)^2} \sqrt{c+d x^2} \left(\sqrt{d} \sqrt{\frac{d x^2}{c}+1} \left(8 a \left(c+d x^2\right)+3 b x \left(c+2 d x^2\right)\right)-3 b c^{3/2} \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)\right)}{24 d^{3/2} (a+b x) \sqrt{\frac{d x^2}{c}+1}}","-\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,"(Sqrt[(a + b*x)^2]*Sqrt[c + d*x^2]*(Sqrt[d]*Sqrt[1 + (d*x^2)/c]*(8*a*(c + d*x^2) + 3*b*x*(c + 2*d*x^2)) - 3*b*c^(3/2)*ArcSinh[(Sqrt[d]*x)/Sqrt[c]]))/(24*d^(3/2)*(a + b*x)*Sqrt[1 + (d*x^2)/c])","A",1
42,1,85,148,0.0626658,"\int \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2],x]","\frac{\sqrt{(a+b x)^2} \left(\sqrt{c+d x^2} \left(3 a d x+2 b \left(c+d x^2\right)\right)+3 a c \sqrt{d} \log \left(\sqrt{d} \sqrt{c+d x^2}+d x\right)\right)}{6 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,"(Sqrt[(a + b*x)^2]*(Sqrt[c + d*x^2]*(3*a*d*x + 2*b*(c + d*x^2)) + 3*a*c*Sqrt[d]*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]]))/(6*d*(a + b*x))","A",1
43,1,139,160,0.1428405,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{x} \, dx","Integrate[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/x,x]","\frac{\sqrt{(a+b x)^2} \left(\sqrt{d} \sqrt{\frac{d x^2}{c}+1} \left((2 a+b x) \sqrt{c+d x^2}-2 a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)\right)+b \sqrt{c} \sqrt{c+d x^2} \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)\right)}{2 \sqrt{d} (a+b x) \sqrt{\frac{d x^2}{c}+1}}","\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,"(Sqrt[(a + b*x)^2]*(b*Sqrt[c]*Sqrt[c + d*x^2]*ArcSinh[(Sqrt[d]*x)/Sqrt[c]] + Sqrt[d]*Sqrt[1 + (d*x^2)/c]*((2*a + b*x)*Sqrt[c + d*x^2] - 2*a*Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])))/(2*Sqrt[d]*(a + b*x)*Sqrt[1 + (d*x^2)/c])","A",1
44,1,118,156,0.2807249,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{x^2} \, dx","Integrate[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/x^2,x]","\frac{\sqrt{(a+b x)^2} \left(\frac{(b x-a) \sqrt{c+d x^2}}{x}+\frac{a \sqrt{c} \sqrt{d} \sqrt{\frac{d x^2}{c}+1} \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)}{\sqrt{c+d x^2}}-b \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)\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,"(Sqrt[(a + b*x)^2]*(((-a + b*x)*Sqrt[c + d*x^2])/x + (a*Sqrt[c]*Sqrt[d]*Sqrt[1 + (d*x^2)/c]*ArcSinh[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[c + d*x^2] - b*Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]))/(a + b*x)","A",1
45,1,126,161,0.1164117,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{x^3} \, dx","Integrate[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/x^3,x]","-\frac{\sqrt{(a+b x)^2} \sqrt{c+d x^2} \left(c (a+2 b x) \sqrt{\frac{d x^2}{c}+1}+a d x^2 \tanh ^{-1}\left(\sqrt{\frac{d x^2}{c}+1}\right)-2 b \sqrt{c} \sqrt{d} x^2 \sinh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)\right)}{2 c x^2 (a+b x) \sqrt{\frac{d x^2}{c}+1}}","-\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,"-1/2*(Sqrt[(a + b*x)^2]*Sqrt[c + d*x^2]*(c*(a + 2*b*x)*Sqrt[1 + (d*x^2)/c] - 2*b*Sqrt[c]*Sqrt[d]*x^2*ArcSinh[(Sqrt[d]*x)/Sqrt[c]] + a*d*x^2*ArcTanh[Sqrt[1 + (d*x^2)/c]]))/(c*x^2*(a + b*x)*Sqrt[1 + (d*x^2)/c])","A",1
46,1,198,317,0.2748496,"\int x^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2} \, dx","Integrate[x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2],x]","\frac{\sqrt{(a+b x)^2} \left(-\frac{5 \left(2 a d \left(4 c d-5 e^2\right)+b \left(7 e^3-12 c d e\right)\right) \left(\left(4 c d-e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right)+2 \sqrt{d} (2 d x+e) \sqrt{c+x (d x+e)}\right)}{256 d^{7/2}}+\frac{(c+x (d x+e))^{3/2} (10 a d (6 d x-5 e)-32 b c d+7 b e (5 e-6 d x))}{48 d^2}+b x^2 (c+x (d x+e))^{3/2}\right)}{5 d (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(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} \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,"(Sqrt[(a + b*x)^2]*(b*x^2*(c + x*(e + d*x))^(3/2) + ((c + x*(e + d*x))^(3/2)*(-32*b*c*d + 7*b*e*(5*e - 6*d*x) + 10*a*d*(-5*e + 6*d*x)))/(48*d^2) - (5*(2*a*d*(4*c*d - 5*e^2) + b*(-12*c*d*e + 7*e^3))*(2*Sqrt[d]*(e + 2*d*x)*Sqrt[c + x*(e + d*x)] + (4*c*d - e^2)*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])]))/(256*d^(7/2))))/(5*d*(a + b*x))","A",1
47,1,147,227,0.1307386,"\int x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2} \, dx","Integrate[x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2],x]","\frac{\sqrt{(a+b x)^2} \left((c+x (d x+e))^{3/2} (8 a d+6 b d x-5 b e)-\frac{3 \left(8 a d e+4 b c d-5 b e^2\right) \left(\left(4 c d-e^2\right) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right)+2 \sqrt{d} (2 d x+e) \sqrt{c+x (d x+e)}\right)}{16 d^{3/2}}\right)}{24 d^2 (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} (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(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,"(Sqrt[(a + b*x)^2]*((8*a*d - 5*b*e + 6*b*d*x)*(c + x*(e + d*x))^(3/2) - (3*(4*b*c*d + 8*a*d*e - 5*b*e^2)*(2*Sqrt[d]*(e + 2*d*x)*Sqrt[c + x*(e + d*x)] + (4*c*d - e^2)*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])]))/(16*d^(3/2))))/(24*d^2*(a + b*x))","A",1
48,1,134,198,0.1274853,"\int \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2],x]","\frac{\sqrt{(a+b x)^2} \left(2 \sqrt{d} \sqrt{c+x (d x+e)} \left(6 a d (2 d x+e)+b \left(8 c d+8 d^2 x^2+2 d e x-3 e^2\right)\right)+3 \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+x (d x+e)}}\right)\right)}{48 d^{5/2} (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,"(Sqrt[(a + b*x)^2]*(2*Sqrt[d]*Sqrt[c + x*(e + d*x)]*(6*a*d*(e + 2*d*x) + b*(8*c*d - 3*e^2 + 2*d*e*x + 8*d^2*x^2)) + 3*(2*a*d - b*e)*(4*c*d - e^2)*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])]))/(48*d^(5/2)*(a + b*x))","A",1
49,1,149,211,0.1978943,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2}}{x} \, dx","Integrate[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/x,x]","\frac{\sqrt{(a+b x)^2} \left(\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+x (d x+e)}}\right)+2 \sqrt{d} \left(\sqrt{c+x (d x+e)} (4 a d+b (2 d x+e))-4 a \sqrt{c} d \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+x (d x+e)}}\right)\right)\right)}{8 d^{3/2} (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,"(Sqrt[(a + b*x)^2]*((4*b*c*d + 4*a*d*e - b*e^2)*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])] + 2*Sqrt[d]*(Sqrt[c + x*(e + d*x)]*(4*a*d + b*(e + 2*d*x)) - 4*a*Sqrt[c]*d*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + x*(e + d*x)])])))/(8*d^(3/2)*(a + b*x))","A",1
50,1,155,202,0.1894217,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2}}{x^2} \, dx","Integrate[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/x^2,x]","\frac{\sqrt{(a+b x)^2} \left(\sqrt{c} x (2 a d+b e) \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right)+\sqrt{d} \left(2 \sqrt{c} (b x-a) \sqrt{c+x (d x+e)}-x (a e+2 b c) \tanh ^{-1}\left(\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+x (d x+e)}}\right)\right)\right)}{2 \sqrt{c} \sqrt{d} x (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,"(Sqrt[(a + b*x)^2]*(Sqrt[c]*(2*a*d + b*e)*x*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])] + Sqrt[d]*(2*Sqrt[c]*(-a + b*x)*Sqrt[c + x*(e + d*x)] - (2*b*c + a*e)*x*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + x*(e + d*x)])])))/(2*Sqrt[c]*Sqrt[d]*x*(a + b*x))","A",1
51,1,161,215,0.2183048,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+e x+d x^2}}{x^3} \, dx","Integrate[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + e*x + d*x^2])/x^3,x]","-\frac{\sqrt{(a+b x)^2} \left(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+x (d x+e)}}\right)+2 \sqrt{c} \sqrt{c+x (d x+e)} (2 a c+a e x+4 b c x)-8 b c^{3/2} \sqrt{d} x^2 \tanh ^{-1}\left(\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right)\right)}{8 c^{3/2} x^2 (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,"-1/8*(Sqrt[(a + b*x)^2]*(2*Sqrt[c]*(2*a*c + 4*b*c*x + a*e*x)*Sqrt[c + x*(e + d*x)] - 8*b*c^(3/2)*Sqrt[d]*x^2*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])] + (4*a*c*d + 4*b*c*e - a*e^2)*x^2*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + x*(e + d*x)])]))/(c^(3/2)*x^2*(a + b*x))","A",1
52,1,516,452,2.4995404,"\int \frac{x^2 \sqrt{a+c x^2}}{d+e x+f x^2} \, dx","Integrate[(x^2*Sqrt[a + c*x^2])/(d + e*x + f*x^2),x]","\frac{\frac{2 f \left(\frac{a^{3/2} \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{c}}+a x+c x^3\right)}{\sqrt{a+c x^2}}+\frac{\left(\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}+e\right) \left(\sqrt{4 a f^2-2 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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)-\sqrt{c} \left(\sqrt{e^2-4 d f}-e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)\right)}{f}+\frac{\left(e \sqrt{e^2-4 d f}-2 d f+e^2\right) \left(\sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)+\sqrt{c} \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)\right)}{f \sqrt{e^2-4 d f}}-2 \sqrt{a+c x^2} \left(\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}+e\right)-2 \sqrt{a+c x^2} \left(\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}+e\right)}{4 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 + (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2] - 2*(e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2] + (2*f*(a*x + c*x^3 + (a^(3/2)*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[c]))/Sqrt[a + c*x^2] + ((e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*(-(Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]]) + Sqrt[4*a*f^2 - 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[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f + ((e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])*(Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/(f*Sqrt[e^2 - 4*d*f]))/(4*f^2)","A",1
53,1,422,395,1.5242742,"\int \frac{x \sqrt{a+c x^2}}{d+e x+f x^2} \, dx","Integrate[(x*Sqrt[a + c*x^2])/(d + e*x + f*x^2),x]","-\frac{\frac{\left(\sqrt{e^2-4 d f}-e\right) \sqrt{4 a f^2-2 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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)}{\sqrt{e^2-4 d f}}+\frac{e \sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f}}+\sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)+4 \sqrt{c} e \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)-4 f \sqrt{a+c x^2}}{4 f^2}","-\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,"-1/4*(-4*f*Sqrt[a + c*x^2] + 4*Sqrt[c]*e*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + ((-e + Sqrt[e^2 - 4*d*f])*Sqrt[4*a*f^2 - 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[4*a*f^2 - 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[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])] + (e*Sqrt[4*a*f^2 + 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[4*a*f^2 + 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])/f^2","A",1
54,1,282,298,0.362601,"\int \frac{\sqrt{a+c x^2}}{d+e x+f x^2} \, dx","Integrate[Sqrt[a + c*x^2]/(d + e*x + f*x^2),x]","\frac{-\sqrt{4 a f^2-2 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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)+\sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)+2 \sqrt{c} \sqrt{e^2-4 d f} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{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(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,"(2*Sqrt[c]*Sqrt[e^2 - 4*d*f]*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[4*a*f^2 - 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[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])] + Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(2*f*Sqrt[e^2 - 4*d*f])","A",1
55,1,314,358,0.6512199,"\int \frac{\sqrt{a+c x^2}}{x \left(d+e x+f x^2\right)} \, dx","Integrate[Sqrt[a + c*x^2]/(x*(d + e*x + f*x^2)),x]","\frac{\left(\sqrt{e^2-4 d f}+e\right) \sqrt{4 a f^2-2 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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)+\left(\sqrt{e^2-4 d f}-e\right) \sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)-4 \sqrt{a} f \sqrt{e^2-4 d f} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{4 d f \sqrt{e^2-4 d f}}","\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,"((e + Sqrt[e^2 - 4*d*f])*Sqrt[4*a*f^2 - 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[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])] + (-e + Sqrt[e^2 - 4*d*f])*Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])] - 4*Sqrt[a]*f*Sqrt[e^2 - 4*d*f]*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(4*d*f*Sqrt[e^2 - 4*d*f])","A",1
56,1,569,382,3.0333219,"\int \frac{\sqrt{a+c x^2}}{x^2 \left(d+e x+f x^2\right)} \, dx","Integrate[Sqrt[a + c*x^2]/(x^2*(d + e*x + f*x^2)),x]","\frac{\frac{\left(e \sqrt{e^2-4 d f}-2 d f+e^2\right) \left(\sqrt{c} \left(\sqrt{e^2-4 d f}-e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)-\sqrt{4 a f^2-2 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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)\right)}{f \sqrt{e^2-4 d f}}+\frac{\left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right) \left(\sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)+\sqrt{c} \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)\right)}{f \sqrt{e^2-4 d f}}+2 \sqrt{a+c x^2} \left(\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}+e\right)+2 \sqrt{a+c x^2} \left(\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}+e\right)-\frac{4 d \left(-\sqrt{a} \sqrt{c} x \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)+a+c x^2\right)}{x \sqrt{a+c x^2}}-4 e \left(\sqrt{a+c x^2}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)\right)}{4 d^2}","-\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,"(2*(e + (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2] + 2*(e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2] - (4*d*(a + c*x^2 - Sqrt[a]*Sqrt[c]*x*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(x*Sqrt[a + c*x^2]) + ((e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])*(Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[4*a*f^2 - 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[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/(f*Sqrt[e^2 - 4*d*f]) + ((e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])*(Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/(f*Sqrt[e^2 - 4*d*f]) - 4*e*(Sqrt[a + c*x^2] - Sqrt[a]*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]))/(4*d^2)","A",1
57,1,642,507,2.6065063,"\int \frac{\sqrt{a+c x^2}}{x^3 \left(d+e x+f x^2\right)} \, dx","Integrate[Sqrt[a + c*x^2]/(x^3*(d + e*x + f*x^2)),x]","\frac{-\frac{2 d^2 \left(c x^2 \sqrt{\frac{c x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{c x^2}{a}+1}\right)+a+c x^2\right)}{x^2 \sqrt{a+c x^2}}+\frac{\left(\frac{e \left(e^2-3 d f\right)}{\sqrt{e^2-4 d f}}-d f+e^2\right) \left(\sqrt{4 a f^2-2 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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)-\sqrt{c} \left(\sqrt{e^2-4 d f}-e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)\right)}{f}+\frac{\left(-\frac{e \left(e^2-3 d f\right)}{\sqrt{e^2-4 d f}}-d f+e^2\right) \left(\sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)+\sqrt{c} \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)\right)}{f}-2 \sqrt{a+c x^2} \left(-\frac{e \left(e^2-3 d f\right)}{\sqrt{e^2-4 d f}}-d f+e^2\right)-2 \sqrt{a+c x^2} \left(\frac{e \left(e^2-3 d f\right)}{\sqrt{e^2-4 d f}}-d f+e^2\right)+4 \left(e^2-d f\right) \left(\sqrt{a+c x^2}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)\right)+\frac{4 d e \left(-\sqrt{a} \sqrt{c} x \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)+a+c x^2\right)}{x \sqrt{a+c x^2}}}{4 d^3}","-\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{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+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,"(-2*(e^2 - d*f - (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2] - 2*(e^2 - d*f + (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2] + (4*d*e*(a + c*x^2 - Sqrt[a]*Sqrt[c]*x*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(x*Sqrt[a + c*x^2]) + ((e^2 - d*f + (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*(-(Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]]) + Sqrt[4*a*f^2 - 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[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f + ((e^2 - d*f - (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*(Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f + 4*(e^2 - d*f)*(Sqrt[a + c*x^2] - Sqrt[a]*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]) - (2*d^2*(a + c*x^2 + c*x^2*Sqrt[1 + (c*x^2)/a]*ArcTanh[Sqrt[1 + (c*x^2)/a]]))/(x^2*Sqrt[a + c*x^2]))/(4*d^3)","A",1
58,1,793,795,3.4492123,"\int \frac{x^2 \left(a+c x^2\right)^{3/2}}{d+e x+f x^2} \, dx","Integrate[(x^2*(a + c*x^2)^(3/2))/(d + e*x + f*x^2),x]","\frac{3 f \sqrt{a+c x^2} \left(\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{c} \sqrt{\frac{c x^2}{a}+1}}+5 a x+2 c x^3\right)-\frac{3 \left(\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}+e\right) \left(\frac{2 \left(2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \left(-\sqrt{4 a f^2-2 c e \sqrt{e^2-4 d f}-4 c d f+2 c e^2} \tanh ^{-1}\left(\frac{2 a f+c x \left(\sqrt{e^2-4 d f}-e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)+\sqrt{c} \left(\sqrt{e^2-4 d f}-e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+2 f \sqrt{a+c x^2}\right)}{f^2}+\frac{2 \sqrt{c} \sqrt{a+c x^2} \left(\sqrt{e^2-4 d f}-e\right) \left(\sqrt{c} x \sqrt{\frac{c x^2}{a}+1}+\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right)}{\sqrt{\frac{c x^2}{a}+1}}\right)}{2 f}+\frac{3 \left(\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}+e\right) \left(\frac{2 \left(2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \left(\sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)+\sqrt{c} \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)-2 f \sqrt{a+c x^2}\right)}{f^2}+\frac{2 \sqrt{c} \sqrt{a+c x^2} \left(\sqrt{e^2-4 d f}+e\right) \left(\sqrt{c} x \sqrt{\frac{c x^2}{a}+1}+\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right)}{\sqrt{\frac{c x^2}{a}+1}}\right)}{2 f}-4 \left(a+c x^2\right)^{3/2} \left(\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}+e\right)-4 \left(a+c x^2\right)^{3/2} \left(\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}+e\right)}{24 f^2}","-\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,"(-4*(e + (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*(a + c*x^2)^(3/2) - 4*(e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*(a + c*x^2)^(3/2) + 3*f*Sqrt[a + c*x^2]*(5*a*x + 2*c*x^3 + (3*a^(3/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[c]*Sqrt[1 + (c*x^2)/a])) - (3*(e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*((2*Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/Sqrt[1 + (c*x^2)/a] + (2*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))*(2*f*Sqrt[a + c*x^2] + Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[2*c*e^2 - 4*c*d*f + 4*a*f^2 - 2*c*e*Sqrt[e^2 - 4*d*f]]*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f^2))/(2*f) + (3*(e + (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*((2*Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/Sqrt[1 + (c*x^2)/a] + (2*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*(-2*f*Sqrt[a + c*x^2] + Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f^2))/(2*f))/(24*f^2)","A",1
59,1,755,553,2.011058,"\int \frac{x \left(a+c x^2\right)^{3/2}}{d+e x+f x^2} \, dx","Integrate[(x*(a + c*x^2)^(3/2))/(d + e*x + f*x^2),x]","\frac{8 f^3 \left(a+c x^2\right)^{5/2} \sqrt{\frac{c x^2}{a}+1} \left(\sqrt{e^2-4 d f}-e\right)+8 f^3 \left(a+c x^2\right)^{5/2} \sqrt{\frac{c x^2}{a}+1} \left(\sqrt{e^2-4 d f}+e\right)+3 \left(e-\sqrt{e^2-4 d f}\right) \left(2 \sqrt{c} f^2 \sqrt{a+c x^2} \left(e-\sqrt{e^2-4 d f}\right) \left(a \sqrt{c} x \left(\frac{c x^2}{a}+1\right)^{3/2}+\sqrt{a} \left(a+c x^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right)-a \left(\frac{c x^2}{a}+1\right)^{3/2} \left(4 a f^2+c \left(e-\sqrt{e^2-4 d f}\right)^2\right) \left(-\sqrt{4 a f^2-2 c e \sqrt{e^2-4 d f}-4 c d f+2 c e^2} \tanh ^{-1}\left(\frac{2 a f+c x \left(\sqrt{e^2-4 d f}-e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)+\sqrt{c} \left(\sqrt{e^2-4 d f}-e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+2 f \sqrt{a+c x^2}\right)\right)-3 \left(\sqrt{e^2-4 d f}+e\right) \left(2 \sqrt{c} f^2 \sqrt{a+c x^2} \left(\sqrt{e^2-4 d f}+e\right) \left(a \sqrt{c} x \left(\frac{c x^2}{a}+1\right)^{3/2}+\sqrt{a} \left(a+c x^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right)-a \left(\frac{c x^2}{a}+1\right)^{3/2} \left(4 a f^2+c \left(\sqrt{e^2-4 d f}+e\right)^2\right) \left(-\sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)-\sqrt{c} \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+2 f \sqrt{a+c x^2}\right)\right)}{48 a f^4 \left(\frac{c x^2}{a}+1\right)^{3/2} \sqrt{e^2-4 d 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{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{\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{\left(a+c x^2\right)^{3/2}}{3 f}",1,"(8*f^3*(-e + Sqrt[e^2 - 4*d*f])*(a + c*x^2)^(5/2)*Sqrt[1 + (c*x^2)/a] + 8*f^3*(e + Sqrt[e^2 - 4*d*f])*(a + c*x^2)^(5/2)*Sqrt[1 + (c*x^2)/a] + 3*(e - Sqrt[e^2 - 4*d*f])*(2*Sqrt[c]*f^2*(e - Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(a*Sqrt[c]*x*(1 + (c*x^2)/a)^(3/2) + Sqrt[a]*(a + c*x^2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]) - a*(4*a*f^2 + c*(e - Sqrt[e^2 - 4*d*f])^2)*(1 + (c*x^2)/a)^(3/2)*(2*f*Sqrt[a + c*x^2] + Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[2*c*e^2 - 4*c*d*f + 4*a*f^2 - 2*c*e*Sqrt[e^2 - 4*d*f]]*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])) - 3*(e + Sqrt[e^2 - 4*d*f])*(2*Sqrt[c]*f^2*(e + Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(a*Sqrt[c]*x*(1 + (c*x^2)/a)^(3/2) + Sqrt[a]*(a + c*x^2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]) - a*(4*a*f^2 + c*(e + Sqrt[e^2 - 4*d*f])^2)*(1 + (c*x^2)/a)^(3/2)*(2*f*Sqrt[a + c*x^2] - Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])))/(48*a*f^4*Sqrt[e^2 - 4*d*f]*(1 + (c*x^2)/a)^(3/2))","A",1
60,1,603,484,1.0734421,"\int \frac{\left(a+c x^2\right)^{3/2}}{d+e x+f x^2} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x + f*x^2),x]","\frac{\frac{2 \left(2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \left(-\sqrt{4 a f^2-2 c e \sqrt{e^2-4 d f}-4 c d f+2 c e^2} \tanh ^{-1}\left(\frac{2 a f+c x \left(\sqrt{e^2-4 d f}-e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)+\sqrt{c} \left(\sqrt{e^2-4 d f}-e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+2 f \sqrt{a+c x^2}\right)}{f^2}+\frac{2 \left(2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right) \left(\sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)+\sqrt{c} \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)-2 f \sqrt{a+c x^2}\right)}{f^2}+\frac{2 \sqrt{c} \sqrt{a+c x^2} \left(\sqrt{e^2-4 d f}-e\right) \left(\sqrt{c} x \sqrt{\frac{c x^2}{a}+1}+\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right)}{\sqrt{\frac{c x^2}{a}+1}}+\frac{2 \sqrt{c} \sqrt{a+c x^2} \left(\sqrt{e^2-4 d f}+e\right) \left(\sqrt{c} x \sqrt{\frac{c x^2}{a}+1}+\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right)}{\sqrt{\frac{c x^2}{a}+1}}}{8 f \sqrt{e^2-4 d f}}","-\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,"((2*Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/Sqrt[1 + (c*x^2)/a] + (2*Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/Sqrt[1 + (c*x^2)/a] + (2*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))*(2*f*Sqrt[a + c*x^2] + Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[2*c*e^2 - 4*c*d*f + 4*a*f^2 - 2*c*e*Sqrt[e^2 - 4*d*f]]*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f^2 + (2*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*(-2*f*Sqrt[a + c*x^2] + Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f^2)/(8*f*Sqrt[e^2 - 4*d*f])","A",1
61,1,746,496,1.5751237,"\int \frac{\left(a+c x^2\right)^{3/2}}{x \left(d+e x+f x^2\right)} \, dx","Integrate[(a + c*x^2)^(3/2)/(x*(d + e*x + f*x^2)),x]","-\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{a \sqrt{a f^2+\frac{1}{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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{2 d f}-\frac{a e \sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{4 d f \sqrt{e^2-4 d f}}-\frac{\left(c d \left(\sqrt{e^2-4 d f}-e\right)-a f \left(\sqrt{e^2-4 d f}+e\right)\right) \sqrt{4 a f^2-2 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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)}{4 d f^2 \sqrt{e^2-4 d f}}-\frac{c \sqrt{a f^2+\frac{1}{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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{2 f^2}-\frac{c e \sqrt{4 a f^2+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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{4 f^2 \sqrt{e^2-4 d f}}+\frac{c \sqrt{a+c x^2}}{f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{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{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,"(c*Sqrt[a + c*x^2])/f - (c^(3/2)*e*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/f^2 - ((c*d*(-e + Sqrt[e^2 - 4*d*f]) - a*f*(e + Sqrt[e^2 - 4*d*f]))*Sqrt[4*a*f^2 - 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[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(4*d*f^2*Sqrt[e^2 - 4*d*f]) - (c*Sqrt[a*f^2 + (c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))/2]*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(2*f^2) + (a*Sqrt[a*f^2 + (c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))/2]*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(2*d*f) - (c*e*Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(4*f^2*Sqrt[e^2 - 4*d*f]) - (a*e*Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(4*d*f*Sqrt[e^2 - 4*d*f]) - (a^(3/2)*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d","A",1
62,1,885,604,4.15877,"\int \frac{\left(a+c x^2\right)^{3/2}}{x^2 \left(d+e x+f x^2\right)} \, dx","Integrate[(a + c*x^2)^(3/2)/(x^2*(d + e*x + f*x^2)),x]","\frac{-x \left(2 \sqrt{a} \sqrt{c} d f \sqrt{e^2-4 d f} \sqrt{c x^2+a} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)+\sqrt{\frac{c x^2}{a}+1} \left(-2 c \sqrt{4 a f^2+2 c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \tanh ^{-1}\left(\frac{2 a f-c \left(e+\sqrt{e^2-4 d f}\right) x}{\sqrt{4 a f^2+2 c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \sqrt{c x^2+a}}\right) d^2-4 \sqrt{c} (c d-a f) \sqrt{e^2-4 d f} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right) d+2 a f \sqrt{4 a f^2+2 c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \tanh ^{-1}\left(\frac{2 a f-c \left(e+\sqrt{e^2-4 d f}\right) x}{\sqrt{4 a f^2+2 c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \sqrt{c x^2+a}}\right) d+2 c f \sqrt{e^2-4 d f} x \sqrt{c x^2+a} d+\left(2 c d^2+a \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)\right) \sqrt{4 a f^2-2 c \left(-e^2+\sqrt{e^2-4 d f} e+2 d f\right)} \tanh ^{-1}\left(\frac{2 a f+c \left(\sqrt{e^2-4 d f}-e\right) x}{\sqrt{4 a f^2-2 c \left(-e^2+\sqrt{e^2-4 d f} e+2 d f\right)} \sqrt{c x^2+a}}\right)+\sqrt{2} a e \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)} \tanh ^{-1}\left(\frac{2 a f-c \left(e+\sqrt{e^2-4 d f}\right) x}{\sqrt{4 a f^2+2 c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \sqrt{c x^2+a}}\right)-a e^2 \sqrt{4 a f^2+2 c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \tanh ^{-1}\left(\frac{2 a f-c \left(e+\sqrt{e^2-4 d f}\right) x}{\sqrt{4 a f^2+2 c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \sqrt{c x^2+a}}\right)-4 a^{3/2} e f \sqrt{e^2-4 d f} \tanh ^{-1}\left(\frac{\sqrt{c x^2+a}}{\sqrt{a}}\right)\right)\right)-4 a d f \sqrt{e^2-4 d f} \sqrt{c x^2+a} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x^2}{a}\right)}{4 d^2 f \sqrt{e^2-4 d f} x \sqrt{\frac{c x^2}{a}+1}}","\frac{a^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^2}-\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 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,"(-(x*(2*Sqrt[a]*Sqrt[c]*d*f*Sqrt[e^2 - 4*d*f]*Sqrt[a + c*x^2]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]] + Sqrt[1 + (c*x^2)/a]*(2*c*d*f*Sqrt[e^2 - 4*d*f]*x*Sqrt[a + c*x^2] - 4*Sqrt[c]*d*(c*d - a*f)*Sqrt[e^2 - 4*d*f]*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + (2*c*d^2 + a*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*Sqrt[4*a*f^2 - 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[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])] + Sqrt[2]*a*e*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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])] - 2*c*d^2*Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])] - a*e^2*Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])] + 2*a*d*f*Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])] - 4*a^(3/2)*e*f*Sqrt[e^2 - 4*d*f]*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]))) - 4*a*d*f*Sqrt[e^2 - 4*d*f]*Sqrt[a + c*x^2]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((c*x^2)/a)])/(4*d^2*f*Sqrt[e^2 - 4*d*f]*x*Sqrt[1 + (c*x^2)/a])","C",1
63,1,904,668,3.2232619,"\int \frac{\left(a+c x^2\right)^{3/2}}{x^3 \left(d+e x+f x^2\right)} \, dx","Integrate[(a + c*x^2)^(3/2)/(x^3*(d + e*x + f*x^2)),x]","\frac{\frac{6 c d^2 \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{c x^2}{a}+1\right) \left(c x^2+a\right)^{5/2}}{a^2}-5 \left(e^2-\frac{\left(e^2-3 d f\right) e}{\sqrt{e^2-4 d f}}-d f\right) \left(c x^2+a\right)^{3/2}-5 \left(e^2+\frac{\left(e^2-3 d f\right) e}{\sqrt{e^2-4 d f}}-d f\right) \left(c x^2+a\right)^{3/2}+\frac{30 a d e \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x^2}{a}\right) \sqrt{c x^2+a}}{x \sqrt{\frac{c x^2}{a}+1}}+\frac{15 \left(-e^2-\frac{\left(e^2-3 d f\right) e}{\sqrt{e^2-4 d f}}+d f\right) \left(\frac{2 \sqrt{c} \left(\sqrt{e^2-4 d f}-e\right) \sqrt{c x^2+a} \left(\sqrt{c} \sqrt{\frac{c x^2}{a}+1} x+\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right)}{\sqrt{\frac{c x^2}{a}+1}}+\frac{2 \left(2 a f^2+c \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right)\right) \left(2 \sqrt{c x^2+a} f+\sqrt{c} \left(\sqrt{e^2-4 d f}-e\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right)-\sqrt{2 c e^2-2 c \sqrt{e^2-4 d f} e+4 a f^2-4 c d f} \tanh ^{-1}\left(\frac{2 a f+c \left(\sqrt{e^2-4 d f}-e\right) x}{\sqrt{4 a f^2-2 c \left(-e^2+\sqrt{e^2-4 d f} e+2 d f\right)} \sqrt{c x^2+a}}\right)\right)}{f^2}\right)}{8 f}-\frac{15 \left(-e^2+\frac{\left(e^2-3 d f\right) e}{\sqrt{e^2-4 d f}}+d f\right) \left(\frac{2 \sqrt{c} \left(e+\sqrt{e^2-4 d f}\right) \sqrt{c x^2+a} \left(\sqrt{c} \sqrt{\frac{c x^2}{a}+1} x+\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right)}{\sqrt{\frac{c x^2}{a}+1}}+\frac{2 \left(2 a f^2+c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)\right) \left(-2 \sqrt{c x^2+a} f+\sqrt{c} \left(e+\sqrt{e^2-4 d f}\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right)+\sqrt{4 a f^2+2 c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \tanh ^{-1}\left(\frac{2 a f-c \left(e+\sqrt{e^2-4 d f}\right) x}{\sqrt{4 a f^2+2 c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)} \sqrt{c x^2+a}}\right)\right)}{f^2}\right)}{8 f}+10 \left(e^2-d f\right) \left(\sqrt{c x^2+a} \left(c x^2+4 a\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c x^2+a}}{\sqrt{a}}\right)\right)}{30 d^3}","-\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{\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 \sqrt{a+c x^2} \left(e^2-d f\right)}{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{\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{\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,"(-5*(e^2 - d*f - (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*(a + c*x^2)^(3/2) - 5*(e^2 - d*f + (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*(a + c*x^2)^(3/2) + (15*(-e^2 + d*f - (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*((2*Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/Sqrt[1 + (c*x^2)/a] + (2*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))*(2*f*Sqrt[a + c*x^2] + Sqrt[c]*(-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[2*c*e^2 - 4*c*d*f + 4*a*f^2 - 2*c*e*Sqrt[e^2 - 4*d*f]]*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f^2))/(8*f) - (15*(-e^2 + d*f + (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*((2*Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/Sqrt[1 + (c*x^2)/a] + (2*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*(-2*f*Sqrt[a + c*x^2] + Sqrt[c]*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[4*a*f^2 + 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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]))/f^2))/(8*f) + 10*(e^2 - d*f)*(Sqrt[a + c*x^2]*(4*a + c*x^2) - 3*a^(3/2)*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]) + (30*a*d*e*Sqrt[a + c*x^2]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((c*x^2)/a)])/(x*Sqrt[1 + (c*x^2)/a]) + (6*c*d^2*(a + c*x^2)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (c*x^2)/a])/a^2)/(30*d^3)","C",1
64,1,378,380,1.3332,"\int \frac{x^3}{\sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[x^3/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{\frac{\sqrt{2} \left(\left(e^2-d f\right) \left(\sqrt{e^2-4 d f}-e\right)+2 d e f\right) \tanh ^{-1}\left(\frac{2 a f+c x \left(\sqrt{e^2-4 d f}-e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 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} \left(e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2+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{2 e \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c}}-\frac{2 f \sqrt{a+c x^2}}{c}}{2 f^2}","-\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,"-1/2*((-2*f*Sqrt[a + c*x^2])/c + (2*e*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/Sqrt[c] + (Sqrt[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[4*a*f^2 - 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]*(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[4*a*f^2 + 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])]))/f^2","A",1
65,1,334,344,0.7007594,"\int \frac{x^2}{\sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[x^2/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\frac{\sqrt{2} \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{a+c x^2} \sqrt{4 a f^2-2 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} \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{a+c x^2} \sqrt{4 a f^2+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{2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c}}}{2 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,"((2*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/Sqrt[c] + (Sqrt[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[4*a*f^2 - 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]*(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[4*a*f^2 + 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])]))/(2*f)","A",1
66,1,275,294,0.3198248,"\int \frac{x}{\sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[x/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\sqrt{2} \left(-\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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)}{2 \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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{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}}","\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,"(Sqrt[2]*(-1/2*((-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(2*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])))/Sqrt[e^2 - 4*d*f]","A",1
67,1,247,266,0.2909377,"\int \frac{1}{\sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","\frac{2 \sqrt{2} f \left(\frac{\tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{2 \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{2 a f+c x \left(\sqrt{e^2-4 d f}-e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)}{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}}","\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,"(2*Sqrt[2]*f*(-1/2*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]/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[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])]/(2*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])))/Sqrt[e^2 - 4*d*f]","A",1
68,1,319,330,0.7419226,"\int \frac{1}{x \sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(x*Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\frac{\sqrt{2} f \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{a+c x^2} \sqrt{4 a f^2-2 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 \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{a+c x^2} \sqrt{4 a f^2+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{2 \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a}}}{2 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,"((Sqrt[2]*f*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 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*(-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 + 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])]) - (2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/Sqrt[a])/(2*d)","A",1
69,1,356,367,0.8382753,"\int \frac{1}{x^2 \sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(x^2*Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{\frac{\sqrt{2} 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{a+c x^2} \sqrt{4 a f^2-2 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 \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{a+c x^2} \sqrt{4 a f^2+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{2 d \sqrt{a+c x^2}}{a x}-\frac{2 e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a}}}{2 d^2}","-\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,"-1/2*((2*d*Sqrt[a + c*x^2])/(a*x) + (Sqrt[2]*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[4*a*f^2 - 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*(-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[4*a*f^2 + 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])]) - (2*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/Sqrt[a])/d^2","A",1
70,1,460,457,1.5690677,"\int \frac{1}{x^3 \sqrt{a+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(x^3*Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{\frac{c d^2 \sqrt{a+c x^2} \left(\frac{a}{c x^2}-\frac{\tanh ^{-1}\left(\sqrt{\frac{c x^2}{a}+1}\right)}{\sqrt{\frac{c x^2}{a}+1}}\right)}{a^2}+\frac{2 \left(e^2-d f\right) \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{\sqrt{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) \tanh ^{-1}\left(\frac{2 a f+c x \left(\sqrt{e^2-4 d f}-e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 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 \left(-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2+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{2 d e \sqrt{a+c x^2}}{a x}}{2 d^3}","\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 a^{3/2} d}-\frac{\left(e^2-d f\right) \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d^3}+\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{e \sqrt{a+c x^2}}{a d^2 x}-\frac{\sqrt{a+c x^2}}{2 a d x^2}",1,"-1/2*((-2*d*e*Sqrt[a + c*x^2])/(a*x) - (Sqrt[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[4*a*f^2 - 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*(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[4*a*f^2 + 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])]) + (2*(e^2 - d*f)*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/Sqrt[a] + (c*d^2*Sqrt[a + c*x^2]*(a/(c*x^2) - ArcTanh[Sqrt[1 + (c*x^2)/a]]/Sqrt[1 + (c*x^2)/a]))/a^2)/d^3","A",1
71,1,577,499,2.6958742,"\int \frac{x^3}{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[x^3/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{\left(-\frac{e \left(e^2-3 d f\right)}{\sqrt{e^2-4 d f}}-d f+e^2\right) \left(2 a f+c x \left(e-\sqrt{e^2-4 d f}\right)\right)}{a f^2 \sqrt{a+c x^2} \left(4 a f^2+c \left(e-\sqrt{e^2-4 d f}\right)^2\right)}+\frac{\left(\frac{e \left(e^2-3 d f\right)}{\sqrt{e^2-4 d f}}-d f+e^2\right) \left(2 a f+c x \left(\sqrt{e^2-4 d f}+e\right)\right)}{a f^2 \sqrt{a+c x^2} \left(4 a f^2+c \left(\sqrt{e^2-4 d f}+e\right)^2\right)}+\frac{\sqrt{2} \left(-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right) \tanh ^{-1}\left(\frac{2 a f+c x \left(\sqrt{e^2-4 d f}-e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}-\frac{\sqrt{2} \left(e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right) \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}-\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]) + ((e^2 - d*f - (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*(2*a*f + c*(e - Sqrt[e^2 - 4*d*f])*x))/(a*f^2*(4*a*f^2 + c*(e - Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + c*x^2]) + ((e^2 - d*f + (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*(2*a*f + c*(e + Sqrt[e^2 - 4*d*f])*x))/(a*f^2*(4*a*f^2 + c*(e + Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + c*x^2]) + (Sqrt[2]*(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[4*a*f^2 - 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]*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))^(3/2)) - (Sqrt[2]*(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[4*a*f^2 + 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]*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))^(3/2))","A",1
72,1,509,410,2.0725059,"\int \frac{x^2}{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[x^2/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{-\frac{\left(\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}+e\right) \left(2 a f+c x \left(e-\sqrt{e^2-4 d f}\right)\right)}{a \sqrt{a+c x^2} \left(4 a f^2+c \left(e-\sqrt{e^2-4 d f}\right)^2\right)}-\frac{\left(\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}+e\right) \left(2 a f+c x \left(\sqrt{e^2-4 d f}+e\right)\right)}{a \sqrt{a+c x^2} \left(4 a f^2+c \left(\sqrt{e^2-4 d f}+e\right)^2\right)}+\frac{\sqrt{2} f^2 \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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}+\frac{\sqrt{2} f^2 \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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}+\frac{x}{a \sqrt{a+c x^2}}}{f}","-\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,"(x/(a*Sqrt[a + c*x^2]) - ((e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*(2*a*f + c*(e - Sqrt[e^2 - 4*d*f])*x))/(a*(4*a*f^2 + c*(e - Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + c*x^2]) - ((e + (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*(2*a*f + c*(e + Sqrt[e^2 - 4*d*f])*x))/(a*(4*a*f^2 + c*(e + Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + c*x^2]) + (Sqrt[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[4*a*f^2 - 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]*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))^(3/2)) + (Sqrt[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[4*a*f^2 + 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]*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))^(3/2)))/f","A",1
73,1,457,411,0.8060181,"\int \frac{x}{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[x/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{\left(1-\frac{e}{\sqrt{e^2-4 d f}}\right) \left(2 a f+c x \left(e-\sqrt{e^2-4 d f}\right)\right)}{a \sqrt{a+c x^2} \left(4 a f^2+c \left(e-\sqrt{e^2-4 d f}\right)^2\right)}+\frac{\left(\frac{e}{\sqrt{e^2-4 d f}}+1\right) \left(2 a f+c x \left(\sqrt{e^2-4 d f}+e\right)\right)}{a \sqrt{a+c x^2} \left(4 a f^2+c \left(\sqrt{e^2-4 d f}+e\right)^2\right)}+\frac{\sqrt{2} f^2 \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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}-\frac{\sqrt{2} f^2 \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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}","\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,"((1 - e/Sqrt[e^2 - 4*d*f])*(2*a*f + c*(e - Sqrt[e^2 - 4*d*f])*x))/(a*(4*a*f^2 + c*(e - Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + c*x^2]) + ((1 + e/Sqrt[e^2 - 4*d*f])*(2*a*f + c*(e + Sqrt[e^2 - 4*d*f])*x))/(a*(4*a*f^2 + c*(e + Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + c*x^2]) + (Sqrt[2]*f^2*(e - Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 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]*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))^(3/2)) - (Sqrt[2]*f^2*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 + 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]*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))^(3/2))","A",1
74,1,320,416,2.0613478,"\int \frac{1}{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{c (a (e-f x)+c d x)}{a \sqrt{a+c x^2} \left(a^2 f^2+a c \left(e^2-2 d f\right)+c^2 d^2\right)}-\frac{2 \sqrt{2} f^3 \tanh ^{-1}\left(\frac{2 a f+c x \left(\sqrt{e^2-4 d f}-e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}+\frac{2 \sqrt{2} f^3 \tanh ^{-1}\left(\frac{2 a f-c x \left(\sqrt{e^2-4 d f}+e\right)}{\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}","-\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*(c*d*x + a*(e - f*x)))/(a*(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))*Sqrt[a + c*x^2]) - (2*Sqrt[2]*f^3*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 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]*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))^(3/2)) + (2*Sqrt[2]*f^3*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 + 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]*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))^(3/2))","A",1
75,1,497,526,3.4968913,"\int \frac{1}{x \left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(x*(a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{-\frac{f \left(\frac{e}{\sqrt{e^2-4 d f}}+1\right) \left(2 a f+c x \left(e-\sqrt{e^2-4 d f}\right)\right)}{a \sqrt{a+c x^2} \left(4 a f^2+c \left(e-\sqrt{e^2-4 d f}\right)^2\right)}-\frac{f \left(1-\frac{e}{\sqrt{e^2-4 d f}}\right) \left(2 a f+c x \left(\sqrt{e^2-4 d f}+e\right)\right)}{a \sqrt{a+c x^2} \left(4 a f^2+c \left(\sqrt{e^2-4 d f}+e\right)^2\right)}+\frac{\sqrt{2} f^3 \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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}+\frac{\sqrt{2} f^3 \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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}+\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c x^2}{a}+1\right)}{a \sqrt{a+c x^2}}}{d}","-\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,"(-((f*(1 + e/Sqrt[e^2 - 4*d*f])*(2*a*f + c*(e - Sqrt[e^2 - 4*d*f])*x))/(a*(4*a*f^2 + c*(e - Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + c*x^2])) - (f*(1 - e/Sqrt[e^2 - 4*d*f])*(2*a*f + c*(e + Sqrt[e^2 - 4*d*f])*x))/(a*(4*a*f^2 + c*(e + Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + c*x^2]) + (Sqrt[2]*f^3*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 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]*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))^(3/2)) + (Sqrt[2]*f^3*(-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 + 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]*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))^(3/2)) + Hypergeometric2F1[-1/2, 1, 1/2, 1 + (c*x^2)/a]/(a*Sqrt[a + c*x^2]))/d","C",1
76,1,557,618,3.6040205,"\int \frac{1}{x^2 \left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(x^2*(a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","-\frac{\frac{d \left(a+2 c x^2\right)}{a^2 x \sqrt{a+c x^2}}-\frac{f \left(\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}+e\right) \left(2 a f+c x \left(e-\sqrt{e^2-4 d f}\right)\right)}{a \sqrt{a+c x^2} \left(4 a f^2+c \left(e-\sqrt{e^2-4 d f}\right)^2\right)}-\frac{f \left(\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}+e\right) \left(2 a f+c x \left(\sqrt{e^2-4 d f}+e\right)\right)}{a \sqrt{a+c x^2} \left(4 a f^2+c \left(\sqrt{e^2-4 d f}+e\right)^2\right)}+\frac{\sqrt{2} f^3 \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{a+c x^2} \sqrt{4 a f^2-2 c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}+\frac{\sqrt{2} f^3 \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{a+c x^2} \sqrt{4 a f^2+2 c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f} \left(2 a f^2+c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right)\right)^{3/2}}+\frac{e \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c x^2}{a}+1\right)}{a \sqrt{a+c x^2}}}{d^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,"-((-((f*(e + (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*(2*a*f + c*(e - Sqrt[e^2 - 4*d*f])*x))/(a*(4*a*f^2 + c*(e - Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + c*x^2])) - (f*(e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*(2*a*f + c*(e + Sqrt[e^2 - 4*d*f])*x))/(a*(4*a*f^2 + c*(e + Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + c*x^2]) + (d*(a + 2*c*x^2))/(a^2*x*Sqrt[a + c*x^2]) + (Sqrt[2]*f^3*(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[4*a*f^2 - 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]*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))^(3/2)) + (Sqrt[2]*f^3*(-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[4*a*f^2 + 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]*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))^(3/2)) + (e*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (c*x^2)/a])/(a*Sqrt[a + c*x^2]))/d^2)","C",1
77,1,327,392,0.8888244,"\int \frac{x^3 \sqrt{a+b x+c x^2}}{d-f x^2} \, dx","Integrate[(x^3*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]","\frac{-\frac{2 \sqrt{f} \sqrt{a+x (b+c x)} \left(2 c f (4 a+b x)-3 b^2 f+8 c^2 \left(3 d+f x^2\right)\right)}{c^2}-\frac{3 b \sqrt{f} \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+x (b+c x)}}\right)}{c^{5/2}}+24 d \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)-24 d \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{48 f^{5/2}}","-\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 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{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{\left(a+b x+c x^2\right)^{3/2}}{3 c f}",1,"((-2*Sqrt[f]*Sqrt[a + x*(b + c*x)]*(-3*b^2*f + 2*c*f*(4*a + b*x) + 8*c^2*(3*d + f*x^2)))/c^2 - (3*b*Sqrt[f]*(8*c^2*d + b^2*f - 4*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2) + 24*d*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] - 24*d*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(48*f^(5/2))","A",1
78,1,302,316,0.4852071,"\int \frac{x^2 \sqrt{a+b x+c x^2}}{d-f x^2} \, dx","Integrate[(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+x (b+c x)}}\right)-2 \sqrt{c} \left(-2 c \sqrt{d} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)-2 c \sqrt{d} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+f (b+2 c x) \sqrt{a+x (b+c x)}\right)}{8 c^{3/2} f^2}","-\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,"((-8*c^2*d + b^2*f - 4*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - 2*Sqrt[c]*(f*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - 2*c*Sqrt[d]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] - 2*c*Sqrt[d]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]))/(8*c^(3/2)*f^2)","A",1
79,1,272,282,0.2944317,"\int \frac{x \sqrt{a+b x+c x^2}}{d-f x^2} \, dx","Integrate[(x*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]","\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)-\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)-2 \sqrt{f} \sqrt{a+x (b+c x)}-\frac{b \sqrt{f} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}}{2 f^{3/2}}","-\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,"(-2*Sqrt[f]*Sqrt[a + x*(b + c*x)] - (b*Sqrt[f]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] - Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(2*f^(3/2))","A",1
80,1,253,266,0.1663616,"\int \frac{\sqrt{a+b x+c x^2}}{d-f x^2} \, dx","Integrate[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}+b \sqrt{d}-b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)-2 \sqrt{c} \sqrt{d} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{2 \sqrt{d} 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,"(-2*Sqrt[c]*Sqrt[d]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + 2*c*Sqrt[d]*x - b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(2*Sqrt[d]*f)","A",1
81,1,255,267,0.2953108,"\int \frac{\sqrt{a+b x+c x^2}}{x \left(d-f x^2\right)} \, dx","Integrate[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}+b \sqrt{d}-b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)-\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)+2 \sqrt{a} \sqrt{f} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{2 d \sqrt{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 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,"-1/2*(2*Sqrt[a]*Sqrt[f]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] + Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + 2*c*Sqrt[d]*x - b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] - Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(d*Sqrt[f])","A",1
82,1,275,286,0.4313108,"\int \frac{\sqrt{a+b x+c x^2}}{x^2 \left(d-f x^2\right)} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(x^2*(d - f*x^2)),x]","\frac{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)+\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)-\frac{2 \sqrt{d} \sqrt{a+x (b+c x)}}{x}-\frac{b \sqrt{d} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{\sqrt{a}}}{2 d^{3/2}}","\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,"((-2*Sqrt[d]*Sqrt[a + x*(b + c*x)])/x - (b*Sqrt[d]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/Sqrt[a] + Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(2*d^(3/2))","A",1
83,1,316,353,0.5215504,"\int \frac{\sqrt{a+b x+c x^2}}{x^3 \left(d-f x^2\right)} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(x^3*(d - f*x^2)),x]","\frac{x^2 \left(b^2 d-4 a (2 a f+c d)\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{a} \left(-2 a \sqrt{f} x^2 \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)+2 a \sqrt{f} x^2 \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+d (2 a+b x) \sqrt{a+x (b+c x)}\right)}{8 a^{3/2} d^2 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,"((b^2*d - 4*a*(c*d + 2*a*f))*x^2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] - 2*Sqrt[a]*(d*(2*a + b*x)*Sqrt[a + x*(b + c*x)] - 2*a*Sqrt[f]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*x^2*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + 2*a*Sqrt[f]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*x^2*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]))/(8*a^(3/2)*d^2*x^2)","A",1
84,1,447,501,1.4926489,"\int \frac{x^3 \left(a+b x+c x^2\right)^{3/2}}{d-f x^2} \, dx","Integrate[(x^3*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x]","\frac{-\frac{\sqrt{f} \sqrt{a+x (b+c x)} \left(24 c^2 f \left(16 a^2 f+7 a b f x+b^2 \left(10 d+f x^2\right)\right)-30 b^2 c f^2 (10 a+b x)+16 c^3 f \left(160 a d+48 a f x^2+70 b d x+33 b f x^3\right)+45 b^4 f^2+128 c^4 \left(15 d^2+5 d f x^2+3 f^2 x^4\right)\right)}{c^3}+960 d \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)-960 d \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{1920 f^{7/2}}+\frac{b \left(16 c^2 f \left(3 a^2 f+b^2 d\right)-24 a b^2 c f^2-192 a c^3 d f+3 b^4 f^2-384 c^4 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{256 c^{7/2} f^3}","\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{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{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{b (b+2 c x) \left(a+b x+c x^2\right)^{3/2}}{16 c^2 f}-\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{d \left(a+b x+c x^2\right)^{3/2}}{3 f^2}-\frac{\left(a+b x+c x^2\right)^{5/2}}{5 c f}",1,"(b*(-384*c^4*d^2 - 192*a*c^3*d*f + 3*b^4*f^2 - 24*a*b^2*c*f^2 + 16*c^2*f*(b^2*d + 3*a^2*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(256*c^(7/2)*f^3) + (-((Sqrt[f]*Sqrt[a + x*(b + c*x)]*(45*b^4*f^2 - 30*b^2*c*f^2*(10*a + b*x) + 16*c^3*f*(160*a*d + 70*b*d*x + 48*a*f*x^2 + 33*b*f*x^3) + 128*c^4*(15*d^2 + 5*d*f*x^2 + 3*f^2*x^4) + 24*c^2*f*(16*a^2*f + 7*a*b*f*x + b^2*(10*d + f*x^2))))/c^3) + 960*d*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] - 960*d*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(1920*f^(7/2))","A",1
85,1,395,417,1.0171553,"\int \frac{x^2 \left(a+b x+c x^2\right)^{3/2}}{d-f x^2} \, dx","Integrate[(x^2*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x]","\frac{-\left(\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+x (b+c x)}}\right)\right)-2 \sqrt{c} \left(f \sqrt{a+x (b+c x)} \left(4 b c \left(5 a f+20 c d+6 c f x^2\right)+8 c^2 x \left(5 a f+4 c d+2 c f x^2\right)-3 b^3 f+2 b^2 c f x\right)+32 c^2 \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}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+32 c^2 \sqrt{d} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)\right)}{128 c^{5/2} f^3}","-\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,"(-((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 + x*(b + c*x)])]) - 2*Sqrt[c]*(f*Sqrt[a + x*(b + c*x)]*(-3*b^3*f + 2*b^2*c*f*x + 8*c^2*x*(4*c*d + 5*a*f + 2*c*f*x^2) + 4*b*c*(20*c*d + 5*a*f + 6*c*f*x^2)) + 32*c^2*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]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + 32*c^2*Sqrt[d]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]))/(128*c^(5/2)*f^3)","A",1
86,1,330,349,0.8049721,"\int \frac{x \left(a+b x+c x^2\right)^{3/2}}{d-f x^2} \, dx","Integrate[(x*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x]","\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+x (b+c x)}}\right)}{16 c^{3/2} f^2}-\frac{\sqrt{f} \sqrt{a+x (b+c x)} \left(2 c f (16 a+7 b x)+3 b^2 f+8 c^2 \left(3 d+f x^2\right)\right)-12 c \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+12 c \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{24 c f^{5/2}}","-\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 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,"(b*(-24*c^2*d + b^2*f - 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(16*c^(3/2)*f^2) - (Sqrt[f]*Sqrt[a + x*(b + c*x)]*(3*b^2*f + 2*c*f*(16*a + 7*b*x) + 8*c^2*(3*d + f*x^2)) - 12*c*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + 12*c*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(24*c*f^(5/2))","A",1
87,1,298,315,0.8121345,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{d-f x^2} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d - f*x^2),x]","-\frac{\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+x (b+c x)}}\right)}{\sqrt{c}}+\frac{4 \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{d}}+\frac{4 \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{d}}+2 f (5 b+2 c x) \sqrt{a+x (b+c x)}}{8 f^2}","-\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,"-1/8*(2*f*(5*b + 2*c*x)*Sqrt[a + x*(b + c*x)] + ((8*c^2*d + 3*b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + (4*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[d] + (4*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[d])/f^2","A",1
88,1,755,469,0.5047989,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{x \left(d-f x^2\right)} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(x*(d - f*x^2)),x]","-\frac{2 a^{3/2} f^{3/2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)+2 c d \sqrt{f} \sqrt{a+x (b+c x)}-a f \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+a f \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)+3 b \sqrt{c} d \sqrt{f} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)+b \sqrt{d} \sqrt{f} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+b \sqrt{d} \sqrt{f} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)-c d \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+c d \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \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{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{\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{\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,"-1/2*(2*c*d*Sqrt[f]*Sqrt[a + x*(b + c*x)] + 2*a^(3/2)*f^(3/2)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] + 3*b*Sqrt[c]*d*Sqrt[f]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - c*d*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + b*Sqrt[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]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] - a*f*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + c*d*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + b*Sqrt[d]*Sqrt[f]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + a*f*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(d*f^(3/2))","A",1
89,1,765,463,0.6381255,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{x^2 \left(d-f x^2\right)} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(x^2*(d - f*x^2)),x]","-\frac{2 c^{3/2} d^{3/2} x \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)+2 a \sqrt{d} f \sqrt{a+x (b+c x)}+c d x \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+c d x \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)+3 \sqrt{a} b \sqrt{d} f x \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)-b \sqrt{d} \sqrt{f} x \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+b \sqrt{d} \sqrt{f} x \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)+a f x \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)+a f x \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 d^{3/2} f 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}",1,"-1/2*(2*a*Sqrt[d]*f*Sqrt[a + x*(b + c*x)] + 3*Sqrt[a]*b*Sqrt[d]*f*x*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] + 2*c^(3/2)*d^(3/2)*x*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + c*d*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*x*ArcTanh[(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] - b*Sqrt[d]*Sqrt[f]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*x*ArcTanh[(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + a*f*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*x*ArcTanh[(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + c*d*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*x*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + b*Sqrt[d]*Sqrt[f]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*x*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + a*f*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*x*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(d^(3/2)*f*x)","A",1
90,1,303,614,0.9703318,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{x^3 \left(d-f x^2\right)} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(x^3*(d - f*x^2)),x]","-\frac{\frac{\left(4 a (2 a f+3 c d)+3 b^2 d\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{\sqrt{a}}-\frac{4 \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{f}}+\frac{4 \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{f}}+\frac{2 d (2 a+5 b x) \sqrt{a+x (b+c x)}}{x^2}}{8 d^2}","-\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{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{\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 \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,"-1/8*((2*d*(2*a + 5*b*x)*Sqrt[a + x*(b + c*x)])/x^2 + ((3*b^2*d + 4*a*(3*c*d + 2*a*f))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/Sqrt[a] - (4*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[f] + (4*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[f])/d^2","A",1
91,1,181,189,0.5834573,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{1-x^2} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(1 - x^2),x]","\frac{1}{8} \left(-\frac{\left(4 c (3 a+2 c)+3 b^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}-2 (5 b+2 c x) \sqrt{a+x (b+c x)}-4 (a-b+c)^{3/2} \tanh ^{-1}\left(\frac{2 a+b (x-1)-2 c x}{2 \sqrt{a-b+c} \sqrt{a+x (b+c x)}}\right)+4 (a+b+c)^{3/2} \tanh ^{-1}\left(\frac{2 a+b x+b+2 c x}{2 \sqrt{a+b+c} \sqrt{a+x (b+c x)}}\right)\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,"(-2*(5*b + 2*c*x)*Sqrt[a + x*(b + c*x)] - 4*(a - b + c)^(3/2)*ArcTanh[(2*a + b*(-1 + x) - 2*c*x)/(2*Sqrt[a - b + c]*Sqrt[a + x*(b + c*x)])] - ((3*b^2 + 4*c*(3*a + 2*c))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + 4*(a + b + c)^(3/2)*ArcTanh[(2*a + b + b*x + 2*c*x)/(2*Sqrt[a + b + c]*Sqrt[a + x*(b + c*x)])])/8","A",1
92,1,75,75,0.03787,"\int \frac{\sqrt{-1-x+x^2}}{1-x^2} \, dx","Integrate[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,"-1/2*ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])] + ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])] + ArcTanh[(1 + 3*x)/(2*Sqrt[-1 - x + x^2])]/2","A",1
93,1,120,130,0.1610385,"\int \frac{\left(x+x^2\right)^{3/2}}{1+x^2} \, dx","Integrate[(x + x^2)^(3/2)/(1 + x^2),x]","\frac{\sqrt{x} \sqrt{x+1} \left(2 \sqrt{x+1} x^{3/2}+5 \sqrt{x+1} \sqrt{x}+4 (-1+i)^{3/2} \tan ^{-1}\left(\sqrt{-1+i} \sqrt{\frac{x}{x+1}}\right)-5 \sinh ^{-1}\left(\sqrt{x}\right)+4 (1+i)^{3/2} \tanh ^{-1}\left(\sqrt{1+i} \sqrt{\frac{x}{x+1}}\right)\right)}{4 \sqrt{x (x+1)}}","\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,"(Sqrt[x]*Sqrt[1 + x]*(5*Sqrt[x]*Sqrt[1 + x] + 2*x^(3/2)*Sqrt[1 + x] - 5*ArcSinh[Sqrt[x]] + 4*(-1 + I)^(3/2)*ArcTan[Sqrt[-1 + I]*Sqrt[x/(1 + x)]] + 4*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + I]*Sqrt[x/(1 + x)]]))/(4*Sqrt[x*(1 + x)])","C",1
94,1,300,369,1.708664,"\int \frac{x^4}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Integrate[x^4/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{-\frac{\left(-4 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+x (b+c x)}}\right)}{c^{5/2}}-\frac{2 f (2 c x-3 b) \sqrt{a+x (b+c x)}}{c^2}+\frac{4 d^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}+\frac{4 d^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}}{8 f^2}","-\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,"((-2*f*(-3*b + 2*c*x)*Sqrt[a + x*(b + c*x)])/c^2 - ((8*c^2*d + 3*b^2*f - 4*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2) + (4*d^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f] + (4*d^(3/2)*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f])/(8*f^2)","A",1
95,1,325,287,1.1584403,"\int \frac{x^3}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Integrate[x^3/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{\frac{b \sqrt{f} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{c^{3/2}}+\frac{d \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{d \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}-\frac{2 \sqrt{f} x^2}{\sqrt{a+x (b+c x)}}-\frac{2 b \sqrt{f} x}{c \sqrt{a+x (b+c x)}}-\frac{2 a \sqrt{f}}{c \sqrt{a+x (b+c x)}}}{2 f^{3/2}}","\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,"((-2*a*Sqrt[f])/(c*Sqrt[a + x*(b + c*x)]) - (2*b*Sqrt[f]*x)/(c*Sqrt[a + x*(b + c*x)]) - (2*Sqrt[f]*x^2)/Sqrt[a + x*(b + c*x)] + (b*Sqrt[f]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(3/2) + (d*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f] - (d*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f])/(2*f^(3/2))","A",1
96,1,250,266,0.4252482,"\int \frac{x^2}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Integrate[x^2/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{\sqrt{d} \left(\frac{\tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \sqrt{d}-b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)-\frac{2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}}{2 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,"((-2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + Sqrt[d]*(ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + 2*c*Sqrt[d]*x - b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f] + ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]))/(2*f)","A",1
97,1,211,220,0.1733718,"\int \frac{x}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Integrate[x/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","-\frac{\frac{\tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}}{2 \sqrt{f}}","\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,"-1/2*(ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f] + ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])/Sqrt[f]","A",1
98,1,209,220,0.0956005,"\int \frac{1}{\sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Integrate[1/(Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{\frac{\tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}+\frac{\tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}}{2 \sqrt{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]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f] + ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f])/(2*Sqrt[d])","A",1
99,1,252,267,0.4429573,"\int \frac{1}{x \sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Integrate[1/(x*Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{\sqrt{f} \left(\frac{\tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{\tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \sqrt{d}-b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)-\frac{2 \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{\sqrt{a}}}{2 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,"((-2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/Sqrt[a] + Sqrt[f]*(-(ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + 2*c*Sqrt[d]*x - b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]))/(2*d)","A",1
100,1,325,291,1.0328102,"\int \frac{1}{x^2 \sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Integrate[1/(x^2*Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{\frac{b \sqrt{d} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{a^{3/2}}+\frac{f \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}+\frac{f \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}-\frac{2 b \sqrt{d}}{a \sqrt{a+x (b+c x)}}-\frac{2 c \sqrt{d} x}{a \sqrt{a+x (b+c x)}}-\frac{2 \sqrt{d}}{x \sqrt{a+x (b+c x)}}}{2 d^{3/2}}","\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,"((-2*b*Sqrt[d])/(a*Sqrt[a + x*(b + c*x)]) - (2*Sqrt[d])/(x*Sqrt[a + x*(b + c*x)]) - (2*c*Sqrt[d]*x)/(a*Sqrt[a + x*(b + c*x)]) + (b*Sqrt[d]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/a^(3/2) + (f*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f] + (f*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f])/(2*d^(3/2))","A",1
101,1,314,376,2.0094679,"\int \frac{1}{x^3 \sqrt{a+b x+c x^2} \left(d-f x^2\right)} \, dx","Integrate[1/(x^3*Sqrt[a + b*x + c*x^2]*(d - f*x^2)),x]","\frac{2 \sqrt{a} \left(\frac{2 a^2 f^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{2 a^2 f^{3/2} \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}-\frac{d (2 a-3 b x) \sqrt{a+x (b+c x)}}{x^2}\right)+\left(4 a (c d-2 a f)-3 b^2 d\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{8 a^{5/2} d^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,"((-3*b^2*d + 4*a*(c*d - 2*a*f))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[a]*(-((d*(2*a - 3*b*x)*Sqrt[a + x*(b + c*x)])/x^2) + (2*a^2*f^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f] - (2*a^2*f^(3/2)*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]))/(8*a^(5/2)*d^2)","A",1
102,1,562,466,1.4050978,"\int \frac{x^4}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Integrate[x^4/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","\frac{\frac{f \left(a \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{c} \left(a (b-2 c x)+b c x^2\right) \sqrt{a+x (b+c x)}\right)}{a c^{3/2} \left(4 a c-b^2\right)}+\frac{d^{3/2} f \left(\frac{\left(b^2-4 a c\right) \left(a f+b \sqrt{d} \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\left(4 a c-b^2\right) \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{2 \left(b^2-4 a c\right) \left((a f+c d)^2-b^2 d f\right)}+\frac{2 d (b+2 c x)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}-\frac{2 f x^3 \left(-2 a c+b^2+b c x\right)}{a \left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}-\frac{2 d^2 \left(-b c (3 a f+c d)-2 c^2 x (a f+c d)+b^3 f+b^2 c f x\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(b^2 d f-(a f+c d)^2\right)}}{f^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*d*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]) - (2*f*x^3*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]) - (2*d^2*(b^3*f - b*c*(c*d + 3*a*f) + b^2*c*f*x - 2*c^2*(c*d + a*f)*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + x*(b + c*x)]) + (f*(-2*Sqrt[c]*(b*c*x^2 + a*(b - 2*c*x))*Sqrt[a + x*(b + c*x)] + a*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(a*c^(3/2)*(-b^2 + 4*a*c)) + (d^(3/2)*f*(((b^2 - 4*a*c)*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f] + ((-b^2 + 4*a*c)*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]))/(2*(b^2 - 4*a*c)*(-(b^2*d*f) + (c*d + a*f)^2)))/f^2","A",1
103,1,414,341,1.196761,"\int \frac{x^3}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Integrate[x^3/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","\frac{1}{2} \left(\frac{8 a^3 f+4 a^2 (b f x+2 c d)-4 a b d (b-3 c x)-4 b^3 d x}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(f \left(b^2 d-a^2 f\right)-2 a c d f-c^2 d^2\right)}-\frac{d \log \left(\sqrt{d} \sqrt{f}-f x\right)}{\sqrt{f} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}-\frac{d \log \left(\sqrt{d} \sqrt{f}+f x\right)}{\sqrt{f} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{d \log \left(\sqrt{d} \left(2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}+2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x\right)\right)}{\sqrt{f} \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2}}+\frac{d \log \left(\sqrt{d} \left(2 \left(\sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+a \sqrt{f}+c \sqrt{d} x\right)+b \left(\sqrt{d}+\sqrt{f} x\right)\right)\right)}{\sqrt{f} \left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2}}\right)","-\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,"((8*a^3*f - 4*b^3*d*x - 4*a*b*d*(b - 3*c*x) + 4*a^2*(2*c*d + b*f*x))/((b^2 - 4*a*c)*(-(c^2*d^2) - 2*a*c*d*f + f*(b^2*d - a^2*f))*Sqrt[a + x*(b + c*x)]) - (d*Log[Sqrt[d]*Sqrt[f] - f*x])/(Sqrt[f]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) - (d*Log[Sqrt[d]*Sqrt[f] + f*x])/(Sqrt[f]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (d*Log[Sqrt[d]*(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x + 2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(Sqrt[f]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (d*Log[Sqrt[d]*(b*(Sqrt[d] + Sqrt[f]*x) + 2*(a*Sqrt[f] + c*Sqrt[d]*x + Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)]))])/(Sqrt[f]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)))/2","A",1
104,1,352,297,0.4024301,"\int \frac{x^2}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Integrate[x^2/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","\frac{2 \left(\frac{a^2 f (b+2 c x)+a c d (2 c x-b)-b^2 c d x}{\sqrt{a+x (b+c x)}}+\frac{\sqrt{d} \left(b^2-4 a c\right) \left(a f+b \sqrt{d} \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{4 \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\sqrt{d} \left(4 a c-b^2\right) \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{4 \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\left(b^2-4 a c\right) \left((a f+c d)^2-b^2 d f\right)}","\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*((-(b^2*c*d*x) + a*c*d*(-b + 2*c*x) + a^2*f*(b + 2*c*x))/Sqrt[a + x*(b + c*x)] + ((b^2 - 4*a*c)*Sqrt[d]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(4*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ((-b^2 + 4*a*c)*Sqrt[d]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(4*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])))/((b^2 - 4*a*c)*(-(b^2*d*f) + (c*d + a*f)^2))","A",1
105,1,356,299,0.3533227,"\int \frac{x}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Integrate[x/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","\frac{2 \left(\frac{2 a^2 c f+a \left(b^2 (-f)-b c f x+2 c^2 d\right)+b c^2 d x}{\sqrt{a+x (b+c x)}}-\frac{\sqrt{f} \left(b^2-4 a c\right) \left(a f+b \sqrt{d} \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{4 \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\sqrt{f} \left(4 a c-b^2\right) \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{4 \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\left(b^2-4 a c\right) \left((a f+c d)^2-b^2 d f\right)}","-\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*((2*a^2*c*f + b*c^2*d*x + a*(2*c^2*d - b^2*f - b*c*f*x))/Sqrt[a + x*(b + c*x)] - ((b^2 - 4*a*c)*Sqrt[f]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(4*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ((-b^2 + 4*a*c)*Sqrt[f]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(4*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])))/((b^2 - 4*a*c)*(-(b^2*d*f) + (c*d + a*f)^2))","A",1
106,1,360,310,0.3993163,"\int \frac{1}{\left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Integrate[1/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","\frac{2 \left(\frac{f \left(b^2-4 a c\right) \left(a f+b \sqrt{d} \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{-2 a \sqrt{f}+b \left(\sqrt{d}-\sqrt{f} x\right)+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{4 \sqrt{d} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{f \left(4 a c-b^2\right) \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{-2 \left(a \sqrt{f}+c \sqrt{d} x\right)-b \left(\sqrt{d}+\sqrt{f} x\right)}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{4 \sqrt{d} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}+\frac{-b c (3 a f+c d)-2 c^2 x (a f+c d)+b^3 f+b^2 c f x}{\sqrt{a+x (b+c x)}}\right)}{\left(b^2-4 a c\right) \left((a f+c d)^2-b^2 d f\right)}","-\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^3*f - b*c*(c*d + 3*a*f) + b^2*c*f*x - 2*c^2*(c*d + a*f)*x)/Sqrt[a + x*(b + c*x)] + ((b^2 - 4*a*c)*f*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*(Sqrt[d] - Sqrt[f]*x))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(4*Sqrt[d]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]) + ((-b^2 + 4*a*c)*f*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-2*(a*Sqrt[f] + c*Sqrt[d]*x) - b*(Sqrt[d] + Sqrt[f]*x))/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/(4*Sqrt[d]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f])))/((b^2 - 4*a*c)*(-(b^2*d*f) + (c*d + a*f)^2))","A",1
107,1,436,394,0.8659465,"\int \frac{1}{x \left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Integrate[1/(x*(a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","\frac{-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{a^{3/2}}-\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)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(b^2 d f-(a f+c d)^2\right)}+\frac{f^{3/2} \left(\left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)+\left(a f+b \sqrt{d} \sqrt{f}+c d\right)^{3/2} \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)\right)}{2 \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d} \left((a f+c d)^2-b^2 d f\right)}+\frac{2 \left(-2 a c+b^2+b c x\right)}{a \left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}}{d}","-\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)*Sqrt[a + x*(b + c*x)]) - (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)*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + x*(b + c*x)]) - ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]/a^(3/2) + (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]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])] + (c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])]))/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*(-(b^2*d*f) + (c*d + a*f)^2)))/d","A",1
108,1,488,454,1.1928629,"\int \frac{1}{x^2 \left(a+b x+c x^2\right)^{3/2} \left(d-f x^2\right)} \, dx","Integrate[1/(x^2*(a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]","\frac{\frac{3 b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{a^{5/2}}+\frac{2 \left(8 a c-3 b^2\right) \sqrt{a+x (b+c x)}}{a^2 x}-\frac{f^2 \left(\frac{\left(b^2-4 a c\right) \left(a f+b \sqrt{d} \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}\right)}{\sqrt{a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d}}+\frac{\left(4 a c-b^2\right) \left(a f+b \left(-\sqrt{d}\right) \sqrt{f}+c d\right) \tanh ^{-1}\left(\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right)}{\sqrt{d} \left((a f+c d)^2-b^2 d f\right)}+\frac{4 \left(-2 a c+b^2+b c x\right)}{a x \sqrt{a+x (b+c x)}}-\frac{4 f \left(-b c (3 a f+c d)-2 c^2 x (a f+c d)+b^3 f+b^2 c f x\right)}{\sqrt{a+x (b+c x)} \left(b^2 d f-(a f+c d)^2\right)}}{2 d \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{\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{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,"((4*(b^2 - 2*a*c + b*c*x))/(a*x*Sqrt[a + x*(b + c*x)]) - (4*f*(b^3*f - b*c*(c*d + 3*a*f) + b^2*c*f*x - 2*c^2*(c*d + a*f)*x))/((b^2*d*f - (c*d + a*f)^2)*Sqrt[a + x*(b + c*x)]) + (2*(-3*b^2 + 8*a*c)*Sqrt[a + x*(b + c*x)])/(a^2*x) + (3*b*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/a^(5/2) - (f^2*(((b^2 - 4*a*c)*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f] + ((-b^2 + 4*a*c)*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + 2*c*Sqrt[d]*x + b*Sqrt[f]*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]))/(Sqrt[d]*(-(b^2*d*f) + (c*d + a*f)^2)))/(2*(b^2 - 4*a*c)*d)","A",1
109,1,552,761,2.3121684,"\int \frac{x^2 \sqrt{a+b x+c x^2}}{d+e x+f x^2} \, dx","Integrate[(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+x (b+c x)}}\right) \left(4 c f (a f-b e)-b^2 f^2+8 c^2 \left(e^2-d f\right)\right)}{8 c^{3/2} f^3}+\frac{f \sqrt{e^2-4 d f} \sqrt{a+x (b+c x)} (b f-4 c e+2 c f x)+\sqrt{2} c \left(e \sqrt{e^2-4 d f}-2 d f+e^2\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)} \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)+\sqrt{2} c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\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)} \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{4 c f^3 \sqrt{e^2-4 d f}}","-\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^2-4 d e^2 f+2 d e f \sqrt{e^2-4 d f}-e^3 \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^2-4 d e^2 f-2 d e f \sqrt{e^2-4 d f}+e^3 \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,"((-(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 + x*(b + c*x)])])/(8*c^(3/2)*f^3) + (f*Sqrt[e^2 - 4*d*f]*(-4*c*e + b*f + 2*c*f*x)*Sqrt[a + x*(b + c*x)] + Sqrt[2]*c*(e^2 - 2*d*f + e*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 - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])] + Sqrt[2]*c*(-e^2 + 2*d*f + e*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 + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(4*c*f^3*Sqrt[e^2 - 4*d*f])","A",1
110,1,496,549,1.7757943,"\int \frac{x \sqrt{a+b x+c x^2}}{d+e x+f x^2} \, dx","Integrate[(x*Sqrt[a + b*x + c*x^2])/(d + e*x + f*x^2),x]","\frac{4 f \sqrt{e^2-4 d f} \sqrt{a+x (b+c x)}-\sqrt{2} \left(\sqrt{e^2-4 d f}+e\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)} \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)-\sqrt{2} \left(\sqrt{e^2-4 d f}-e\right) \sqrt{f \left(2 a f+b \sqrt{e^2-4 d f}+b (-e)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left(2 a f+b \sqrt{e^2-4 d f}+b (-e)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{4 f^2 \sqrt{e^2-4 d f}}-\frac{(2 c e-b f) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{2 \sqrt{c} f^2}","-\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,"-1/2*((2*c*e - b*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(Sqrt[c]*f^2) + (4*f*Sqrt[e^2 - 4*d*f]*Sqrt[a + x*(b + c*x)] - Sqrt[2]*(e + 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 - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])] - Sqrt[2]*(-e + Sqrt[e^2 - 4*d*f])*Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*ArcTanh[(4*a*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + x*(b + c*x)])])/(4*f^2*Sqrt[e^2 - 4*d*f])","A",1
111,1,417,431,0.7376529,"\int \frac{\sqrt{a+b x+c x^2}}{d+e x+f x^2} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x + f*x^2),x]","\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-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)-\sqrt{f \left(2 a f+b \sqrt{e^2-4 d f}+b (-e)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left(2 a f+b \sqrt{e^2-4 d f}+b (-e)\right)+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{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\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 + x*(b + c*x)])])/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 - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])] - Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*ArcTanh[(4*a*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + x*(b + c*x)])])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f])","A",1
112,1,454,523,1.3092616,"\int \frac{\sqrt{a+b x+c x^2}}{x \left(d+e x+f x^2\right)} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(x*(d + e*x + f*x^2)),x]","\frac{\left(\sqrt{e^2-4 d f}-e\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)} \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)+\left(\sqrt{e^2-4 d f}+e\right) \sqrt{f \left(2 a f+b \sqrt{e^2-4 d f}+b (-e)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)} \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left(2 a f+b \sqrt{e^2-4 d f}+b (-e)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{2 \sqrt{2} d f \sqrt{e^2-4 d f}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\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 + x*(b + c*x)])])/d) + ((-e + 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 - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])] + (e + Sqrt[e^2 - 4*d*f])*Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*ArcTanh[(4*a*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + x*(b + c*x)])])/(2*Sqrt[2]*d*f*Sqrt[e^2 - 4*d*f])","A",1
113,1,520,736,1.6827233,"\int \frac{\sqrt{a+b x+c x^2}}{x^2 \left(d+e x+f x^2\right)} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(x^2*(d + e*x + f*x^2)),x]","\frac{(2 a e-b d) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{2 \sqrt{a} d^2}-\frac{4 d f \sqrt{e^2-4 d f} \sqrt{a+x (b+c x)}+\sqrt{2} x \left(e \sqrt{e^2-4 d f}+2 d f-e^2\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)} \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)+\sqrt{2} x \left(e \sqrt{e^2-4 d f}-2 d f+e^2\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)} \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{4 d^2 f x \sqrt{e^2-4 d f}}","-\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,"((-(b*d) + 2*a*e)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(2*Sqrt[a]*d^2) - (4*d*f*Sqrt[e^2 - 4*d*f]*Sqrt[a + x*(b + c*x)] + Sqrt[2]*(-e^2 + 2*d*f + e*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]))]*x*ArcTanh[(4*a*f - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])] + Sqrt[2]*(e^2 - 2*d*f + e*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]))]*x*ArcTanh[(4*a*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(4*d^2*f*Sqrt[e^2 - 4*d*f]*x)","A",1
114,1,550,545,2.2007852,"\int \frac{x^3}{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[x^3/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{\frac{b f \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{c^{3/2}}+\frac{\sqrt{2} \left(-\frac{e \left(e^2-3 d f\right)}{\sqrt{e^2-4 d f}}-d f+e^2\right) \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\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{\sqrt{2} \left(e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right) \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{\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{2 e \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}-\frac{2 f \sqrt{a+x (b+c x)}}{c}}{2 f^2}","-\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,"-1/2*((-2*f*Sqrt[a + x*(b + c*x)])/c + (2*e*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + (b*f*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(3/2) + (Sqrt[2]*(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 - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(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]))]) + (Sqrt[2]*(e^2 - d*f - (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/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","A",1
115,1,468,463,0.9468841,"\int \frac{x^2}{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[x^2/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\frac{\sqrt{2} \left(e \sqrt{e^2-4 d f}-2 d f+e^2\right) \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{\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{2} \left(\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}+e\right) \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\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 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}}{2 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,"((2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + (Sqrt[2]*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(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]))]) + (Sqrt[2]*(e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/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*f)","A",1
116,1,407,402,0.9735608,"\int \frac{x}{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[x/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","\frac{-\frac{\left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{\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{\left(1-\frac{e}{\sqrt{e^2-4 d f}}\right) \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\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)}}}{\sqrt{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 - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(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]))])) - ((1 - e/Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/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]))])/Sqrt[2]","A",1
117,1,376,374,0.789116,"\int \frac{1}{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\sqrt{2} f \left(\frac{\tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\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{\tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left(2 a f+b \sqrt{e^2-4 d f}+b (-e)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{f \left(2 a f+b \sqrt{e^2-4 d f}+b (-e)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f}}","\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 - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])]/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 + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + x*(b + c*x)])]/Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]))/Sqrt[e^2 - 4*d*f]","A",1
118,1,450,451,2.4136256,"\int \frac{1}{x \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(x*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\frac{\sqrt{2} f \left(\frac{\left(\sqrt{e^2-4 d f}-e\right) \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\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{\left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left(2 a f+b \sqrt{e^2-4 d f}+b (-e)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{f \left(2 a f+b \sqrt{e^2-4 d f}+b (-e)\right)+c \left(-e \sqrt{e^2-4 d f}-2 d f+e^2\right)}}\right)}{\sqrt{e^2-4 d f}}-\frac{2 \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{\sqrt{a}}}{2 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,"((-2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/Sqrt[a] + (Sqrt[2]*f*(((-e + Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/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]))] + ((e + Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + x*(b + c*x)])])/Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]))/Sqrt[e^2 - 4*d*f])/(2*d)","A",1
119,1,533,543,1.4076336,"\int \frac{1}{x^2 \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(x^2*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","-\frac{-\frac{b d \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{a^{3/2}}+\frac{\sqrt{2} f \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right) \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{\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{2} f \left(\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}+e\right) \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\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 d \sqrt{a+x (b+c x)}}{a x}-\frac{2 e \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{\sqrt{a}}}{2 d^2}","\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,"-1/2*((2*d*Sqrt[a + x*(b + c*x)])/(a*x) - (b*d*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/a^(3/2) - (2*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/Sqrt[a] + (Sqrt[2]*f*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(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]))]) + (Sqrt[2]*f*(e + (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/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]))])/d^2","A",1
120,1,669,679,2.1099201,"\int \frac{1}{x^3 \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(x^3*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2)),x]","\frac{\frac{d^2 \left(\left(4 a c x-3 b^2 x\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)+6 \sqrt{a} b \sqrt{a+x (b+c x)}\right)}{a^{5/2} x}-\frac{4 b d e \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{a^{3/2}}-\frac{4 d^2 \sqrt{a+x (b+c x)}}{a x^2}-\frac{8 \left(e^2-d f\right) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{\sqrt{a}}+\frac{4 \sqrt{2} f \left(\frac{e \left(e^2-3 d f\right)}{\sqrt{e^2-4 d f}}-d f+e^2\right) \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\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{4 \sqrt{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) \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{\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{8 d e \sqrt{a+x (b+c x)}}{a x}}{8 d^3}","-\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{\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{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{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,"((-4*d^2*Sqrt[a + x*(b + c*x)])/(a*x^2) + (8*d*e*Sqrt[a + x*(b + c*x)])/(a*x) - (4*b*d*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/a^(3/2) - (8*(e^2 - d*f)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/Sqrt[a] + (d^2*(6*Sqrt[a]*b*Sqrt[a + x*(b + c*x)] + (-3*b^2*x + 4*a*c*x)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))/(a^(5/2)*x) - (4*Sqrt[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[(4*a*f - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(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]))]) + (4*Sqrt[2]*f*(e^2 - d*f + (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/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]))])/(8*d^3)","A",1
121,1,1066,779,2.623957,"\int \frac{x^3}{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[x^3/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{\frac{4 \left(2 f a^3+(-2 c d-b e+2 c e x+b f x) a^2+b (b (d-e x)-3 c d x) a+b^3 d x\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}+\frac{\sqrt{2} \left(c \left(\sqrt{e^2-4 d f}-e\right) d^2+b \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right) d+a \left(-e^3+\sqrt{e^2-4 d f} e^2+3 d f e-d f \sqrt{e^2-4 d f}\right)\right) \log \left(-e-2 f x+\sqrt{e^2-4 d f}\right)}{\sqrt{e^2-4 d f} \sqrt{c \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right)+f \left(2 a f+b \left(\sqrt{e^2-4 d f}-e\right)\right)}}+\frac{\sqrt{2} \left(c \left(e+\sqrt{e^2-4 d f}\right) d^2-b \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right) d+a \left(e^3+\sqrt{e^2-4 d f} e^2-3 d f e-d f \sqrt{e^2-4 d f}\right)\right) \log \left(e+2 f x+\sqrt{e^2-4 d f}\right)}{\sqrt{e^2-4 d f} \sqrt{c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)+f \left(2 a f-b \left(e+\sqrt{e^2-4 d f}\right)\right)}}-\frac{\sqrt{2} \left(c \left(e+\sqrt{e^2-4 d f}\right) d^2-b \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right) d+a \left(e^3+\sqrt{e^2-4 d f} e^2-3 d f e-d f \sqrt{e^2-4 d f}\right)\right) \log \left(-4 a f+2 c e x+2 c \sqrt{e^2-4 d f} x+b \left(e-2 f x+\sqrt{e^2-4 d f}\right)-2 \sqrt{2} \sqrt{c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)+f \left(2 a f-b \left(e+\sqrt{e^2-4 d f}\right)\right)} \sqrt{a+x (b+c x)}\right)}{\sqrt{e^2-4 d f} \sqrt{c \left(e^2+\sqrt{e^2-4 d f} e-2 d f\right)+f \left(2 a f-b \left(e+\sqrt{e^2-4 d f}\right)\right)}}-\frac{\sqrt{2} \left(c \left(\sqrt{e^2-4 d f}-e\right) d^2+b \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right) d+a \left(-e^3+\sqrt{e^2-4 d f} e^2+3 d f e-d f \sqrt{e^2-4 d f}\right)\right) \log \left(b \left(-e+2 f x+\sqrt{e^2-4 d f}\right)+2 \left(2 a f-c e x+c \sqrt{e^2-4 d f} x+\sqrt{2} \sqrt{f \left(-e b+\sqrt{e^2-4 d f} b+2 a f\right)+c \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right)} \sqrt{a+x (b+c x)}\right)\right)}{\sqrt{e^2-4 d f} \sqrt{c \left(e^2-\sqrt{e^2-4 d f} e-2 d f\right)+f \left(2 a f+b \left(\sqrt{e^2-4 d f}-e\right)\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)}","\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,"((4*(2*a^3*f + b^3*d*x + a^2*(-2*c*d - b*e + 2*c*e*x + b*f*x) + a*b*(-3*c*d*x + b*(d - e*x))))/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]) + (Sqrt[2]*(c*d^2*(-e + Sqrt[e^2 - 4*d*f]) + b*d*(e^2 - 2*d*f - 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]))*Log[-e + Sqrt[e^2 - 4*d*f] - 2*f*x])/(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]))]) + (Sqrt[2]*(c*d^2*(e + Sqrt[e^2 - 4*d*f]) - b*d*(e^2 - 2*d*f + 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]))*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x])/(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]))]) - (Sqrt[2]*(c*d^2*(e + Sqrt[e^2 - 4*d*f]) - b*d*(e^2 - 2*d*f + 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]))*Log[-4*a*f + 2*c*e*x + 2*c*Sqrt[e^2 - 4*d*f]*x + b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x) - 2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)]])/(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]))]) - (Sqrt[2]*(c*d^2*(-e + Sqrt[e^2 - 4*d*f]) + b*d*(e^2 - 2*d*f - 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]))*Log[b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x) + 2*(2*a*f - c*e*x + c*Sqrt[e^2 - 4*d*f]*x + Sqrt[2]*Sqrt[f*(-(b*e) + 2*a*f + b*Sqrt[e^2 - 4*d*f]) + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + x*(b + c*x)])])/(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*(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",1
122,1,1097,609,6.5850982,"\int \frac{x^2}{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[x^2/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{16 \sqrt{2} f \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \left(e+\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e-\sqrt{e^2-4 d f}\right)-\left(2 c \left(e-\sqrt{e^2-4 d f}\right)-2 b f\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right) \left(c x^2+b x+a\right)^{3/2}}{\left(4 a f^2-2 b \left(e-\sqrt{e^2-4 d f}\right) f+c \left(e-\sqrt{e^2-4 d f}\right)^2\right) \left(16 a f^2-8 b \left(e-\sqrt{e^2-4 d f}\right) f+4 c \left(e-\sqrt{e^2-4 d f}\right)^2\right) (a+x (b+c x))^{3/2}}+\frac{16 \sqrt{2} f \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \left(e-\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}\right) \tanh ^{-1}\left(\frac{4 a f-b \left(e+\sqrt{e^2-4 d f}\right)-\left(2 c \left(e+\sqrt{e^2-4 d f}\right)-2 b f\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right) \left(c x^2+b x+a\right)^{3/2}}{\left(4 a f^2-2 b \left(e+\sqrt{e^2-4 d f}\right) f+c \left(e+\sqrt{e^2-4 d f}\right)^2\right) \left(16 a f^2-8 b \left(e+\sqrt{e^2-4 d f}\right) f+4 c \left(e+\sqrt{e^2-4 d f}\right)^2\right) (a+x (b+c x))^{3/2}}-\frac{2 \left(e-\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}\right) \left(2 f b^2-c \left(e-\sqrt{e^2-4 d f}\right) b-4 a c f+2 c \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) x\right) \left(c x^2+b x+a\right)}{\left(b^2-4 a c\right) f \left(4 a f^2-2 b \left(e-\sqrt{e^2-4 d f}\right) f+c \left(e-\sqrt{e^2-4 d f}\right)^2\right) (a+x (b+c x))^{3/2}}-\frac{2 \left(e+\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}\right) \left(2 f b^2-c \left(e+\sqrt{e^2-4 d f}\right) b-4 a c f+2 c \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) x\right) \left(c x^2+b x+a\right)}{\left(b^2-4 a c\right) f \left(4 a f^2-2 b \left(e+\sqrt{e^2-4 d f}\right) f+c \left(e+\sqrt{e^2-4 d f}\right)^2\right) (a+x (b+c x))^{3/2}}+\frac{4 (b+2 c x) \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/2}}{c f (a+x (b+c x))^{3/2} \sqrt{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}","-\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*(e - (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*(2*b^2*f - 4*a*c*f - b*c*(e - Sqrt[e^2 - 4*d*f]) + 2*c*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)*(a + b*x + c*x^2))/((b^2 - 4*a*c)*f*(4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2)) - (2*(e + (e^2 - 2*d*f)/Sqrt[e^2 - 4*d*f])*(2*b^2*f - 4*a*c*f - b*c*(e + Sqrt[e^2 - 4*d*f]) + 2*c*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)*(a + b*x + c*x^2))/((b^2 - 4*a*c)*f*(4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2)) + (4*(b + 2*c*x)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/2))/(c*f*(a + x*(b + c*x))^(3/2)*Sqrt[1 - (b + 2*c*x)^2/(b^2 - 4*a*c)]) + (16*Sqrt[2]*f*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*(e + (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*(a + b*x + c*x^2)^(3/2)*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) - (-2*b*f + 2*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*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/((4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2)*(16*a*f^2 - 8*b*f*(e - Sqrt[e^2 - 4*d*f]) + 4*c*(e - Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2)) + (16*Sqrt[2]*f*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*(e - (-e^2 + 2*d*f)/Sqrt[e^2 - 4*d*f])*(a + b*x + c*x^2)^(3/2)*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) - (-2*b*f + 2*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*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/((4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2)*(16*a*f^2 - 8*b*f*(e + Sqrt[e^2 - 4*d*f]) + 4*c*(e + Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2))","A",1
123,1,770,609,5.6838571,"\int \frac{x}{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[x/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{2 \left(1-\frac{e}{\sqrt{e^2-4 d f}}\right) \left(2 c \left(c x \left(\sqrt{e^2-4 d f}-e\right)-2 a f\right)+2 b^2 f+b c \left(\sqrt{e^2-4 d f}-e+2 f x\right)\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(4 a f^2+2 b f \left(\sqrt{e^2-4 d f}-e\right)+c \left(e-\sqrt{e^2-4 d f}\right)^2\right)}+\frac{2 \left(\frac{e}{\sqrt{e^2-4 d f}}+1\right) \left(-2 c \left(2 a f+c x \left(\sqrt{e^2-4 d f}+e\right)\right)+2 b^2 f-b c \left(\sqrt{e^2-4 d f}+e-2 f x\right)\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(4 a f^2-2 b f \left(\sqrt{e^2-4 d f}+e\right)+c \left(\sqrt{e^2-4 d f}+e\right)^2\right)}-\frac{\sqrt{2} f^2 \left(\sqrt{e^2-4 d f}+e\right) \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{\sqrt{e^2-4 d 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)^{3/2}}-\frac{\sqrt{2} f^2 \left(\sqrt{e^2-4 d f}-e\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)} \tanh ^{-1}\left(\frac{4 a f+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{\sqrt{e^2-4 d f} \left(f \left(b \left(e-\sqrt{e^2-4 d f}\right)-2 a f\right)+c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)\right)^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*(1 - e/Sqrt[e^2 - 4*d*f])*(2*b^2*f + b*c*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x) + 2*c*(-2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)))/((b^2 - 4*a*c)*(4*a*f^2 + c*(e - Sqrt[e^2 - 4*d*f])^2 + 2*b*f*(-e + Sqrt[e^2 - 4*d*f]))*Sqrt[a + x*(b + c*x)]) + (2*(1 + e/Sqrt[e^2 - 4*d*f])*(2*b^2*f - b*c*(e + Sqrt[e^2 - 4*d*f] - 2*f*x) - 2*c*(2*a*f + c*(e + Sqrt[e^2 - 4*d*f])*x)))/((b^2 - 4*a*c)*(4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + x*(b + c*x)]) - (Sqrt[2]*f^2*(e + Sqrt[e^2 - 4*d*f])*ArcTanh[(4*a*f - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(Sqrt[e^2 - 4*d*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])))^(3/2)) - (Sqrt[2]*f^2*(-e + 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 + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(Sqrt[e^2 - 4*d*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])))^2)","A",1
124,1,700,666,5.0417334,"\int \frac{1}{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{2 f \left(-\frac{2 \left(-2 c \left(2 a f+c x \left(\sqrt{e^2-4 d f}+e\right)\right)+2 b^2 f-b c \left(\sqrt{e^2-4 d f}+e-2 f x\right)\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(4 a f^2-2 b f \left(\sqrt{e^2-4 d f}+e\right)+c \left(\sqrt{e^2-4 d f}+e\right)^2\right)}+\frac{2 c \left(c x \left(\sqrt{e^2-4 d f}-e\right)-2 a f\right)+2 b^2 f+b c \left(\sqrt{e^2-4 d f}-e+2 f x\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \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)}+\frac{\sqrt{2} f^2 \tanh ^{-1}\left(\frac{4 a f-b \left(\sqrt{e^2-4 d f}+e-2 f x\right)-2 c x \left(\sqrt{e^2-4 d f}+e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{\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)^{3/2}}-\frac{\sqrt{2} f^2 \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+b \left(\sqrt{e^2-4 d f}-e+2 f x\right)+2 c x \left(\sqrt{e^2-4 d f}-e\right)}{2 \sqrt{2} \sqrt{a+x (b+c x)} \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)}}\right)}{\left(f \left(b \left(e-\sqrt{e^2-4 d f}\right)-2 a f\right)+c \left(e \sqrt{e^2-4 d f}+2 d f-e^2\right)\right)^2}\right)}{\sqrt{e^2-4 d f}}","\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^3 (-f)+b^2 c e\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*f*((2*b^2*f + b*c*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x) + 2*c*(-2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x))/((b^2 - 4*a*c)*(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])))*Sqrt[a + x*(b + c*x)]) - (2*(2*b^2*f - b*c*(e + Sqrt[e^2 - 4*d*f] - 2*f*x) - 2*c*(2*a*f + c*(e + Sqrt[e^2 - 4*d*f])*x)))/((b^2 - 4*a*c)*(4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2)*Sqrt[a + x*(b + c*x)]) + (Sqrt[2]*f^2*ArcTanh[(4*a*f - 2*c*(e + Sqrt[e^2 - 4*d*f])*x - b*(e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(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])))^(3/2) - (Sqrt[2]*f^2*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 + 2*c*(-e + Sqrt[e^2 - 4*d*f])*x + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*Sqrt[2]*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]))]*Sqrt[a + x*(b + c*x)])])/(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))/Sqrt[e^2 - 4*d*f]","A",1
125,1,1121,816,6.5603008,"\int \frac{1}{x \left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)} \, dx","Integrate[1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]","\frac{16 \sqrt{2} \left(\frac{e f}{\sqrt{e^2-4 d f}}+f\right) \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \left(c x^2+b x+a\right)^{3/2} \tanh ^{-1}\left(\frac{4 a f-b \left(e-\sqrt{e^2-4 d f}\right)-\left(2 c \left(e-\sqrt{e^2-4 d f}\right)-2 b f\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right) f^2}{d \left(4 a f^2-2 b \left(e-\sqrt{e^2-4 d f}\right) f+c \left(e-\sqrt{e^2-4 d f}\right)^2\right) \left(16 a f^2-8 b \left(e-\sqrt{e^2-4 d f}\right) f+4 c \left(e-\sqrt{e^2-4 d f}\right)^2\right) (a+x (b+c x))^{3/2}}-\frac{16 \sqrt{2} \left(\frac{e f}{\sqrt{e^2-4 d f}}-f\right) \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \left(c x^2+b x+a\right)^{3/2} \tanh ^{-1}\left(\frac{4 a f-b \left(e+\sqrt{e^2-4 d f}\right)-\left(2 c \left(e+\sqrt{e^2-4 d f}\right)-2 b f\right) x}{2 \sqrt{2} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right) f^2}{d \left(4 a f^2-2 b \left(e+\sqrt{e^2-4 d f}\right) f+c \left(e+\sqrt{e^2-4 d f}\right)^2\right) \left(16 a f^2-8 b \left(e+\sqrt{e^2-4 d f}\right) f+4 c \left(e+\sqrt{e^2-4 d f}\right)^2\right) (a+x (b+c x))^{3/2}}-\frac{2 \left(\frac{e}{\sqrt{e^2-4 d f}}+1\right) \left(2 f b^2-c \left(e-\sqrt{e^2-4 d f}\right) b-4 a c f+2 c \left(b f-c \left(e-\sqrt{e^2-4 d f}\right)\right) x\right) \left(c x^2+b x+a\right) f}{\left(b^2-4 a c\right) d \left(4 a f^2-2 b \left(e-\sqrt{e^2-4 d f}\right) f+c \left(e-\sqrt{e^2-4 d f}\right)^2\right) (a+x (b+c x))^{3/2}}-\frac{2 \left(1-\frac{e}{\sqrt{e^2-4 d f}}\right) \left(2 f b^2-c \left(e+\sqrt{e^2-4 d f}\right) b-4 a c f+2 c \left(b f-c \left(e+\sqrt{e^2-4 d f}\right)\right) x\right) \left(c x^2+b x+a\right) f}{\left(b^2-4 a c\right) d \left(4 a f^2-2 b \left(e+\sqrt{e^2-4 d f}\right) f+c \left(e+\sqrt{e^2-4 d f}\right)^2\right) (a+x (b+c x))^{3/2}}-\frac{\left(c x^2+b x+a\right)^{3/2} \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{c x^2+b x+a}}\right)}{a^{3/2} d (a+x (b+c x))^{3/2}}+\frac{2 \left(b^2+c x b-2 a c\right) \left(c x^2+b x+a\right)}{a \left(b^2-4 a c\right) d (a+x (b+c x))^{3/2}}","\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*x + c*x^2))/(a*(b^2 - 4*a*c)*d*(a + x*(b + c*x))^(3/2)) - (2*f*(1 + e/Sqrt[e^2 - 4*d*f])*(2*b^2*f - 4*a*c*f - b*c*(e - Sqrt[e^2 - 4*d*f]) + 2*c*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)*(a + b*x + c*x^2))/((b^2 - 4*a*c)*d*(4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2)) - (2*f*(1 - e/Sqrt[e^2 - 4*d*f])*(2*b^2*f - 4*a*c*f - b*c*(e + Sqrt[e^2 - 4*d*f]) + 2*c*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)*(a + b*x + c*x^2))/((b^2 - 4*a*c)*d*(4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2)) - ((a + b*x + c*x^2)^(3/2)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(a^(3/2)*d*(a + x*(b + c*x))^(3/2)) + (16*Sqrt[2]*f^2*(f + (e*f)/Sqrt[e^2 - 4*d*f])*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*(a + b*x + c*x^2)^(3/2)*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) - (-2*b*f + 2*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*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(d*(4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2)*(16*a*f^2 - 8*b*f*(e - Sqrt[e^2 - 4*d*f]) + 4*c*(e - Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2)) - (16*Sqrt[2]*f^2*(-f + (e*f)/Sqrt[e^2 - 4*d*f])*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*(a + b*x + c*x^2)^(3/2)*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) - (-2*b*f + 2*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*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(d*(4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2)*(16*a*f^2 - 8*b*f*(e + Sqrt[e^2 - 4*d*f]) + 4*c*(e + Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2))","A",1
126,1,210,140,0.5265268,"\int \frac{x^4}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Integrate[x^4/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\frac{1}{24} \left(-6 \sqrt{-x^2-4 x-3} x+60 \sqrt{-x^2-4 x-3}-\sqrt{1-2 i \sqrt{2}} \left(4 \sqrt{2}+7 i\right) \tanh ^{-1}\left(\frac{-i \sqrt{2} x+2 x-2 i \sqrt{2}+2}{\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)-\sqrt{1+2 i \sqrt{2}} \left(4 \sqrt{2}-7 i\right) \tanh ^{-1}\left(\frac{\left(2+i \sqrt{2}\right) x+2 i \sqrt{2}+2}{\sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)+132 \sin ^{-1}(x+2)\right)","-\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,"(60*Sqrt[-3 - 4*x - x^2] - 6*x*Sqrt[-3 - 4*x - x^2] + 132*ArcSin[2 + x] - Sqrt[1 - (2*I)*Sqrt[2]]*(7*I + 4*Sqrt[2])*ArcTanh[(2 - (2*I)*Sqrt[2] + 2*x - I*Sqrt[2]*x)/(Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])] - Sqrt[1 + (2*I)*Sqrt[2]]*(-7*I + 4*Sqrt[2])*ArcTanh[(2 + (2*I)*Sqrt[2] + (2 + I*Sqrt[2])*x)/(Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])])/24","C",1
127,1,192,115,0.4380717,"\int \frac{x^3}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Integrate[x^3/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\frac{1}{8} \left(-4 \left(\sqrt{-x^2-4 x-3}+4 \sin ^{-1}(x+2)\right)+\frac{\left(5 \sqrt{2}-2 i\right) \tanh ^{-1}\left(\frac{i \sqrt{2} x+2 x+2 i \sqrt{2}+2}{\sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)}{\sqrt{1-2 i \sqrt{2}}}+\frac{\left(5 \sqrt{2}+2 i\right) \tanh ^{-1}\left(\frac{\left(2-i \sqrt{2}\right) x-2 i \sqrt{2}+2}{\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)}{\sqrt{1+2 i \sqrt{2}}}\right)","-\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,"(-4*(Sqrt[-3 - 4*x - x^2] + 4*ArcSin[2 + x]) + ((-2*I + 5*Sqrt[2])*ArcTanh[(2 + (2*I)*Sqrt[2] + 2*x + I*Sqrt[2]*x)/(Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])])/Sqrt[1 - (2*I)*Sqrt[2]] + ((2*I + 5*Sqrt[2])*ArcTanh[(2 - (2*I)*Sqrt[2] + (2 - I*Sqrt[2])*x)/(Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])])/Sqrt[1 + (2*I)*Sqrt[2]])/8","C",1
128,1,159,98,0.1872171,"\int \frac{x^2}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Integrate[x^2/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\frac{1}{4} \left(-i \sqrt{1-2 i \sqrt{2}} \tanh ^{-1}\left(\frac{i \sqrt{2} x+2 x+2 i \sqrt{2}+2}{\sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)+i \sqrt{1+2 i \sqrt{2}} \tanh ^{-1}\left(\frac{\left(2-i \sqrt{2}\right) x-2 i \sqrt{2}+2}{\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)+2 \sin ^{-1}(x+2)\right)","-\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,"(2*ArcSin[2 + x] - I*Sqrt[1 - (2*I)*Sqrt[2]]*ArcTanh[(2 + (2*I)*Sqrt[2] + 2*x + I*Sqrt[2]*x)/(Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])] + I*Sqrt[1 + (2*I)*Sqrt[2]]*ArcTanh[(2 - (2*I)*Sqrt[2] + (2 - I*Sqrt[2])*x)/(Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])])/4","C",1
129,1,174,68,0.167583,"\int \frac{x}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Integrate[x/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\frac{\left(1-i \sqrt{2}\right) \sqrt{1-2 i \sqrt{2}} \tanh ^{-1}\left(\frac{\left(2-i \sqrt{2}\right) x-2 i \sqrt{2}+2}{\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)+\left(1+i \sqrt{2}\right) \sqrt{1+2 i \sqrt{2}} \tanh ^{-1}\left(\frac{\left(2+i \sqrt{2}\right) x+2 i \sqrt{2}+2}{\sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)}{6 \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,"((1 - I*Sqrt[2])*Sqrt[1 - (2*I)*Sqrt[2]]*ArcTanh[(2 - (2*I)*Sqrt[2] + (2 - I*Sqrt[2])*x)/(Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])] + (1 + I*Sqrt[2])*Sqrt[1 + (2*I)*Sqrt[2]]*ArcTanh[(2 + (2*I)*Sqrt[2] + (2 + I*Sqrt[2])*x)/(Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])])/(6*Sqrt[2])","C",1
130,1,150,95,0.0996119,"\int \frac{1}{\sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Integrate[1/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\frac{1}{6} i \left(\sqrt{1-2 i \sqrt{2}} \tanh ^{-1}\left(\frac{\left(2-i \sqrt{2}\right) x-2 i \sqrt{2}+2}{\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)-\sqrt{1+2 i \sqrt{2}} \tanh ^{-1}\left(\frac{\left(2+i \sqrt{2}\right) x+2 i \sqrt{2}+2}{\sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)\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,"(I/6)*(Sqrt[1 - (2*I)*Sqrt[2]]*ArcTanh[(2 - (2*I)*Sqrt[2] + (2 - I*Sqrt[2])*x)/(Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])] - Sqrt[1 + (2*I)*Sqrt[2]]*ArcTanh[(2 + (2*I)*Sqrt[2] + (2 + I*Sqrt[2])*x)/(Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])])","C",1
131,1,200,130,0.4372284,"\int \frac{1}{x \sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Integrate[1/(x*Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\frac{1}{54} \left(-6 \sqrt{3} \tan ^{-1}\left(\frac{2 x+3}{\sqrt{3} \sqrt{-x^2-4 x-3}}\right)-3 \sqrt{1-2 i \sqrt{2}} \left(\sqrt{2}+2 i\right) \tanh ^{-1}\left(\frac{\left(2-i \sqrt{2}\right) x-2 i \sqrt{2}+2}{\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)-3 \sqrt{1+2 i \sqrt{2}} \left(\sqrt{2}-2 i\right) \tanh ^{-1}\left(\frac{\left(2+i \sqrt{2}\right) x+2 i \sqrt{2}+2}{\sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)\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,"(-6*Sqrt[3]*ArcTan[(3 + 2*x)/(Sqrt[3]*Sqrt[-3 - 4*x - x^2])] - 3*Sqrt[1 - (2*I)*Sqrt[2]]*(2*I + Sqrt[2])*ArcTanh[(2 - (2*I)*Sqrt[2] + (2 - I*Sqrt[2])*x)/(Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])] - 3*Sqrt[1 + (2*I)*Sqrt[2]]*(-2*I + Sqrt[2])*ArcTanh[(2 + (2*I)*Sqrt[2] + (2 + I*Sqrt[2])*x)/(Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])])/54","C",1
132,1,225,151,0.4053188,"\int \frac{1}{x^2 \sqrt{-3-4 x-x^2} \left(3+4 x+2 x^2\right)} \, dx","Integrate[1/(x^2*Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]","\frac{3 \left(\sqrt{-x^2-4 x-3}+2 \sqrt{3} x \tan ^{-1}\left(\frac{2 x+3}{\sqrt{3} \sqrt{-x^2-4 x-3}}\right)\right)+\sqrt{1-2 i \sqrt{2}} \left(2 \sqrt{2}+i\right) x \tanh ^{-1}\left(\frac{-i \sqrt{2} x+2 x-2 i \sqrt{2}+2}{\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)+\sqrt{1+2 i \sqrt{2}} \left(2 \sqrt{2}-i\right) x \tanh ^{-1}\left(\frac{\left(2+i \sqrt{2}\right) x+2 i \sqrt{2}+2}{\sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)}{27 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)",1,"(3*(Sqrt[-3 - 4*x - x^2] + 2*Sqrt[3]*x*ArcTan[(3 + 2*x)/(Sqrt[3]*Sqrt[-3 - 4*x - x^2])]) + Sqrt[1 - (2*I)*Sqrt[2]]*(I + 2*Sqrt[2])*x*ArcTanh[(2 - (2*I)*Sqrt[2] + 2*x - I*Sqrt[2]*x)/(Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])] + Sqrt[1 + (2*I)*Sqrt[2]]*(-I + 2*Sqrt[2])*x*ArcTanh[(2 + (2*I)*Sqrt[2] + (2 + I*Sqrt[2])*x)/(Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])])/(27*x)","C",1
133,1,87,149,0.1908058,"\int (2+3 x)^2 \left(30+31 x-12 x^2\right)^2 \sqrt{6+17 x+12 x^2} \, dx","Integrate[(2 + 3*x)^2*(30 + 31*x - 12*x^2)^2*Sqrt[6 + 17*x + 12*x^2],x]","\frac{12 \sqrt{12 x^2+17 x+6} \left(171228266496 x^7-732816211968 x^6-1190083166208 x^5+3438453030912 x^4+8974844476416 x^3+7899203409792 x^2+3132157281976 x+474999091769\right)-878185 \sqrt{3} \tanh ^{-1}\left(\frac{24 x+17}{4 \sqrt{36 x^2+51 x+18}}\right)}{12683575296}","-\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,"(12*Sqrt[6 + 17*x + 12*x^2]*(474999091769 + 3132157281976*x + 7899203409792*x^2 + 8974844476416*x^3 + 3438453030912*x^4 - 1190083166208*x^5 - 732816211968*x^6 + 171228266496*x^7) - 878185*Sqrt[3]*ArcTanh[(17 + 24*x)/(4*Sqrt[18 + 51*x + 36*x^2])])/12683575296","A",1
134,1,72,103,0.03281,"\int (2+3 x) \left(30+31 x-12 x^2\right) \sqrt{6+17 x+12 x^2} \, dx","Integrate[(2 + 3*x)*(30 + 31*x - 12*x^2)*Sqrt[6 + 17*x + 12*x^2],x]","\frac{485 \sqrt{3} \tanh ^{-1}\left(\frac{24 x+17}{4 \sqrt{36 x^2+51 x+18}}\right)+12 \sqrt{12 x^2+17 x+6} \left(-884736 x^4+1963008 x^3+6837888 x^2+5455144 x+1353611\right)}{1474560}","-\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,"(12*Sqrt[6 + 17*x + 12*x^2]*(1353611 + 5455144*x + 6837888*x^2 + 1963008*x^3 - 884736*x^4) + 485*Sqrt[3]*ArcTanh[(17 + 24*x)/(4*Sqrt[18 + 51*x + 36*x^2])])/1474560","A",1
135,1,37,28,0.0976383,"\int \frac{\sqrt{6+17 x+12 x^2}}{(2+3 x) \left(30+31 x-12 x^2\right)} \, dx","Integrate[Sqrt[6 + 17*x + 12*x^2]/((2 + 3*x)*(30 + 31*x - 12*x^2)),x]","\frac{1}{42} \log \left(84 \sqrt{12 x^2+17 x+6}+291 x+206\right)-\frac{1}{42} \log (10-3 x)","\frac{1}{42} \tanh ^{-1}\left(\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right)",1,"-1/42*Log[10 - 3*x] + Log[206 + 291*x + 84*Sqrt[6 + 17*x + 12*x^2]]/42","A",1
136,1,114,84,0.2378097,"\int \frac{\sqrt{6+17 x+12 x^2}}{(2+3 x)^2 \left(30+31 x-12 x^2\right)^2} \, dx","Integrate[Sqrt[6 + 17*x + 12*x^2]/((2 + 3*x)^2*(30 + 31*x - 12*x^2)^2),x]","\frac{\sqrt{12 x^2+17 x+6} \left(97 \left(36 x^3-69 x^2-152 x-60\right) \tanh ^{-1}\left(\frac{7 \sqrt{3 x+2}}{6 \sqrt{4 x+3}}\right)-42 \sqrt{3 x+2} \sqrt{4 x+3} \left(37644 x^2-98767 x-88978\right)\right)}{1613472 (3 x-10) (3 x+2)^{3/2} (4 x+3)^{3/2}}","-\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,"(Sqrt[6 + 17*x + 12*x^2]*(-42*Sqrt[2 + 3*x]*Sqrt[3 + 4*x]*(-88978 - 98767*x + 37644*x^2) + 97*(-60 - 152*x - 69*x^2 + 36*x^3)*ArcTanh[(7*Sqrt[2 + 3*x])/(6*Sqrt[3 + 4*x])]))/(1613472*(-10 + 3*x)*(2 + 3*x)^(3/2)*(3 + 4*x)^(3/2))","A",1
137,1,131,139,0.3667768,"\int \frac{\sqrt{6+17 x+12 x^2}}{(2+3 x)^3 \left(30+31 x-12 x^2\right)^3} \, dx","Integrate[Sqrt[6 + 17*x + 12*x^2]/((2 + 3*x)^3*(30 + 31*x - 12*x^2)^3),x]","\frac{\sqrt{12 x^2+17 x+6} \left(40325 \left(-36 x^3+69 x^2+152 x+60\right)^2 \tanh ^{-1}\left(\frac{7 \sqrt{3 x+2}}{6 \sqrt{4 x+3}}\right)+42 \sqrt{3 x+2} \sqrt{4 x+3} \left(706089565584 x^5-3206824169544 x^4-1096520427663 x^3+9848047480070 x^2+10124325497244 x+2773753482408\right)\right)}{318770436096 (10-3 x)^2 (3 x+2)^{5/2} (4 x+3)^{5/2}}","-\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,"(Sqrt[6 + 17*x + 12*x^2]*(42*Sqrt[2 + 3*x]*Sqrt[3 + 4*x]*(2773753482408 + 10124325497244*x + 9848047480070*x^2 - 1096520427663*x^3 - 3206824169544*x^4 + 706089565584*x^5) + 40325*(60 + 152*x + 69*x^2 - 36*x^3)^2*ArcTanh[(7*Sqrt[2 + 3*x])/(6*Sqrt[3 + 4*x])]))/(318770436096*(10 - 3*x)^2*(2 + 3*x)^(5/2)*(3 + 4*x)^(5/2))","A",1
138,1,13,15,0.0068078,"\int (-3+2 x) \left(-3 x+x^2\right)^{2/3} \, dx","Integrate[(-3 + 2*x)*(-3*x + x^2)^(2/3),x]","\frac{3}{5} ((x-3) x)^{5/3}","\frac{3}{5} \left(x^2-3 x\right)^{5/3}",1,"(3*((-3 + x)*x)^(5/3))/5","A",1
139,1,13,16,0.0031133,"\int ((-3+x) x)^{2/3} (-3+2 x) \, dx","Integrate[((-3 + x)*x)^(2/3)*(-3 + 2*x),x]","\frac{3}{5} ((x-3) x)^{5/3}","\frac{3}{5} (-((3-x) x))^{5/3}",1,"(3*((-3 + x)*x)^(5/3))/5","A",1
140,1,13,15,0.0057311,"\int \frac{x \left(9-9 x+2 x^2\right)}{\sqrt[3]{-3 x+x^2}} \, dx","Integrate[(x*(9 - 9*x + 2*x^2))/(-3*x + x^2)^(1/3),x]","\frac{3}{5} ((x-3) x)^{5/3}","\frac{3}{5} \left(x^2-3 x\right)^{5/3}",1,"(3*((-3 + x)*x)^(5/3))/5","A",1
141,1,13,15,0.0038216,"\int \frac{x \left(9-9 x+2 x^2\right)}{\sqrt[3]{(-3+x) x}} \, dx","Integrate[(x*(9 - 9*x + 2*x^2))/((-3 + x)*x)^(1/3),x]","\frac{3}{5} ((x-3) x)^{5/3}","\frac{3}{5} \left(x^2-3 x\right)^{5/3}",1,"(3*((-3 + x)*x)^(5/3))/5","A",1
142,1,268,242,0.5595674,"\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","Integrate[(g + h*x)/((-((c*g^2)/h^2) + 9*c*x^2)^(1/3)*(g^2 + 3*h^2*x^2)),x]","\frac{h^2 x \left(-h x \sqrt[3]{1-\frac{9 h^2 x^2}{g^2}} F_1\left(1;\frac{1}{3},1;2;\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right)-\frac{2 g^5 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right)}{\left(g^2+3 h^2 x^2\right) \left(g^2 F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right)+2 h^2 x^2 \left(F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right)-F_1\left(\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right)\right)\right)}\right) \left(c \left(9 x^2-\frac{g^2}{h^2}\right)\right)^{2/3}}{2 c g^2 \left(g^2-9 h^2 x^2\right)}","\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,"(h^2*x*(c*(-(g^2/h^2) + 9*x^2))^(2/3)*(-(h*x*(1 - (9*h^2*x^2)/g^2)^(1/3)*AppellF1[1, 1/3, 1, 2, (9*h^2*x^2)/g^2, (-3*h^2*x^2)/g^2]) - (2*g^5*AppellF1[1/2, 1/3, 1, 3/2, (9*h^2*x^2)/g^2, (-3*h^2*x^2)/g^2])/((g^2 + 3*h^2*x^2)*(g^2*AppellF1[1/2, 1/3, 1, 3/2, (9*h^2*x^2)/g^2, (-3*h^2*x^2)/g^2] + 2*h^2*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (9*h^2*x^2)/g^2, (-3*h^2*x^2)/g^2] + AppellF1[3/2, 4/3, 1, 5/2, (9*h^2*x^2)/g^2, (-3*h^2*x^2)/g^2])))))/(2*c*g^2*(g^2 - 9*h^2*x^2))","C",0
143,0,0,488,0.5353541,"\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","Integrate[(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]","\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","\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,"Integrate[(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]","F",-1